††footnotetext:  PG and CS were partially supported by DFG grant SE 1114/5-2.
TR and AS acknowledge support by DFG grant RE 3567/1-1. UW was supported by
NSF grant DMS-2100288 and by Simons Foundation Collaboration Grant for
Mathematicians #580839.
2010 _Mathematics Subject Classification._ 32C38, 14F10, 32S40
Keywords: Tautological system, Fourier–Laplace transformation, mixed Hodge
module, Lie group, homogeneous space, representation

# Tautological systems, homogeneous spaces and the holonomic rank problem

Paul Görlach, Thomas Reichelt, Christian Sevenheck, Avi Steiner and Uli
Walther

###### Abstract

Many GKZ-systems that arise from a geometric setting can be endowed with the
structure of mixed Hodge modules. We generalize this fundamental result to the
tautological systems associated to homogeneous spaces by giving a functorial
construction for them. As an application, we solve the holonomic rank problem
for such tautological systems in full generality.

###### Contents

  1. 1 Introduction
  2. 2 Mixed Hodge modules on line bundles
    1. 2.1 The $\mathscr{D}$-module $\mathcal{O}_{L^{*}}^{\beta}$
    2. 2.2 Mixed Hodge module structure on $\mathcal{O}_{L^{*}}^{\beta}$
  3. 3 Fourier–Laplace transformation
    1. 3.1 Fourier–Laplace transformation on vector bundles
    2. 3.2 Fourier–Laplace transform of extensions of $\mathcal{O}_{L^{*}}^{\beta}$
    3. 3.3 Twisted cohomology of hyperplane sections
  4. 4 Equivariant constructions and tautological systems
    1. 4.1 Vector fields from group actions
    2. 4.2 $\mathscr{D}$-modules from group actions
  5. 5 Non-vanishing criteria
    1. 5.1 $\mathcal{A}_{Y}$-modules
    2. 5.2 Transitive group actions
    3. 5.3 Torsion line bundles
    4. 5.4 Equivariant line bundles from characters
    5. 5.5 Left-right transforms on $\mathcal{A}_{Y}$-modules
    6. 5.6 Cyclic right $\mathscr{D}_{Y}$-modules from group actions
    7. 5.7 Sufficient criterion for non-zero tautological systems
  6. 6 Total spaces of line bundles
    1. 6.1 Equivariant vector fields on line bundles
    2. 6.2 Canonical sheaf on total spaces of line bundles
    3. 6.3 Constructing $\mathcal{O}_{L^{*}}^{\beta}$ from group actions
  7. 7 Necessary criterion for non-zero tautological systems
    1. 7.1 Some notation
    2. 7.2 Representation-theoretic proof
      1. 7.2.1 The Casimir element
      2. 7.2.2 Differential operators on affine varieties
      3. 7.2.3 Proof of 7.1
    3. 7.3 Geometric interpretation of $\beta(\mathbf{e})$
      1. 7.3.1 Parabolic subgroups
      2. 7.3.2 Equivariant line bundles on homogeneous spaces
      3. 7.3.3 Proof of 7.8
  8. 8 Tautological systems associated to homogeneous spaces
    1. 8.1 Fourier transform of tautological systems from homogeneous spaces
    2. 8.2 Localization property of $\widehat{\tau}$
    3. 8.3 Colocalization property of $\widehat{\tau}$
    4. 8.4 Tautological systems as mixed Hodge modules

##  1 Introduction

The purpose of this paper is to investigate differential systems that one can
naturally associate to group actions on smooth algebraic varieties, and more
specifically to representations of algebraic groups. A well-known and widely
studied case is when the group is an algebraic torus, in which case the
corresponding $\mathscr{D}$-modules are known as GKZ-systems (see, e.g.,
[RSSW21] for an overview on the algebraic aspects of this theory). Consider
now the following data: a complex algebraic group acting linearly on a finite-
dimensional vector space, an invariant subvariety of this space, and a
homomorphism from the Lie algebra of the group into the complex numbers. To
this situation is naturally attached a _tautological system_ , which is an
equivariant $\mathscr{D}$-module on the dual vector space. This construction
seems to go back to [Hot98, Section 4.], but has been considered more recently
in a series of papers by Bloch, Huang, Lian, Song, Yau and Zhu ([LSY13,
BHL+14, LY13, HLZ16]). One of their main motivations comes from mirror
symmetry, understood in the classical sense of recovering enumerative geometry
information (i.e., quantum cohomology) of certain symplectic varieties by
period integral computations of their mirror families (or oscillating
integrals in the non-Calabi–Yau case). While the case of complete
intersections inside toric varieties can be considered as settled (at least
under sufficient positivity assumptions, cf. [Giv98, Iri11, RS17] and the
respective bibliography trees), it is a longstanding and challenging problem
to establish mirror symmetry, expressed as an equivalence of
$\mathscr{D}$-modules (possibly with additional structures, such as Hodge
modules or irregular variants of them) for non-toric varieties. An important
class of examples arise from homogeneous spaces; for a partial list of known
results on mirror symmetry in that context see [Rie08, MR20, LT17]. A common
feature of these papers is that the mirror of a Fano manifold which is a
homogeneous space for some group $G^{\vee}$ consists of a Landau–Ginzburg
potential, constructed via Lie theoretic methods from the Langlands dual group
$G$ of $G^{\vee}$. When restricted to a torus inside $G$, such a potential
function can be expressed as a Laurent polynomial. Describing, and then
studying, an appropriate partial compactification of this mirror Laurent
polynomial is a major and central problem in the area; the toric situation is
considered for example in [RS17] and, from a very different point of view, in
[CPS22].

Motivated by these results and problems, a question of fundamental importance
is, for a given homogeneous space $X=G/P$, to describe the differential system
satisfied by periods of families of hyperplane sections, for an appropriate
embedding of $X$ into a projective space. Such a differential system would be
the analogue of a GKZ-$\mathscr{D}$-module, and should yield (by dimensional
reduction) the mirror $\mathscr{D}$-module considered in the papers mentioned
above. Our main findings paraphrased in 1.2 below, give criteria to determine
when tautological systems arise as such $\mathscr{D}$-modules in a setting
where we allow $G$ to be any linear algebraic group, and where the
representation will be in the space of sections of some equivariant line
bundle $L$ on $X$.

Our investigations show that one needs to impose rather delicate conditions on
the bundle $L$ and the parameter Lie algebra homomorphism mentioned above in
order to obtain a non-zero tautological system. If these conditions—which we
make explicit—hold true, we show that the corresponding tautological system
has a functorial description, and thus naturally underlies a mixed Hodge
module. We determine its possible weights, and we show how to compute its
solution rank at any point. In particular, we determine its holonomic rank in
terms of the dimension of the cohomology of a natural family of (complements
of) hyperplane sections of $X$. The latter result gives a complete solution
for arbitrary line bundles to the holonomic rank problem raised in [BHL+14].
We also show that in many cases (depending on the value of the parameter Lie
algebra homomorphism), the monodromy representation defined by the smooth part
of the tautological system is irreducible. Besides applications to mirror
symmetry, our results should also lay the foundations for further study of
Hodge theory of various differential modules constructed from representations
of algebraic groups, such as Frenkel–Gross connections (see [FG09]) or
generalized Kloosterman $\mathscr{D}$-modules ([HNY13]).

In the remainder of this introduction, we are going to describe our main
results in more detail, and we give an overview on the content of this paper.
The main character, the tautological system, is defined below. In terms of
notation, for a vector space $V$ and its dual space $W:=V^{\vee}$, we denote
the Fourier–Laplace transformation functor
$\operatorname{\textup{FL}}^{V}\colon\textup{Mod}(\mathscr{D}_{V})\rightarrow\textup{Mod}(\mathscr{D}_{W})$
(see Section 3.1 below for more details about Fourier–Laplace transformations
on arbitrary vector bundles). For now, $G^{\prime}$ can be any linear
algebraic group, but in the later parts we will consider a group $G$ acting
transitively on a variety $X$, and $G^{\prime}$ will denote the product
${\mathds{C}}^{*}\times G$, acting on equivariant line bundles $L\rightarrow
X$.

######  Definition 1.1.

Let $\rho\colon G^{\prime}\to\operatorname{\textup{GL}}(V)$ be a finite-
dimensional rational representation of an algebraic group and denote the
induced Lie algebra representation by
$\operatorname{d\\!}\rho\colon\mathfrak{g}^{\prime}\to\mathfrak{gl}(V)$. Let
$\overline{Y}$ be a $G^{\prime}$-invariant closed subvariety of $V$. For a Lie
algebra homomorphism $\beta\colon\mathfrak{g}^{\prime}\to\mathds{C}$, define
the left $\mathscr{D}_{V}$-module

$\hat{\tau}(\rho,\overline{Y},\beta):=\mathscr{D}_{V}/\mathopen{}\mathclose{{}\left(\mathscr{D}_{V}\mathcal{I}+\mathscr{D}_{V}\mathopen{}\mathclose{{}\left(Z_{V}(\xi)-\beta^{\prime}(\xi)\mid\xi\in\mathfrak{g}^{\prime}}\right)}\right),$

where $\mathcal{I}\subseteq\mathcal{O}_{V}$ is the vanishing ideal of
$\overline{Y}$, where $Z_{V}(\xi)$ denotes the vector field on $V$ given by
the infinitesimal action of $\mathfrak{g}^{\prime}$ (see 4.3 for a detailed
discussion), and where
$\beta^{\prime}(\xi):=\operatorname{trace}(\operatorname{d\\!}\rho(\xi))-\beta(\xi)$.

Its Fourier–Laplace transform

$\tau(\rho,\overline{Y},\beta):=\operatorname{\textup{FL}}^{V}(\hat{\tau}(\rho,\overline{Y},\beta))$

is a left $\mathscr{D}_{V^{\vee}}$-module called the tautological system
associated to $\rho$, $\overline{Y}$ and $\beta$.

The next statement summarizes our main results. To state them, assume that $X$
is a smooth projective variety, and that $G$ is a reductive and connected
linear algebraic group that acts transitively on $X$. Suppose that
$L\rightarrow X$ is a $G$-equivariant line bundle on $X$, with sheaf of
sections $\mathscr{L}$, which we assume to be very ample. We put
$G^{\prime}:={\mathds{C}}^{*}\times G$, and we define an action of
$G^{\prime}$ on $L$ by letting the ${\mathds{C}}^{*}$-factor act via inverse
scaling in the fibres of $L$ (see 5.9 for a more precise and more general
description). Setting $V:=H^{0}(X,\mathscr{L})^{\vee}$, we obtain a
representation $G^{\prime}\rightarrow\operatorname{\textup{GL}}(V)$. Moreover,
since $L$ is very ample, the linear system $|\mathscr{L}|$ yields an embedding
$g\colon X\hookrightarrow\mathds{P}V$. Let $\hat{X}\subseteq V$ be the affine
cone; this is a $G^{\prime}$-invariant subvariety. Notice that there is an
isomorphism $L^{*}\cong\hat{X}\setminus\\{0\\}$, where $L^{*}$ is the
complement of the zero section of $L\rightarrow X$ and we write $\iota\colon
L^{*}\hookrightarrow V$ for the corresponding locally closed embedding
obtained by composing this isomorphism with the embedding
$\hat{X}\setminus\\{0\\}\hookrightarrow V$. Choose any Lie algebra
homomorphism
$\beta\colon\mathfrak{g}^{\prime}={\mathds{C}}\mathbf{e}\oplus\mathfrak{g}\rightarrow{\mathds{C}}$
with $\beta_{|\mathfrak{g}}=0$ (this is forced on $\beta$ if $G$ is
semisimple, since then $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$), i.e.,
choose a number $\beta(\mathbf{e})\in{\mathds{C}}$.

######  Theorem 1.2 (8.4, 8.20 and 8.21).

In the above situation, the following statements hold true.

  1. 1.

Let $\beta(\mathbf{e})\notin{\mathds{Z}}$. We have

    1. (a)

$\tau(\rho,\hat{X},\beta)=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{lcl}\operatorname{\textup{FL}}^{V}(\iota_{+}\mathcal{O}_{L^{*}}^{\ell/k})&&\textup{if
}\mathscr{L}^{\otimes\ell}\cong\omega^{\otimes(-k)}_{X}\textup{ and
}\beta(\mathbf{e})=\ell/k,\\\ \\\ 0&&\textup{else},\end{array}}\right.$

where $\mathcal{O}_{L^{*}}^{\ell/k}$ is a smooth $\mathscr{D}_{L^{*}}$-module
of rank $1$ on $L^{*}$ (and we denote by
$\underline{{\mathds{C}}}_{L^{*}}^{\ell/k}$ its associated local system) which
underlies a pure complex Hodge module of weight $\dim(X)+1$.

    2. (b)

If $\tau(\rho,\hat{X},\beta)\neq 0$, then it underlies a simple pure complex
Hodge module of weight $\dim(X)+\dim(V^{\vee})$. In particular, the local
system corresponding to the restriction of $\tau(\rho,\hat{X},\beta)$ to the
complement of its singular locus (or, phrased differently, its monodromy
representation) is irreducible.

    3. (c)

The holonomic rank of $\tau(\rho,\hat{X},\beta)$ equals

$\dim_{\mathds{C}}H^{\dim(X)}_{c}\mathopen{}\mathclose{{}\left(X\setminus
Z(\lambda),\underline{{\mathds{C}}}_{\lambda}^{\ell/k}}\right)$

for a generic $\lambda\in V^{\vee}=H^{0}(X,\mathscr{L})$, where $Z(\lambda)$
is the vanishing locus in $X$ of the section $\lambda$, and where
$\underline{{\mathds{C}}}_{\lambda}^{\ell/k}$ is the local system
$\lambda_{|X\setminus
Z(\lambda)}^{*}\underline{{\mathds{C}}}_{L^{*}}^{\ell/k}$.

  2. 2.

Let $\beta(\mathbf{e})\in{\mathds{Z}}_{>0}$. We then have

$\tau(\rho,\hat{X},\beta)=\operatorname{\textup{FL}}^{V}\mathopen{}\mathclose{{}\left(H^{0}\iota_{\dagger}\mathcal{O}_{L^{*}}}\right),$

which underlies a (rational) mixed Hodge module (i.e. an element in
$\textup{MHM}(V^{\vee})$), with weights in
$\mathopen{}\mathclose{{}\left\\{\dim(X)+\dim(V^{\vee}),\,\dim(X)+\dim(V^{\vee})+1}\right\\}$.
Its holonomic rank is given by

$\dim_{\mathds{C}}H^{\dim(X)}(X\setminus Z(\lambda),{\mathds{C}}).$

Since the above theorem is meant only as an overview of our results, we ignore
the case where $\beta(\mathbf{e})\in{\mathds{Z}}_{\leq 0}$ here, as it is
essentially uninteresting (see 8.19 for more details). In a similar spirit, we
only mention the holonomic rank here, whereas 8.21 contains finer results
concering the fibre rank (resp. the solution rank) of the system
$\tau(\rho,\hat{X},\beta)$ at any point. Notice further that the points 1.(a)
and 1.(b) in the above theorem imply in particular that for any given
equivariant line bundle $L$ that gives a non-zero tautological system, some
sufficiently high power of it yields a system with irreducible monodromy
representation.

The main step for achieving these results is to rewrite the Fourier–Laplace
transformation as an operation that involves only functors defined in the
category of mixed Hodge modules. This is done using a strategy that already
appeared in [Rei14], namely, using the Radon transformation for algebraic
$\mathscr{D}_{{\mathds{P}}^{n}}$-modules. Due to the possible non-integrality
of $\beta$ however, we need here a variant of this transformation. This
twisted Radon transformation was used in [RS20, Section 5.2], and the relevant
adaptations are discussed in Section 3.

Another ingredient of our construction is a localization result for the
Fourier–Laplace transform $\hat{\tau}(\rho,\hat{X},\beta)$ of the tautological
system, treated in Section 8.2 and Section 8.3. Although the problem is
similar to the corresponding result in the GKZ-case in [SW09], the techniques
are very much different. It is here where the two cases
$\beta(\mathbf{e})\in{\mathds{Z}}$ resp. $\beta(\mathbf{e})\notin{\mathds{Z}}$
need to be treated separately. While the latter is a relatively simple
argument concerning eigenvalue decomposition for an operator derived from the
Euler vector field on the space $V$, the former is more delicate. Contrary to
the strategy in [SW09] (using so-called Euler-Koszul homology) we study here
various Lie algebroid cohomologies and prove some vanishing theorems about
them.

_Outline:_ Let us give an overview over the various parts of the paper. Notice
that the level of generality is decreasing, in the sense that the results in
the earlier sections apply to much more general situations than the main
result as stated above. In particular, Sections 4 to 6 contain many results of
general interest about $\mathscr{D}$-modules related to group actions on
algebraic varieties.

We start by defining in Section 2 certain Hodge modules on line bundles
$L\rightarrow X$ over smooth varieties (or rather on the complement of the
zero section $L^{*}$). Their underlying $\mathscr{D}$-modules (denoted by
$\mathcal{O}_{L^{*}}^{\beta}$) generalize the twisted structure sheaf
$\mathscr{D}_{{\mathds{C}}^{*}}/\mathscr{D}_{{\mathds{C}}^{*}}(\partial_{t}\cdot
t+\beta)$ (which would correspond to the case where the variety is a point).
Then we study their Fourier–Laplace transforms in Section 3 and show that they
still underly a mixed Hodge module on the dual bundle.

In Section 4, we consider a quite general situation of a smooth algebraic
variety $Y$ endowed with the action of an algebraic group $G^{\prime}$. From
the vector fields induced by this group action, together with a Lie algebra
homomorphism $\beta$, we construct a $\mathscr{D}_{Y}$-module
$\mathscr{N}_{Y}^{\beta}$. Especially important is the case where $Y$ occurs
as an orbit in a vector space $V$ underlying a rational representation
$\rho\colon G^{\prime}\rightarrow\operatorname{\textup{GL}}(V)$. According to
principles outlined above, there is a tautological system
$\tau(\rho,\overline{Y},\beta)$ and its Fourier transform
$\hat{\tau}(\rho,\overline{Y},\beta)$. We relate in 4.11 the restriction of
$\hat{\tau}(\rho,\overline{Y},\beta)$ to $Y$ with the intrinsically defined
module $\mathscr{N}_{Y}^{\beta}$.

In Section 5 we address the question under which hypotheses the tautological
system $\tau(\rho,\overline{Y},\beta)$ and its Fourier transform
$\hat{\tau}(\rho,\overline{Y},\beta)$ are a non-zero
$\mathscr{D}_{V^{\vee}}$\- resp. $\mathscr{D}_{V}$-module. We show that this
issue is closely related to the non-vanishing of the module
$\mathscr{N}_{Y}^{\beta}$. We develop therefore a framework, based on the
formalism of Lie algebroids and their universal enveloping algebras, to study
this problem. The main result in this section is 5.14 which gives a sufficient
criterion for $\hat{\tau}(\rho,\overline{Y},\beta)$ to be non-zero.

Section 6 discusses the more specific case where the variety $Y$ is the total
space of a line bundle over a variety $X$ equipped with an action by a group
$G$. Then $L$ can, with the choice of a character, be made into a
$G^{\prime}$-space, where $G^{\prime}:={\mathds{C}}^{*}\times G$. We give in
6.9 an alternative description, in this equivariant situation, of the module
$\mathcal{O}_{L^{*}}^{\beta}$ from Section 2.

In Section 7, we derive, using standard techniques from representation theory,
a formula for the complex parameter value $\beta(\mathbf{e})\in{\mathds{C}}$
for which the tautological system $\tau(\rho,\hat{X},\beta)$ is non-zero, at
least in the case of a semisimple group $G$. We also give a geometric
interpretation of this formula.

In Section 8, we apply all the previous results in the case where the variety
$X$ is a homogeneous space, and where the representation $\rho$ is in the dual
of the space of sections of an equivariant line bundle on $X$. The affine cone
of $X$ then takes the role of the $G^{\prime}$-invariant space used in the
definition of the tautological system $\tau(\rho,\hat{X},\beta)$. The main
result is then 8.20, showing that if $\beta$ is such that
$\tau(\rho,\hat{X},\beta)\neq 0$, then it underlies a pure complex Hodge
module for $\beta(\mathbf{e})\notin{\mathds{Z}}$ and a rational mixed Hodge
module for $\beta(\mathbf{e})\in{\mathds{Z}}_{>0}$. Moreover, we exhibit in
8.21 a functorial description of $\tau(\rho,\hat{X},\beta)$ as a direct resp.
as a proper direct image of a family of (complements of) hyperplane sections
of $X$, and in consequence solve the holonomic rank problem as stated in
[BHL+14] and [HLZ16] in this generalized situation. A major ingredient
necessary for the formulation of this functorial description is to determine
how the Fourier–Laplace transformation of $\tau(\rho,\hat{X},\beta)$ is
related to its restriction to the complement of the origin. The answer to this
question depends crucially on whether $\beta(\mathbf{e})$ is integral or not;
we treat the two cases separately in Section 8.2 and Section 8.3.

_Notations:_ Throughout, we work over $\mathds{C}$. By _variety_ , we mean an
integral scheme of finite type over $\mathds{C}$. When we talk about _points_
on a variety, we mean closed points unless mentioned otherwise. Our convention
for the projective space of a finite-dimensional vector space $V$ is
$\mathds{P}V:=\operatorname{\textup{Proj}\,}\operatorname{\textup{Sym}}V^{\vee}$,
i.e., $\mathds{P}V$ parameterizes one-dimensional _subspaces_ of $V$.

For a smooth variety $X$, we let $\mathscr{D}_{X}$ be the sheaf of algebraic
differential operators on $X$. If not mentioned otherwise, a
$\mathscr{D}_{X}$-module is a quasi-coherent $\mathcal{O}_{X}$-module equipped
with a left action by $\mathscr{D}_{X}$. The category of such modules is
denoted by $\textup{Mod}_{qc}(\mathscr{D}_{X})$ and the corresponding bounded
derived category by $D_{qc}^{b}(\mathscr{D}_{X})$. Similarly, let
$\textup{Mod}_{h}(\mathscr{D}_{X})$ and $D_{h}^{b}(\mathscr{D}_{X})$ be the
category of holonomic $\mathscr{D}_{X}$-modules and its corresponding bounded
derived category, respectively. Throughout, for a morphism $f\colon X\to Y$
between smooth varieties over $\mathds{C}$, we denote by $f_{+}\colon
D_{qc}^{b}(\mathscr{D}_{X})\to D_{qc}^{b}(\mathscr{D}_{Y})$ and $f^{+}\colon
D_{qc}^{b}(\mathscr{D}_{Y})\to D_{qc}^{b}(\mathscr{D}_{X})$ the functors
defined by

$f_{+}\mathscr{M}:=Rf_{*}(\mathscr{D}_{Y\leftarrow
X}\otimes^{\mathbb{L}}_{\mathscr{D}_{X}}\mathscr{M})\quad\text{and}\quad
f^{+}\mathscr{N}:=\mathscr{D}_{X\to
Y}\otimes^{\mathbb{L}}_{f^{-1}\mathscr{D}_{Y}}f^{-1}\mathscr{N}.$

Moreover, we denote by

${\mathbb{D}}\mathscr{M}:=\omega^{\vee}_{X}\otimes_{\mathcal{O}_{X}}R{\mathcal{H}\\!}om_{\mathscr{D}_{X}}(\mathscr{M},\mathscr{D}_{X})[\dim(X)]$

the duality functor from $D^{b}_{h}(\mathscr{D}_{X})$ to itself; it respects
$\textup{Mod}_{h}(\mathscr{D}_{X})$. We then define the functors

$f_{\dagger}:={\mathbb{D}}\circ
f_{+}\circ{\mathbb{D}}\quad\quad\textup{and}\quad\quad
f^{\dagger}:={\mathbb{D}}\circ f^{+}\circ{\mathbb{D}}.$

For any variety $X$, let $\textup{MHM}(X)$ be the Abelian category of
algebraic (${\mathds{Q}}$-)mixed Hodge modules on $X$ (as defined in [Sai88,
Sai90]) and $D^{b}\textup{MHM}(X)$ its bounded derived category. For any
morphism $f\colon X\rightarrow Y$ the functors $f_{+},f_{\dagger}$ resp.
$f^{\dagger}[\dim(Y)-\dim(X)],f^{+}[\dim(X)-\dim(Y)]$ on
$D^{b}_{h}(\mathscr{D}_{X})$ resp. $D^{b}_{h}(\mathscr{D}_{Y})$ lift to
functors

$f_{*},f_{!}\colon D^{b}\textup{MHM}(X)\rightarrow
D^{b}\textup{MHM}(Y)\quad\quad\textup{resp.}\quad\quad f^{*},f^{!}\colon
D^{b}\textup{MHM}(Y)\rightarrow D^{b}\textup{MHM}(X).$

We also denote by ${\mathbb{D}}$ the functor on $D^{b}\textup{MHM}(X)$ which
lifts the above defined holonomic duality functor on
$D^{b}_{h}(\mathscr{D}_{X})$. Any object $\mathcal{M}\in\textup{MHM}(X)$ is a
tuple $\mathcal{M}=(\mathscr{M},F_{\bullet},W_{\bullet},K)$ where
$\mathscr{M}\in\textup{Mod}_{h}(\mathscr{D}_{X})$ and $W_{\bullet}\mathscr{M}$
is its weight filtration. We denote by $\textup{HM}(X,w)$ (or simply
$\textup{HM}(X)$ if $w$ is clear from the context) the full subcategory of
objects such that $\operatorname{\textup{Gr}}_{l}(\mathscr{M})=0$ for all
$l\neq w$; these are the pure Hodge modules of weight $w$.

We will need an extension of the notion of (${\mathds{Q}}$-)mixed Hodge
modules to the category of complex mixed Hodge modules. It can be constructed
by first defining ${\mathds{R}}$-mixed Hodge modules, see [Moc15, Section
13.5]. Then a filtered $\mathscr{D}$-module $(\mathscr{M},F_{\bullet})$ is
said to underly a complex mixed Hodge module if it is a direct summand of an
${\mathds{R}}$-mixed Hodge module ([DS13, Definition 3.2.1.]). We denote the
corresponding Abelian category by $\textup{MHM}(X,{\mathds{C}})$, by
$D^{b}\textup{MHM}(X,{\mathds{C}})$ its bounded derived category and by
$\textup{HM}(X,{\mathds{C}},w)$ (or $\textup{HM}(X,{\mathds{C}})$ for short)
the category of pure complex Hodge modules of weight $w$. Many of the known
constructions for ${\mathds{R}}$-mixed Hodge modules carry over to the
categories $\textup{MHM}(X,{\mathds{C}})$ and $\textup{HM}(X,{\mathds{C}})$
since they are stable under taking direct summands. The article [DV22, Section
7.1 and Appendix A] contains a more detailed discussion of complex Hodge
modules.

For any variety $X$, write $a_{X}\colon X\rightarrow\\{pt\\}$ for the map to
the point. We denote by ${{}^{H}\\!}{\mathds{C}}_{pt}$ the trivial complex
Hodge structure of dimension $1$. Then

${}^{H\\!}\underline{{\mathds{C}}}_{X}:=a_{X}^{*}{{}^{H}\\!}{\mathds{C}}_{pt}[\dim(X)],$

is a smooth (constant) Hodge module and indeed an object in
$\textup{MHM}(X,{\mathds{C}})$. Notice that our notation differs from the
convention in [Sai88, Sai90], where the (${\mathds{Q}}$-)constant Hodge module
of rank $1$ is denoted by ${{}^{p}}{\mathds{Q}}_{X}^{H}$.

We will further need a particular smooth (but non-constant) complex Hodge
module on the one-dimensional torus ${\mathds{C}}^{*}$. Namely, for any
$\beta\in{\mathds{R}}$, we denote by $\mathcal{O}^{\beta}_{{\mathds{C}}^{*}}$
the $\mathscr{D}_{{\mathds{C}}^{*}}$-module

$\mathcal{O}^{\beta}_{{\mathds{C}}^{*}}=\mathscr{D}_{{\mathds{C}}^{*}}/\mathscr{D}_{{\mathds{C}}^{*}}(\partial_{t}t+\beta).$

We write ${{}^{H}\\!}\underline{{\mathds{C}}}^{\beta}_{{\mathds{C}}^{*}}$ for
the complex Hodge module with underlying $\mathscr{D}$-module equal to
$\mathcal{O}^{\beta}_{{\mathds{C}}^{*}}$ (placed in cohomological degree
zero), and where
$\operatorname{\textup{Gr}}^{F}_{p}\mathcal{O}^{\beta}_{{\mathds{C}}^{*}}=0$
for $p\neq 0$ and
$\operatorname{\textup{Gr}}^{W}_{i}\mathcal{O}^{\beta}_{{\mathds{C}}^{*}}=0$
for $i\neq 1$. Its corresponding perverse sheaf is ${\mathds{V}}[1]$, where
${\mathds{V}}$ is the local system of rank $1$ on ${\mathds{C}}^{*}$ given by
the monodromy with eigenvalue $e^{2\pi\sqrt{-1}\beta}$. Again, in the
conventions of [Sai88, Sai90] this object would have been denoted by
${{}^{p}}{\mathds{C}}_{X}^{H,\beta}$.

_Acknowledgements:_ We thank Michel Brion for his assistance finding
references for the parabolic version of Borel–Weil and for his explanations
about line bundles on homogeneous varieties. We thank Luis Narváez Macarro for
sharing some insights on Lie algebroids with us, especially for his help with
the proof of 5.4.

##  2 Mixed Hodge modules on line bundles

On $\mathds{C}^{*}$, we consider the $\mathscr{D}_{\mathds{C}^{*}}$-module

$\mathcal{O}_{\mathds{C}^{*}}^{\beta}:=\mathscr{D}_{\mathds{C}^{*}}/\mathscr{D}_{\mathds{C}^{*}}\cdot(\partial_{t}t+\beta)$

for every $\beta\in\mathds{C}$, which underlies a polarizable pure complex
Hodge module ${}^{H\\!}\underline{{\mathds{C}}}_{\mathds{C}^{*}}^{\beta}$ when
$\beta\in\mathds{R}$. For a smooth variety $X$, we may then define for every
$\beta\in\mathds{C}$ the $\mathscr{D}_{\mathds{C}^{*}\times X}$-module

$\mathcal{O}_{\mathds{C}^{*}\times
X}^{\beta}:=\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{X}=p^{+}\mathcal{O}_{\mathds{C}^{*}}^{\beta}\otimes_{\mathcal{O}_{\mathds{C}^{*}\\!\times\\!X}}q^{+}\mathcal{O}_{X},$
(1)

where $p\colon\mathds{C}^{*}\times X\to\mathds{C}^{*}$ and
$q\colon\mathds{C}^{*}\times X\to X$ are the projections onto the first and
second factor, respectively.

In this section, we are interested in line bundles $\pi\colon L\to X$ on a
smooth variety and we wish to define on $L^{*}\subseteq L$, the complement of
the zero section of $\pi$, a $\mathscr{D}_{L^{*}}$-module
$\mathcal{O}_{L^{*}}^{\beta}$ by gluing the above construction over a
trivializing open cover of $X$.

We immediately face the following issue: If $\widetilde{\psi}$ is an
automorphism of $\mathds{C}^{*}\times X$ which is given by
$\widetilde{\psi}(\lambda,p)=(\alpha(p)\cdot\lambda,p)$ for some
$\alpha\in\Gamma(X,\mathcal{O}_{X}^{\times})$, then
$\widetilde{\psi}_{+}\mathcal{O}_{\mathds{C}^{*}\times X}^{\beta}$ is in
general not isomorphic to $\mathcal{O}_{\mathds{C}^{*}\times X}^{\beta}$. This
leads to several complications:

  1. 1.

If $L$ is isomorphic to the trivial line bundle $\mathds{C}\times X$, then to
define $\mathcal{O}_{L^{*}}^{\beta}$ we need to choose an isomorphism with
$\mathds{C}\times X$, and the resulting $\mathscr{D}$-module depends on this
choice (with the notable exception of the case $\beta\in\mathds{Z}$, in which
case we get $\mathcal{O}_{L^{*}}^{\beta}\cong\mathcal{O}_{L^{*}}$).

  2. 2.

For a non-trivial line bundle $L\not\cong\mathds{C}\times X$ and a
trivializing open cover $X=\bigcup_{i\in I}U_{i}$, in order to glue all
$\mathcal{O}_{\mathds{C}^{*}\times U_{i}}^{\beta}$, $i\in I$, to a
$\mathscr{D}_{L^{*}}$-module, we need to identify
$\mathcal{O}_{\mathds{C}^{*}\times U_{ij}}^{\beta}$ with
$\widetilde{\psi}_{ij,+}\mathcal{O}_{\mathds{C}^{*}\times U_{ij}}^{\beta}$,
where $\widetilde{\psi}_{ij}\colon\mathds{C}^{*}\times
U_{ij}\to\mathds{C}^{*}\times U_{ij}$ is the transition map over the
intersection $U_{ij}:=U_{i}\cap U_{j}$. However, it turns out that these two
$\mathscr{D}_{\mathds{C}^{*}\times U_{ij}}$-modules are in general isomorphic
(for a suitable choice of transition maps $\widetilde{\psi}_{ij}$) only if
$\beta\in\frac{1}{k}\mathds{Z}$ and $L$ is the $k$-th tensor power of another
line bundle $F$ on $X$.

  3. 3.

Even if $\mathcal{O}_{\mathds{C}^{*}\times U_{ij}}^{\beta}$ and
$\psi_{ij}^{+}\mathcal{O}_{\mathds{C}^{*}\times U_{ij}}^{\beta}$ are
isomorphic, i.e., $\beta\in\frac{1}{k}\mathds{Z}$ and $L=F^{\otimes k}$, then
there may be several isomorphisms between these
$\mathscr{D}_{\mathds{C}^{*}\times U_{ij}}$-modules. leading to non-isomorphic
$\mathscr{D}_{L^{*}}$-modules after gluing, corresponding to the possible
choices of the $k$-th root $F$ of $L$. In fact, one can show that the glued
$\mathscr{D}_{L^{*}}$-modules is independent of the choices only in the case
that there are no $k$-torsion line bundles on $X$.

Given all of these constraints, our approach is as follows: We fix a non-
trivial line bundle $\pi^{F}\colon F\to X$ to begin with. Then, for the tensor
powers $L=F^{\otimes k}$ of this line bundle, we define
$\mathscr{D}_{L^{*}}$-modules $\mathcal{O}_{L^{*}}^{\beta}$ for all
$\beta\in\mathds{C}$ with $k\beta\in\mathds{Z}$ by appropriately gluing (1)
over trivializing open subsets of $X$. The constructed $\mathscr{D}$-module
$\mathcal{O}_{L^{*}}^{\beta}$ still depends on the choice of $F$ as the $k$-th
root of $L$. Only in the case that there is no torsion in the Picard group of
$X$ will the constructed $\mathscr{D}_{L^{*}}$-modules
$\mathcal{O}_{L^{*}}^{\beta}$ always be intrinsic to the line bundles $L$ on
which they are defined.

###  2.1 The $\mathscr{D}$-module $\mathcal{O}_{L^{*}}^{\beta}$

Throughout, let $X$ be a smooth variety and let $\pi^{F}\colon F\to X$ be a
line bundle. Fix $k\in\mathds{Z}$ and let $\pi\colon L:=F^{\otimes k}\to X$ be
the $k$-th tensor power of the line bundle $F$. For negative $k$, this means
the line bundle dual to $F^{\otimes(-k)}$. In particular, we have $L=F^{\vee}$
if $k=-1$. For $k=0$, we have the trivial bundle $L=F^{\otimes
0}=\mathds{C}\times X$. We denote the sheaf of sections of the line bundles
$F$ and $L$ by $\mathscr{F}$ and $\mathscr{L}(=\mathscr{F}^{\otimes k})$,
respectively.

Let $L^{*}\subseteq L$ denote the complement of the zero section of $\pi\colon
L\to X$ (and similarly for other line bundles). In this section, we prove:

######  Proposition 2.1.

Let $\beta\in\mathds{C}$ with $k\beta\in\mathds{Z}$. Up to isomorphism, there
exists a unique $\mathscr{D}_{L^{*}}$-module $\mathcal{O}_{L^{*}}^{\beta}$
with the following local description over open subsets of $X$ trivializing the
line bundle $F$: If $s\in\Gamma(U,\mathscr{F})$ is a local non-vanishing
section of $F\to X$, then under

$\widetilde{\psi}_{s^{k}}\colon\mathds{C}^{*}\times
U\xrightarrow{\cong}\pi^{-1}(U)^{*},\qquad(\lambda,p)\mapsto\lambda\cdot
s^{k}(p),$

we have

$\widetilde{\psi}_{s^{k}}^{+}\big{(}{(\mathcal{O}_{L^{*}}^{\beta})_{|{\pi^{-1}(U)^{*}}}}\big{)}\cong\mathscr{D}_{\mathds{C}^{*}}/\mathscr{D}_{\mathds{C}^{*}}(\partial_{t}t+\beta)\boxtimes\mathcal{O}_{U}.$

The formulation “$\beta\in\mathds{C}$ with $k\beta\in\mathds{Z}$” instead of
$\beta\in\frac{1}{k}\mathds{Z}$ is to also accommodate the case $k=0$ (i.e.,
$L\cong\mathds{C}\times X$) with $\beta\in\mathds{C}$ arbitrary. This is of
relevance for some intermediate results (such as 3.8 or 4.11), but will
however never be the case if $X$ is projective and if we assume that $L$ is
very ample, as we do in our main results (like 8.20).

We will mainly be concerned with the line bundle $L$, and its $k$-th root $F$
should be viewed an additional datum necessary for the definition of
$\mathcal{O}_{L^{*}}^{\beta}$. Later on, we will specialize to cases in which
the Picard group of $X$ is torsion-free. In this case, $F$ is up to
isomorphism already uniquely determined by the line bundle $L$ admitting a
$k$-th root.

We start with the case $k=0$, i.e., the case that $\pi\colon L\to X$ is the
trivial line bundle $F^{\otimes 0}=\mathds{C}\times X$. Then
$L^{*}=\mathds{C}^{*}\times X$ and for any $\beta\in\mathds{C}$, we define the
$\mathscr{D}_{L^{*}}$-module

$\displaystyle\mathcal{O}_{\mathds{C}^{*}\times X}^{\beta}$
$\displaystyle:=\mathscr{D}_{\mathds{C}^{*}}/\mathscr{D}_{\mathds{C}^{*}}(\partial_{t}t+\beta)\boxtimes\mathcal{O}_{X}$
(2)
$\displaystyle\phantom{:}=p^{+}\big{(}\mathscr{D}_{\mathds{C}^{*}}/\mathscr{D}_{\mathds{C}^{*}}(\partial_{t}t+\beta)\big{)}\otimes_{\mathcal{O}_{\mathds{C}^{*}\\!\times\\!X}}q^{+}\mathcal{O}_{X},$

where $p\colon\mathds{C}^{*}\times X\to\mathds{C}^{*}$ and
$q\colon\mathds{C}^{*}\times X\to X$ are the projections onto the first and
second factor, respectively.

From now on, let $k\in\mathds{Z}\setminus\\{0\\}$. We construct
$\mathcal{O}_{L^{*}}^{\beta}$ by gluing the local construction. For this, we
choose an open cover $X=\bigcup_{i\in I}U_{i}$ trivializing the line bundle
$\pi^{F}\colon F\to X$ and therefore also its $k$-th tensor power $L$. We will
for convenience assume that each $U_{i}$ is affine and admits a local
coordinate system. The choice of non-vanishing local sections
$s_{i}\in\Gamma(U_{i},\mathscr{F})$ determines trivializing isomorphisms

$\psi_{s_{i}}\colon\mathds{C}\times
U_{i}\xrightarrow{\cong}\pi^{F,-1}(U_{i}),\qquad(\lambda,p)\mapsto\lambda\cdot
s_{i}(p)$

of line bundles over $U_{i}$, where we view the section $s_{i}$ as a morphism
$U_{i}\to F$ with $\pi^{F}\circ s_{i}=\operatorname{\textup{id}}_{U_{i}}$.

Then, over the intersections $U_{ij}:=U_{i}\cap U_{j}$, the transition maps

$\psi_{s_{i}/s_{j}}\colon\mathds{C}\times U_{ij}\xrightarrow[\cong]{\
{(\psi_{s_{i}})_{|{\mathds{C}\times U_{ij}}}}\
}\pi^{F,-1}(U_{ij})\xrightarrow[\cong]{\
{(\psi_{s_{j}}^{-1})_{|{\pi^{F,-1}(U_{ij})}}}\ }\mathds{C}\times U_{ij}$ (3)

are of the form $\psi_{s_{i}/s_{j}}(\lambda,p)=(\alpha_{ij}(p)\cdot\lambda,p)$
for $\alpha_{ij}:=s_{j}/s_{i}\in\Gamma(U_{ij},\mathcal{O}_{U_{ij}}^{\times})$.
These $(\alpha_{ij})_{i,j\in I}$ can also be viewed as a Čech cocyle
$\alpha\in\check{C}^{1}(\mathcal{U},\mathcal{O}_{X}^{\times})$,
$\mathcal{U}:=(U_{i})_{i\in I}$, representing the isomorphism class of the
line bundle $\mathscr{F}$ in $\operatorname{Pic}(X)\cong
H^{1}(X,\mathcal{O}_{X}^{\times})\cong\check{H}^{1}(\mathcal{U},\mathcal{O}_{X}^{\times})$.

The $k$-th tensor power $L=F^{\otimes k}$ also trivializes over the open sets
$U_{i}$ and the non-vanishing local sections
$s_{i}^{k}\in\Gamma(U_{i},\mathscr{L})$ induce trivializing isomorphisms

$\psi_{s_{i}^{k}}\colon\mathds{C}\times
U_{i}\xrightarrow{\cong}\pi^{-1}(U_{i}),\qquad(\lambda,p)\mapsto\lambda\cdot
s_{i}^{k}(p).$

They restrict to isomorphisms on the complements of the zero sections, which
we denote by

$\widetilde{\psi}_{s_{i}^{k}}\colon\mathds{C}^{*}\times
U_{i}\xrightarrow{\cong}\pi^{-1}(U_{i})^{*}.$

As in (3), the transition maps for $L$ (resp. $L^{*}$) over the intersections
$U_{ij}:=U_{i}\cap U_{j}$ are of the form

$\psi_{s_{i}^{k}/s_{j}^{k}}\colon\mathds{C}\times
U_{ij}\xrightarrow{\cong}\pi^{-1}(U_{ij})\xrightarrow{\cong}\mathds{C}\times
U_{ij},\quad\text{resp.\
}\widetilde{\psi}_{s_{i}^{k}/s_{j}^{k}}\colon\mathds{C}^{*}\times
U_{ij}\xrightarrow{\cong}\pi^{-1}(U_{ij})^{*}\xrightarrow{\cong}\mathds{C}^{*}\times
U_{ij},$

mapping $(\lambda,p)$ to $(\alpha_{ij}^{k}(p)\cdot\lambda,p)$, where we recall
that
$\alpha_{ij}:=s_{i}/s_{j}\in\Gamma(U_{ij},\mathcal{O}_{U_{ij}}^{\times})$.

If the collection of non-vanishing local sections $(s_{i})_{i\in I}$ is fixed,
we will for simplicity denote

$\psi_{i}:=\psi_{s_{i}^{k}},\qquad\psi_{ij}:=\psi_{s_{i}^{k}/s_{j}^{k}},\qquad\widetilde{\psi}_{i}:=\widetilde{\psi}_{s_{i}^{k}},\qquad\widetilde{\psi}_{ij}:=\widetilde{\psi}_{s_{i}^{k}/s_{j}^{k}}.$

Let $\beta\in\frac{1}{k}\mathds{Z}$. For $i\in I$, we define the
$\mathscr{D}_{\pi^{-1}(U_{i})^{*}}$-module

$\mathcal{M}_{i}^{\beta}:=\widetilde{\psi}_{i,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta},$

where the right hand side was defined in (2). We wish to glue these
$\mathcal{M}_{i}^{\beta}$ to a $\mathscr{D}_{L^{*}}$-module, which we will
call $\mathcal{O}_{L^{*}}^{\beta}$.

We recall what needs to be checked in order to glue. If there exists a
$\mathscr{D}_{L^{*}}$-module $\mathcal{M}$ satisfying
${\mathcal{M}_{|{\pi^{-1}(U_{i})^{*}}}}\cong\mathcal{M}_{i}^{\beta}$, then

${(\mathcal{M}_{i}^{\beta})_{|{\pi^{-1}(U_{ij})^{*}}}}\cong{\mathcal{M}_{|{\pi^{-1}(U_{ij})^{*}}}}\cong{(\mathcal{M}_{j}^{\beta})_{|{\pi^{-1}(U_{ij})^{*}}}}.$

Conversely, in order to glue $(\mathcal{M}_{i}^{\beta})_{i\in I}$ to a
$\mathscr{D}_{L^{*}}$-module, we must identify under the transition maps the
restrictions of $\mathcal{M}_{i}^{\beta}$ and $\mathcal{M}_{j}^{\beta}$ to
$\pi^{-1}(U_{ij})^{*}$. More precisely, we need to construct isomorphisms of
$\mathscr{D}_{\mathds{C}^{*}\times U_{ij}}$-modules

$\mu_{ij}\colon\widetilde{\psi}_{ij,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta}\xrightarrow{\cong}\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta}\qquad\text{for }i,j\in I$ (4)

satisfying the compatibility conditions

$\begin{cases}{(\mu_{j\ell})_{|{U}}}\circ\big{(}({(\widetilde{\psi}_{j\ell})_{|{U}}})_{+}{(\mu_{ij})_{|{U}}}\big{)}={(\mu_{i\ell})_{|{U}}}&\text{for
}i,j,\ell\in I,\ U:=\mathds{C}^{*}\\!\times\\!(U_{i}\cap U_{j}\cap
U_{\ell}),\\\\[2.5pt] \mu_{ii}=\operatorname{\textup{id}}&\text{for }i\in
I.\end{cases}$ (5)

In order to construct the gluing data $(\mu_{ji})$, the following explicit
description is key:

######  Lemma 2.2.

Let $U$ be a smooth affine variety which admits a local coordinate system
$(x_{1},\ldots,x_{n})$. Let $\widetilde{\psi}$ be an automorphism on
$\mathds{C}^{*}\times U$ given by
$(\lambda,p)\mapsto(\alpha(p)\cdot\lambda,p)$ for some
$\alpha\in\Gamma(U,\mathcal{O}_{U}^{\times})$. Then, for all
$\beta\in\mathds{C}$, we have

$\displaystyle\mathcal{O}_{\mathds{C}^{*}\times U}^{\beta}$
$\displaystyle=\mathscr{D}_{\mathds{C}^{*}\times
U}/\mathscr{D}_{\mathds{C}^{*}\times
U}\Big{(}\partial_{t}t+\beta,\;\partial_{x_{1}},\dots,\partial_{x_{n}}\Big{)},$
$\displaystyle\widetilde{\psi}_{+}\mathcal{O}_{\mathds{C}^{*}\times
U}^{\beta}$ $\displaystyle=\mathscr{D}_{\mathds{C}^{*}\times
U}/\mathscr{D}_{\mathds{C}^{*}\times
U}\bigg{(}\partial_{t}t+\beta,\;\partial_{x_{\ell}}-(\beta+1)\alpha^{-1}\frac{\partial\alpha}{\partial
x_{\ell}}\mathopen{}\mathclose{{}\left.\vphantom{\Big{\\{}}\;|\;\vphantom{\Big{\\}}}}\right.\ell=1,\dots,n\bigg{)}.$

In terms of these cyclic descriptions, any $\mathscr{D}_{\mathds{C}^{*}\times
U}$-isomorphism $\widetilde{\psi}_{+}\mathcal{O}_{\mathds{C}^{*}\times
U}^{\beta}\xrightarrow{\cong}\mathcal{O}_{\mathds{C}^{*}\times U}^{\beta}$ is
given by right-multiplication with some
$h\in\Gamma(U,\mathcal{O}_{U})^{\times}$ satisfying

$h^{-1}\frac{\partial h}{\partial
x_{\ell}}=(\beta+1)\cdot\alpha^{-1}\frac{\partial\alpha}{\partial
x_{\ell}}\qquad\text{for }\ell=1,\dots,n.$ (6)

###### Proof.

The description of $\mathcal{O}_{\mathds{C}^{*}\times U}^{\beta}$ follows from
$\mathcal{O}_{U}=\mathscr{D}_{U}/\mathscr{D}_{U}(\partial_{x_{1}},\dots,\partial_{x_{n}})$.
Note that the automorphism $\widetilde{\psi}\colon\mathds{C}^{*}\times
U\to\mathds{C}^{*}\times U$ is in coordinates given by mapping
$(t,x_{1},\dots,x_{n})\mapsto(\alpha\cdot t,x_{1},\dots,x_{n})$. With this,
one easily computes $\widetilde{\psi}_{+}\mathcal{O}_{\mathds{C}^{*}\times
U}^{\beta}$ to be the claimed cyclic module.

A $\mathscr{D}_{\mathds{C}^{*}\times U}$-module isomorphism from the cyclic
module $\widetilde{\psi}_{+}\mathcal{O}_{\mathds{C}^{*}\times U}^{\beta}$ to
the cyclic module $\mathcal{O}_{\mathds{C}^{*}\times U}^{\beta}$ must be given
by right-multiplication with an invertible function
$h\in\Gamma(\mathds{C}^{*}\times U,\mathcal{O}_{\mathds{C}^{*}\times
U})^{\times}$. Here, we are using that every operator in
$\Gamma(\mathds{C}^{*}\times U,\mathcal{O}_{\mathds{C}^{*}\times U}^{\beta})$
is represented by a function $h$ which must be invertible if the operator is.
With the explicit descriptions of the cyclic modules from above, one sees that
right-multiplication by $h$ is a well-defined isomorphism between the two
$\mathscr{D}_{\mathds{C}^{*}\times U}$-modules if and only if

$\frac{\partial h}{\partial t}=0\quad\text{and}\quad\frac{\partial h}{\partial
x_{\ell}}=(\beta+1)\alpha^{-1}\frac{\partial\alpha}{\partial
x_{\ell}}h\quad\text{for }\ell=1,\dots,n.$

The first condition is fulfilled if and only if
$h\in\Gamma(U,\mathcal{O}_{U})^{\times}$ and the second condition is precisely
(6). ∎

For $k\neq 0$, we write $\beta\in\frac{1}{k}\mathds{Z}$ as
$\beta=\frac{m}{k}-1$ with $m\in\mathds{Z}$. Given 2.2, we can now define the
gluing isomorphisms $\mu_{ij}$ in (4) to be given by right-multiplication with
$\alpha_{ij}^{m}=s_{i}^{m}/s_{j}^{m}$, since then the equation (6) is
satisfied for $h=\alpha_{ij}^{m}$ with respect to
$\widetilde{\psi}=\widetilde{\psi}_{ij}=\widetilde{\psi}_{s_{i}^{k}/s_{j}^{k}}$.
Moreover, these isomorphisms satisfy the compatibility cocycle condition (5),
since the transition maps of $L$ satisfy
${(\widetilde{\psi}_{j\ell})_{|{U}}}\circ{(\widetilde{\psi}_{ij})_{|{U}}}={(\widetilde{\psi}_{i\ell})_{|{U}}}$
on $U:=\mathds{C}^{*}\times(U_{i}\cap U_{j}\cap U_{\ell})$ for all
$i,j,\ell\in I$.

######  Definition 2.3.

Let $F\to X$ be a line bundle on a smooth variety $X$ and consider its $k$-th
tensor power $L=F^{\otimes k}\to X$ for some $k\in\mathds{Z}$. Let
$\beta\in\mathds{C}$ with $m:=k\beta+1\in\mathds{Z}$. We define
$\mathcal{O}_{L^{*}}^{\beta}$ as the $\mathscr{D}_{L^{*}}$-module that is
obtained by gluing the $\mathscr{D}_{\mathds{C}^{*}\times U_{i}}$-modules
$(\widetilde{\psi}_{i,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta})_{i\in I}$ along the isomorphisms
$\mu_{ij}\colon\widetilde{\psi}_{ij,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta}\xrightarrow{\cong}\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta}$ given by right-multiplication with
$\alpha_{ij}^{m}=s_{i}^{m}/s_{j}^{m}$ with respect to the cyclic description
given in 2.2.

Three remarks are in order: First of all, this construction is independent of
the chosen trivializing open cover on $X$ and the chosen local non-vanishing
sections $s_{i}\in\Gamma(U_{i},\mathscr{F})$ corresponding to the choice of
local trivializing isomorphisms $\psi_{s_{i}}$ of $F$. This means that if we
choose another open cover $(U^{\prime}_{i})_{i\in I^{\prime}}$ of $X$
trivializing $F$ and local non-vanishing sections
$s_{i}^{\prime}\in\Gamma(U^{\prime}_{i},\mathscr{F})$, then the two
$\mathscr{D}_{L^{*}}$-modules we obtain by gluing are isomorphic.

Secondly, if $k^{\prime}\in\mathds{Z}$ divides $k$ and
$k^{\prime}\beta\in\mathds{Z}$, then the construction of
$\mathcal{O}_{L^{*}}^{\beta}$ from the $k$-th root $F$ of $L$ agrees with the
construction from the $k^{\prime}$-th root $F^{\otimes(k/k^{\prime})}$ of $L$.
In this sense, $\mathcal{O}_{L^{*}}^{\beta}$ does not depend on the choice of
$k$ as a denominator for $\beta$.

Thirdly, however, we must emphasize that the $\mathscr{D}_{L^{*}}$-module
$\mathcal{O}_{L^{*}}^{\beta}$ _does_ depend on the choice of the line bundle
$F$ as the $k$-th root of $L$ in the Picard group: If the Picard group has
$k$-torsion, then there exists another line bundle $F^{\prime}$ on $X$ with
$F\not\cong F^{\prime}$, but $F^{\otimes k}\cong(F^{\prime})^{\otimes k}$. In
this case, the two $\mathscr{D}_{L^{*}}$-modules $\mathcal{O}_{F^{\otimes
k,*}}^{\beta}$ and $\mathcal{O}_{(F^{\prime})^{\otimes k,*}}^{\beta}$ are in
general not isomorphic.

###### Proof of 2.1.

As just remarked, the isomorphism class of $\mathcal{O}_{L^{*}}^{\beta}$ does
not depend on the choice of the trivializing open cover for $F\to X$ and the
choice of local non-vanishing sections describing the trivializing
isomorphisms. Thus, a given local non-vanishing section
$s\in\Gamma(U,\mathscr{F})$ may be assumed to be part of the data used in the
gluing. This shows that the construction in 2.3 describes a
$\mathscr{D}_{L^{*}}$-module with the desired property.

It remains to show uniqueness. Since the local description of
$\mathcal{O}_{L^{*}}^{\beta}$ of $F$ on a trivializing open cover is
prescribed, the only part in the construction where we may have a choice was
when defining the gluing data $\mu_{ij}$. According to 2.2, this amounts to
the choice of $h_{ij}\in\Gamma(U_{ij},\mathcal{O}_{U_{ij}}^{\times})$ with

$h_{ij}^{-1}\frac{\partial h_{ij}}{\partial
x_{\ell}}=(\beta+1)\cdot\alpha_{ij}^{-k}\frac{\partial\alpha_{ij}^{k}}{\partial
x_{\ell}}\qquad\text{for }\ell=1,\dots,n.$

Using $k\cdot(\beta+1)=m$, this is equivalent to

$\frac{\partial(h_{ij}/\alpha_{ij}^{m})}{\partial x_{\ell}}=0\qquad\text{for
}\ell=1,\dots,n.$

Hence, $h_{ij}=c_{ij}\alpha_{ij}^{m}$ for some $c_{ij}\in\mathds{C}^{*}$. The
compatibility condition (5) then forces $c_{ij}c_{j\ell}=c_{i\ell}$ and
$c_{ii}=1$ for $i,j,\ell\in I$. This guarantees that we can write
$c_{ij}=\tilde{c}_{i}/\tilde{c}_{j}$ for non-zero constants
$\tilde{c}_{i}\in\mathds{C}^{*}$. The $\mathscr{D}_{L^{*}}$-module obtained
from gluing over $U_{ij}$ via right-multiplication with $\alpha_{ij}^{m}$ is
isomorphic to the one via right-multiplication with $c_{ij}\alpha_{ij}^{m}$.
Indeed, there is an isomorphism locally given on $U_{i}$ by multiplication
with $\tilde{c}_{i}$ in the cyclic representations. ∎

Next, we observe that $\mathcal{O}_{L^{*}}^{\beta}$ only depends on the class
of $\beta$ modulo $\mathds{Z}$:

######  Proposition 2.4.

Let $\beta,\beta^{\prime}\in\mathds{C}$ with
$k\beta,k\beta^{\prime}\in\mathds{Z}$. Then

$\mathcal{O}_{L^{*}}^{\beta}\cong\mathcal{O}_{L^{*}}^{\beta^{\prime}}\quad\Leftrightarrow\quad\beta-\beta^{\prime}\in\mathds{Z}.$

###### Proof.

An isomorphism
$\xi\colon\mathcal{O}_{L^{*}}^{\beta}\xrightarrow{\cong}\mathcal{O}_{L^{*}}^{\beta^{\prime}}$
is given by isomorphisms over the open cover

$\xi_{i}\colon{(\mathcal{O}_{L^{*}}^{\beta})_{|{\pi^{-1}(U_{i})^{*}}}}\xrightarrow{\cong}{(\mathcal{O}_{L^{*}}^{\beta^{\prime}})_{|{\pi^{-1}(U_{i})^{*}}}}$

such that, for all $i,j\in I$, the restrictions of $\xi_{i}$ and $\xi_{j}$ to
$\pi^{-1}(U_{ij})$ agree. By construction, there are isomorphisms

$\chi_{i}\colon{(\mathcal{O}_{L^{*}}^{\beta})_{|{\pi^{-1}(U_{i})^{*}}}}\xrightarrow{\cong}\widetilde{\psi}_{i,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta},\qquad\chi^{\prime}_{i}\colon{(\mathcal{O}_{L^{*}}^{\beta^{\prime}})_{|{\pi^{-1}(U_{i})^{*}}}}\xrightarrow{\cong}\widetilde{\psi}_{i,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta^{\prime}}.$

Therefore, $\xi_{i}$ must be of the form
$(\chi^{\prime}_{i})^{-1}\circ\widetilde{\psi}_{i,+}(\varphi_{i})\circ\chi_{i}$
for some $\varphi_{i}\colon\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta}\xrightarrow{\cong}\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta^{\prime}}$. Using the description as a cyclic
$\mathscr{D}$-module

$\mathcal{O}_{\mathds{C}^{*}\times
U_{i}}^{\beta}=\mathscr{D}_{\mathds{C}^{*}\times
U_{i}}/\mathscr{D}_{\mathds{C}^{*}\times
U_{i}}\big{(}\partial_{t}t+\beta,\;\partial_{x_{1}},\dots,\partial_{x_{n}}\big{)}$

(with respect to a local coordinate system), one sees that
$\mathcal{O}_{\mathds{C}^{*}\times U_{i}}^{\beta}$ and
$\mathcal{O}_{\mathds{C}^{*}\times U_{i}}^{\beta^{\prime}}$ are only
isomorphic in case $\beta-\beta^{\prime}\in\mathds{Z}$, and in that case
$\varphi_{i}$ may be chosen to be right-multiplication with
$t^{\beta^{\prime}-\beta}$. This defines isomorphisms $\xi_{i}$ for
$\beta,\beta^{\prime}$ satisfying $\beta-\beta^{\prime}\in\mathds{Z}$, which
we assume from now on.

In order to show that $\xi_{i}$ and $\xi_{j}$ agree on
$V:=\pi^{-1}(U_{ij})^{*}$, we need to check that the following diagram
commutes:

${\widetilde{\psi}_{ij,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta}}$${\widetilde{\psi}_{ij,+}\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta^{\prime}}}$${\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta}}$${\mathcal{O}_{\mathds{C}^{*}\times
U_{ij}}^{\beta^{\prime}},}$$\scriptstyle{\widetilde{\psi}_{ij,+}({(\varphi_{i})_{|{\mathds{C}^{*}\times
U_{ij}}}})}$$\scriptstyle{\mu_{ij}}$$\scriptstyle{\mu_{ij}^{\prime}}$$\scriptstyle{{(\varphi_{j})_{|{\mathds{C}^{*}\times
U_{ij}}}}}$

where the vertical arrows are
$\widetilde{\psi}_{j}^{+}({\chi_{j}}_{|V}\circ({\chi_{i}}_{|V})^{-1})$ and
$\widetilde{\psi}_{j}^{+}({\chi^{\prime}_{j}}_{|V}\circ({\chi^{\prime}_{i}}_{|V})^{-1})$,
which are the gluing isomorphisms $\mu_{ij}$ and $\mu_{ij}^{\prime}$ in the
constructions of $\mathcal{O}_{L^{*}}^{\beta}$ and
$\mathcal{O}_{L^{*}}^{\beta^{\prime}}$, respectively. In the cyclic
presentations of the involved $\mathscr{D}_{\mathds{C}^{*}\times
U_{ij}}$-modules from 2.2, they are given by right-multiplication with
$\alpha_{ij}^{k\beta+1}$ and $\alpha_{ij}^{k\beta^{\prime}+1}$, respectively.
Meanwhile, ${(\varphi_{i})_{|{\mathds{C}^{*}\times
U_{ij}}}}={(\varphi_{j})_{|{\mathds{C}^{*}\times U_{ij}}}}$ is given by right-
multiplication with $t^{\beta^{\prime}-\beta}$. Applying the functor
$\widetilde{\psi}_{ij,+}$ to this homomorphism, we see that the upper map in
the above diagram is given by right-multiplication with
$(\alpha_{ij}^{-k}t)^{\beta^{\prime}-\beta}$. Since

$\alpha_{ij}^{k(\beta^{\prime}+1)}\cdot(\alpha_{ij}^{-k}t)^{\beta^{\prime}-\beta}=t^{\beta^{\prime}-\beta}\cdot\alpha_{ij}^{k(\beta+1)},$

this shows that the diagram commutes, and therefore
${(\xi_{i})_{|{V}}}={(\xi_{j})_{|{V}}}$. ∎

######  Remark 2.5.

The $\mathcal{O}_{L^{*}}$-module underlying the $\mathscr{D}_{L^{*}}$-module
$\mathcal{O}_{L^{*}}^{\beta}$ for $\beta=\ell/k$ is
$\pi_{0}^{*}\mathscr{F}^{\otimes\ell}$, the $\mathcal{O}$-module pull-back of
$\mathscr{F}^{\otimes\ell}$ under $\pi_{0}\colon
L^{*}\xhookrightarrow{j_{L}}L\xrightarrow{\pi}X$, which is a torsion element
of $\operatorname{Pic}(L^{*})$ of order $k$. We will revisit the construction
of $\mathcal{O}_{L^{*}}^{\beta}$ from this viewpoint in Section 6.3.

There is a short exact sequence

$0\to\pi_{0}^{*}\mathscr{L}\to\Theta_{L^{*}}\to\pi_{0}^{*}\Theta_{X}\to 0$

and $\pi_{0}^{*}\mathscr{L}=\mathcal{O}_{X}E$, where $E$ is the vector field
on $L^{*}$ corresponding to the $\mathds{C}^{*}$-action scaling the fibers,
see Section 6.2. This short exact sequence splits locally over trivializations
$\widetilde{\psi}_{s_{i}^{k}}\colon\mathds{C}^{*}\times
U_{i}\xrightarrow{\cong}\pi_{0}^{-1}(U_{i})$ and the
$\mathscr{D}_{L^{*}}$-module structure on
$\mathcal{O}_{L^{*}}^{\ell/k}=\pi_{0}^{*}\mathscr{F}^{\otimes\ell}$ can be
locally described by
$E\cdot\pi_{0}^{*}s_{i}^{\ell}=\ell\,\pi_{0}^{*}s_{i}^{\ell}$ and
$\pi_{0}^{*}\Theta_{U_{i}}\cdot\pi_{0}^{*}s_{i}^{\ell}=0$.

###  2.2 Mixed Hodge module structure on $\mathcal{O}_{L^{*}}^{\beta}$

In this section, we show that the $\mathscr{D}_{L^{*}}$-module
$\mathcal{O}_{L^{*}}^{\beta}$ constructed in the previous section underlies a
complex pure Hodge module, and we consider its extension to a mixed Hodge
module on $L$.

We continue using the notations from Section 2.1. In particular, we consider a
line bundle $\pi\colon L\to X$ which is the $k$-th tensor power of a line
bundle $\pi^{F}\colon F\to X$ for a fixed integer $k$, and call
$L^{*}\subseteq L$ the complement of the zero section of $L$. Moreover, we fix
an open cover $X=\bigcup_{i\in I}U_{i}$ with a choice of non-vanishing local
sections $s_{i}\in\Gamma(U_{i},\mathscr{F})$ inducing trivializations of $F$
and $L$ over $U_{i}$.

######  Proposition 2.6.

Let $\beta\in\mathds{C}$ be such that $k\beta\in\mathds{Z}$. Let $U\subseteq
X$ be a dense open subset over which the line bundle $F$ trivializes. Then the
$\mathscr{D}_{L^{*}}$-module $\mathcal{O}_{L^{*}}^{\beta}$ is the minimal
extension of its restriction to $\pi^{-1}(U)^{*}\subseteq L^{*}$.

###### Proof.

We deduce from Lemma 2.2 that
$\textup{DR}^{an}(\mathcal{O}^{\beta}_{L^{*}})[\dim(L^{*})]$ is quasi-
isomorphic to a local system. The statement follows now from [HTT08, Corollary
8.2.6] and the algebraic Riemann-Hilbert correspondence. ∎

As an easy consequence, we obtain the following behaviour of the modules
$\mathcal{O}_{L^{*}}^{\beta}$ with respect to the holonomic duality functor.

######  Lemma 2.7.

We have
${\mathbb{D}}\mathcal{O}_{L^{*}}^{\beta}\cong\mathcal{O}_{L^{*}}^{-\beta}$ for
all $\beta\in{\mathds{C}}$ such that $k\beta\in\mathds{Z}$.

###### Proof.

We choose an open dense subset $U\subseteq X$ over which the line bundle $F$
trivializes and pick a non-vanishing section $s\in\Gamma(U,\mathscr{F})$. By
2.1 and 2.6, we have

$\mathcal{O}_{L^{*}}^{\beta}\cong
j_{{\dagger},+}\widetilde{\psi}_{s^{k},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U}),$

where $j$ is the open embedding of $\pi^{-1}(U)^{*}$ into $L^{*}$. Then

$\displaystyle{\mathbb{D}}\mathcal{O}_{L^{*}}^{\beta}$
$\displaystyle\cong{\mathbb{D}}j_{{\dagger},+}\widetilde{\psi}_{s^{k},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U})$
$\displaystyle\cong
j_{{\dagger},+}{\mathbb{D}}\widetilde{\psi}_{s^{k},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U})$
$\displaystyle\cong
j_{{\dagger},+}\widetilde{\psi}_{s^{k},+}({\mathbb{D}}(\mathcal{O}_{\mathds{C}^{*}}^{\beta})\boxtimes\mathcal{O}_{U})$
$\displaystyle\cong
j_{{\dagger},+}\widetilde{\psi}_{s^{k},+}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U})$
$\displaystyle\cong\mathcal{O}_{L^{*}}^{-\beta}.\qed$

######  Corollary 2.8.

Let $\beta\in\mathds{R}$ be such that $k\beta\in\mathds{Z}$. Then the
$\mathscr{D}_{L^{*}}$-module $\mathcal{O}_{L^{*}}^{\beta}$ underlies a
(complex) smooth pure Hodge module on $L^{*}$ that we denote by
${}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{L^{*}}$.

###### Proof.

We choose an open dense subset $U\subseteq X$ over which the line bundle $F$
trivializes and pick a non-vanishing section $s\in\Gamma(U,\mathscr{F})$. By
2.1 and 2.6, we have

$\mathcal{O}_{L^{*}}^{\beta}\cong
j_{{\dagger},+}\widetilde{\psi}_{s^{k},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U}),$

where $j$ is the open embedding of $\pi^{-1}(U)^{*}$ into $L^{*}$.

Using that the functors involved in this formula lift to the category of mixed
Hodge modules, we obtain

${}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{L^{*}}=j_{!*}\widetilde{\psi}_{s^{k},*}(^{H\\!}\underline{{\mathds{C}}}_{\mathds{C}^{*}}^{\beta}\boxtimes\,^{H\\!}\underline{{\mathds{C}}}_{U_{\ell}}).$

Moreover, $j_{!*}$ conserves weights, and so does $\widetilde{\psi}_{s^{k},*}$
since $\widetilde{\psi}_{s^{k}}$ is an isomorphism, therefore, we obtain that
${}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{L^{*}}\in\textup{HM}(L^{*},{\mathds{C}})$,
as claimed. ∎

Denote by $j_{L}\colon L^{*}\hookrightarrow L$ the canonical open embedding.
Then the previous results yield the following.

######  Corollary 2.9.

The $\mathscr{D}_{L}$-modules $j_{L,+}\mathcal{O}^{\beta}_{L^{*}}$ resp.
$j_{L,{\dagger}}\mathcal{O}^{\beta}_{L^{*}}$ underly the mixed Hodge modules
$j_{L,*}{{}^{H\\!}}\underline{{\mathds{C}}}^{\beta}_{L^{*}}$ resp.
$j_{L,!}{{}^{H\\!}}\underline{{\mathds{C}}}^{\beta}_{L^{*}}$ on $L$. If
$\beta\notin{\mathds{Z}}$, then

$j_{L,*}{{}^{H\\!}}\underline{{\mathds{C}}}^{\beta}_{L^{*}}\cong
j_{L,!}{{}^{H\\!}}\underline{{\mathds{C}}}^{\beta}_{L^{*}}\cong
j_{L,!*}{{}^{H\\!}}\underline{{\mathds{C}}}^{\beta}_{L^{*}},$

which is pure of weight $\dim(X)+1$.

###### Proof.

The first statement is obvious from the previous 2.8. The second one follows
from the well-known fact that
$j_{{\mathds{C}},+}\mathcal{O}_{{\mathds{C}}^{*}}^{\beta}\cong
j_{{\mathds{C}},{\dagger}}\mathcal{O}_{{\mathds{C}}^{*}}^{\beta}\cong
j_{{\mathds{C}},{\dagger},+}\mathcal{O}^{\beta}_{{\mathds{C}}^{*}}$ for
$\beta\notin{\mathds{Z}}$, using that for any $i\in I$, we have
$j_{i}^{+}\mathcal{O}_{L^{*}}^{\beta}\cong\widetilde{\psi}_{i,+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{i}})$.
∎

##  3 Fourier–Laplace transformation

The purpose of this section is twofold: First we recall a few basic properties
of general Fourier-Laplace transformations on (not necessarily trivial) vector
bundles. We then apply these constructions to study a (complex of)
$\mathscr{D}$-module(s) that generically computes cohomology groups of
hyperplane sections of projective varieties. These results are used later in
Section 8 for the special case of homogeneous spaces and their corresponding
tautological systems.

###  3.1 Fourier–Laplace transformation on vector bundles

######  Definition 3.1.

Given a vector bundle $E\to X$ on a smooth variety $X$, we consider the
canonical projections $p_{1}\colon E\times_{X}E^{\vee}\to E$ and $p_{2}\colon
E\times_{X}E^{\vee}\to E^{\vee}$. Let $\alpha\colon
E\times_{X}E^{\vee}\to\mathds{C}\times X\to\mathds{C}$ be the natural pairing
and denote
$\mathscr{K}:=\alpha^{+}(\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}+1))$.
The Fourier–Laplace transformation is the functor
$\operatorname{\textup{FL}}_{X}^{E}\colon D_{qc}^{b}(\mathscr{D}_{E})\to
D_{qc}^{b}(\mathscr{D}_{E^{\vee}})$ given as

$\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M}):=p_{2,+}(p_{1}^{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}\mathscr{K}).$

The following three basic properties follow directly from the projection
formula and base change.

######  Lemma 3.2.

Let $\pi\colon E\to X$ be a vector bundle on a smooth variety and let
$\pi^{\vee}\colon E^{\vee}\to X$ be its dual bundle. For $\mathscr{M}\in
D_{qc}^{b}(\mathscr{D}_{E})$ and $\mathscr{N}\in D_{qc}^{b}(\mathscr{D}_{X})$,
we have

$\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E}}\pi^{+}\mathscr{N})\cong\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M})\otimes^{\mathbb{L}}_{\mathcal{O}_{E^{\vee}}}\pi^{\vee,+}\mathscr{N}.$

###### Proof.

We have

$\displaystyle\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M}\otimes_{O_{E}}^{\mathbb{L}}\pi^{+}\mathscr{N})$
$\displaystyle=p_{2,+}(p_{1}^{+}(\mathscr{M}\otimes_{\mathcal{O}_{E}}^{\mathbb{L}}\pi^{+}\mathscr{N})\otimes_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}^{\mathbb{L}}\mathscr{K})$
$\displaystyle\cong
p_{2,+}(p_{1}^{+}\pi^{+}\mathscr{N}\otimes_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}^{\mathbb{L}}p_{1}^{+}\mathscr{M}\otimes_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}^{\mathbb{L}}\mathscr{K})$
$\displaystyle\cong
p_{2,+}(p_{2}^{+}\pi^{\vee,+}\mathscr{N}\otimes_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}^{\mathbb{L}}p_{1}^{+}\mathscr{M}\otimes_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}^{\mathbb{L}}\mathscr{K})$
$\displaystyle\stackrel{{\scriptstyle(\star)}}{{\cong}}\pi^{\vee,+}\mathscr{N}\otimes_{\mathcal{O}_{E^{\vee}}}^{\mathbb{L}}p_{2,+}(p_{1}^{+}\mathscr{M}\otimes_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}^{\mathbb{L}}\mathscr{K})$
$\displaystyle\cong\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M})\otimes_{\mathcal{O}_{E^{\vee}}}^{\mathbb{L}}\pi^{\vee,+}\mathscr{N},$

where $(\star)$ follows from the projection formula in the derived category of
quasi-coherent $\mathscr{D}$-modules. ∎

######  Lemma 3.3.

Let $\varphi\colon E\to F$ be a morphism of vector bundles over $X$ and denote
by $\varphi^{\vee}\colon F^{\vee}\to E^{\vee}$ the induced morphism of dual
vector bundles. Then

$\operatorname{\textup{FL}}_{X}^{F}\circ\varphi_{+}\cong\varphi^{\vee,+}\circ\operatorname{\textup{FL}}_{X}^{E}$

as functors $D_{qc}^{b}(\mathscr{D}_{E})\to
D_{qc}^{b}(\mathscr{D}_{F^{\vee}})$.

###### Proof.

We denote
$\mathscr{K}^{E}:=(\alpha^{E})^{+}(\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}+1))$
and
$\mathscr{K}^{F}:=(\alpha^{F})^{+}(\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}+1))$,
where $\alpha^{E}\colon E\times_{X}E^{\vee}\to\mathds{C}$ and
$\alpha^{F}\colon F\times_{X}F^{\vee}\to\mathds{C}$ are the natural pairings.
Moreover, denote by $q_{1}\colon E\times_{X}F^{\vee}\to E$ and $q_{2}\colon
E\times_{X}F^{\vee}\to F^{\vee}$ the projections onto the first and second
factor. Consider the commutative diagram

${E}$${F}$${E\times_{X}E^{\vee}}$${E\times_{X}F^{\vee}}$${F\times_{X}F^{\vee}}$${E^{\vee}}$${F^{\vee}}$${\phantom{F\times_{X}F^{\vee}},}$$\scriptstyle{\varphi}$$\scriptstyle{p^{E}_{1}}$$\scriptstyle{p^{E}_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{2}}$$\scriptstyle{\operatorname{\textup{id}}_{E}\times\varphi^{\vee}}$$\scriptstyle{\varphi\times\operatorname{\textup{id}}_{F^{\vee}}}$$\scriptstyle{p_{2}^{F}}$$\scriptstyle{p_{1}^{F}}$$\scriptstyle{\varphi^{\vee}}$

whose squares are cartesian. For every $\mathscr{M}\in
D^{b}_{qc}(\mathscr{D}_{E})$, we have:

$\displaystyle\operatorname{\textup{FL}}_{X}^{F}(\varphi_{+}\mathscr{M})$
$\displaystyle=p_{2,+}^{F}(p_{1}^{F,+}\varphi_{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{F\\!\times\\!F^{\vee}}}\mathscr{K}^{F})$
$\displaystyle\cong
p_{2,+}^{F}\big{(}(\varphi\times\operatorname{\textup{id}}_{F^{\vee}})_{+}q_{1}^{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{F\\!\times\\!F^{\vee}}}\mathscr{K}^{F}\big{)}$
(base change) $\displaystyle\cong
p_{2,+}^{F}(\varphi\times\operatorname{\textup{id}}_{F^{\vee}})_{+}\big{(}q_{1}^{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times\\!F^{\vee}}}(\varphi\times\operatorname{\textup{id}}_{F^{\vee}})^{+}\mathscr{K}^{F}\big{)}$
(projection formula) $\displaystyle\cong
q_{2,+}\big{(}q_{1}^{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times\\!F^{\vee}}}(\varphi\times\operatorname{\textup{id}}_{F^{\vee}})^{+}\mathscr{K}^{F}\big{)}$
($q_{2}=p_{2}^{F}\circ(\varphi\times\operatorname{\textup{id}}_{F^{\vee}})$)
$\displaystyle\cong
q_{2,+}\big{(}q_{1}^{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times\\!F^{\vee}}}(\operatorname{\textup{id}}_{E}\times\varphi^{\vee})^{+}\mathscr{K}^{E}\big{)}$
($\alpha^{E}\circ(\operatorname{\textup{id}}_{E}\times\varphi^{\vee})=\alpha^{F}\circ(\varphi\times\operatorname{\textup{id}}_{F^{\vee}})$)
$\displaystyle\cong
q_{2,+}\big{(}(\operatorname{\textup{id}}_{E}\times\varphi^{\vee})^{+}p_{1}^{E,+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times\\!F^{\vee}}}(\operatorname{\textup{id}}_{E}\times\varphi^{\vee})^{+}\mathscr{K}^{E}\big{)}$
($q_{1}=p_{1}^{E}\circ(\operatorname{\textup{id}}_{E}\times\varphi^{\vee})$)
$\displaystyle\cong\varphi^{\vee}_{+}p_{2}^{E,+}(p_{1}^{E,+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times\\!E^{\vee}}}\mathscr{K}^{E})$
(base change)
$\displaystyle=\varphi^{\vee}_{+}\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M}).$

######  Lemma 3.4.

Consider a cartesian square

${E}$${F}$${X}$${Y,}$$\scriptstyle{g}$${\times}$

where the vertical arrows are vector bundles over smooth varieties. Denote by
$g^{\vee}\colon E^{\vee}\to F^{\vee}$ the corresponding morphism of dual
vector bundles. Then

$\operatorname{\textup{FL}}_{Y}^{F}\circ g_{+}\cong
g^{\vee}_{+}\circ\operatorname{\textup{FL}}_{X}^{E}\qquad\text{and}\qquad\operatorname{\textup{FL}}_{X}^{E}\circ
g^{+}\cong g^{\vee,+}\circ\operatorname{\textup{FL}}_{Y}^{F}$

as functors $D_{qc}^{b}(\mathscr{D}_{E})\to
D_{qc}^{b}(\mathscr{D}_{F^{\vee}})$ and $D_{qc}^{b}(\mathscr{D}_{F})\to
D_{qc}^{b}(\mathscr{D}_{E^{\vee}})$, respectively.

###### Proof.

We use notations as in the proof of 3.3. Note that we have the following
commutative diagram with cartesian squares:

${E}$${E\times_{X}E^{\vee}}$${E^{\vee}}$${F}$${F\times_{Y}F^{\vee}}$${F^{\vee}.}$$\scriptstyle{g}$$\scriptstyle{p_{1}^{E}}$$\scriptstyle{p_{2}^{E}}$$\scriptstyle{g\times
g^{\vee}}$$\scriptstyle{g^{\vee}}$$\scriptstyle{p_{1}^{F}}$$\scriptstyle{p_{2}^{F}}$

For every $\mathscr{M}\in D_{qc}^{b}(\mathscr{D}_{E})$, we have:

$\displaystyle\operatorname{\textup{FL}}_{Y}^{F}(g_{+}\mathscr{M})$
$\displaystyle=p_{2,+}^{F}(p_{1}^{F,+}g_{+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{F\\!\times_{\\!Y}\\!F^{\vee}}}\mathscr{K}^{F})$
$\displaystyle\cong p_{2,+}^{F}\big{(}(g\times
g^{\vee})_{+}p_{1}^{E,+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{F\\!\times_{\\!Y}\\!F^{\vee}}}\mathscr{K}^{F}\big{)}$
(base change) $\displaystyle\cong p_{2,+}^{F}(g\times
g^{\vee})_{+}\big{(}p_{1}^{E,+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times_{\\!X}\\!E^{\vee}}}(g\times
g^{\vee})^{+}\mathscr{K}^{F}\big{)}$ (projection formula) $\displaystyle\cong
g^{\vee}_{+}p_{2,+}^{E}\big{(}p_{1}^{E,+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times_{\\!X}\\!E^{\vee}}}(g\times
g^{\vee})^{+}\mathscr{K}^{F}\big{)}$ ($p_{2}^{F}\circ(g\times
g^{\vee})=g^{\vee}\circ p_{2}^{E}$) $\displaystyle\cong
g^{\vee}_{+}p_{2,+}^{E}(p_{1}^{E,+}\mathscr{M}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times_{\\!X}\\!E^{\vee}}}\mathscr{K}^{E})$
($\alpha^{E}=\alpha^{F}\circ(g\times g^{\vee})$)
$\displaystyle=g_{+}^{\vee}\operatorname{\textup{FL}}_{X}^{E}(\mathscr{M}).$

Similarly, for $\mathscr{N}\in D_{qc}^{b}(\mathscr{D}_{F})$, we get:

$\displaystyle\operatorname{\textup{FL}}_{X}^{E}(g^{+}\mathscr{N})$
$\displaystyle=p_{2,+}^{E}(p_{1}^{E,+}g^{+}\mathscr{N}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times_{\\!X}\\!E^{\vee}}}\mathscr{K}^{E})$
$\displaystyle\cong p_{2,+}^{E}\big{(}(g\times
g^{\vee})^{+}p_{1}^{F,+}\mathscr{N}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times_{\\!X}\\!E^{\vee}}}\mathscr{K}^{E}\big{)}$
($g\circ p_{1}^{E}=p_{1}^{F}\circ(g\times g^{\vee})$) $\displaystyle\cong
p_{2,+}^{E}\big{(}(g\times
g^{\vee})^{+}p_{1}^{F,+}\mathscr{N}\otimes^{\mathbb{L}}_{\mathcal{O}_{E\\!\times_{\\!X}\\!E^{\vee}}}(g\times
g^{\vee})^{+}\mathscr{K}^{F}\big{)}$ ($\alpha^{E}=\alpha^{F}\circ(g\times
g^{\vee})$) $\displaystyle\cong
g^{\vee,+}p_{2,+}^{F}\big{(}p_{1}^{F,+}\mathscr{N}\otimes^{\mathbb{L}}_{\mathcal{O}_{F\\!\times_{\\!Y}\\!F^{\vee}}}\mathscr{K}^{F}\big{)}$
(base change)
$\displaystyle=g^{\vee,+}\operatorname{\textup{FL}}_{Y}^{F}(\mathscr{N}).$

In the following, we wish to relate Fourier–Laplace transforms on vector
bundles with classical Fourier–Laplace transforms on a finite-dimensional
vector space (which is the special case of a vector bundle over a point). For
this, we consider the following situation: Let $\pi\colon E\to X$ be a vector
bundle on a smooth variety and denote by $\mathcal{E}$ its sheaf of sections,
i.e.,
$E=\operatorname{Tot}(\mathcal{E}):=\operatorname{\textup{Spec}\,}_{\mathcal{O}_{X}}\operatorname{\textup{Sym}}^{\bullet}\mathcal{E}^{\vee}$.
Let $W\subseteq\Gamma(X,\mathcal{E})$ be a non-zero finite-dimensional vector
space of global sections of $E$ and let $V:=W^{\vee}$ be its dual vector
space. There are natural bundle morphisms $ev\colon W\times X\to E$ and
$ev^{\vee}\colon E^{\vee}\to V\times X$, where $E^{\vee}$ denotes the dual
vector bundle to $E$.

######  Proposition 3.5.

Let $W$ be a finite-dimensional space of global sections of a vector bundle
$E\to X$ on a smooth variety. Let $V$ denote its dual vector space. If
$a_{V}\colon V\times X\to V$ and $a_{W}\colon W\times X\to W$ denote the
projections onto the first factors, we have

$\operatorname{\textup{FL}}^{V}(a_{V,+}ev^{\vee}_{+}\mathscr{M})\cong
a_{W,+}ev^{+}\operatorname{\textup{FL}}_{X}^{E^{\vee}}(\mathscr{M})$

for all $\mathscr{M}\in D_{qc}^{b}(\mathscr{D}_{E^{\vee}})$.

###### Proof.

The claim follows from 3.3 and 3.4 considering the diagram

${E^{\vee}}$${V\times
X}$${V}$${X}$${\operatorname{\textup{Spec}\,}\mathds{C}.}$$\scriptstyle{ev^{\vee}}$$\scriptstyle{a_{V}}$${\times}$

∎

###  3.2 Fourier–Laplace transform of extensions of
$\mathcal{O}_{L^{*}}^{\beta}$

We now determine the Fourier–Laplace transform of the
$\mathscr{D}_{L^{*}}$-modules $\mathcal{O}_{L^{*}}^{\beta}$ defined in Section
2, where $L^{*}$ is the complement of the zero section of a line bundle
$\pi\colon L\to X$. Recall that the definition of
$\mathcal{O}_{L^{*}}^{\beta}$ depends on the choice of a line bundle $F$ with
$L=F^{\otimes k}$ for some $k\in\mathds{Z}$ satisfying $k\beta\in\mathds{Z}$.

Note that the dual line bundle $\pi^{\vee}\colon L^{\vee}\to X$ is the
$(-k)$-th tensor power $F^{\otimes(-k)}$. In what follows, we will consider
the $\mathscr{D}_{L^{\vee,*}}$-module $\mathcal{O}_{L^{\vee,*}}^{-\beta}$
whose definition we always base on the choice of $F$ as a $(-k)$-th root of
$L^{\vee}$ (or, equivalently, based on $F^{\vee}$ as a $k$-th root of
$L^{\vee}$).

We denote by $j_{L}\colon L^{*}\hookrightarrow L$ and $j_{L^{\vee}}\colon
L^{\vee,*}\hookrightarrow L^{\vee}$ the open embeddings from the complements
of the zero section into $L$ and $L^{\vee}$, respectively.

######  Proposition 3.6.

Let $\beta\in\mathds{C}$ with $k\beta\in\mathds{Z}$. Then

$\operatorname{\textup{FL}}_{X}^{L}(j_{L,+}\mathcal{O}_{L^{*}}^{\beta})\cong
j_{L^{\vee},{\dagger}}\mathcal{O}_{L^{\vee,*}}^{-\beta}.$

###### Proof.

Denote
$\mathscr{M}:=\operatorname{\textup{FL}}_{X}^{L}(j_{L,+}\mathcal{O}_{L^{*}}^{\beta})$
and $\mathscr{N}:=j_{L^{\vee},{\dagger}}\mathcal{O}_{L^{\vee,*}}^{-\beta}$. We
construct the isomorphism between $\mathscr{M}$ and $\mathscr{N}$ locally over
the trivializing open cover. To this end, we need to produce isomorphisms
$\xi_{i}\colon{\mathscr{M}_{|{\pi^{\vee,-1}(U_{i})}}}\to{\mathscr{N}_{|{\pi^{\vee,-1}(U_{i})}}}$
for all $i\in I$ such that the restrictions of $\xi_{i}$ and $\xi_{j}$ to
$\pi^{\vee,-1}(U_{ij})$ agree for all $i,j\in I$.

By construction, the restriction of $\mathcal{O}_{L^{*}}^{\beta}$ to
$\pi^{-1}(U_{i})^{*}$ is isomorphic to
$\widetilde{\psi}_{i,+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{i}})$,
where by $\widetilde{\psi}_{i}=\widetilde{\psi}_{s_{i}^{k}}$ we denote the
restriction to the complement of the zero section of the trivializing
isomorphism $\psi_{i}=\psi_{s_{i}^{k}}\colon\mathds{C}\times
U_{i}\xrightarrow{\cong}\pi^{-1}(U_{i})$ corresponding to the nonvanishing
section $s_{i}^{k}\in\Gamma(U_{i},\mathscr{L})$. Analogously, denote by

$\widetilde{\psi}^{\vee}_{i}:=\widetilde{\psi}_{s_{i}^{-k}}\colon\mathds{C}^{*}\times
U_{i}\xrightarrow{\cong}\pi^{\vee,-1}(U_{i})^{*}\qquad\text{and}\qquad\psi^{\vee}_{i}:=\psi_{s_{i}^{-k}}\colon\mathds{C}\times
U_{i}\xrightarrow{\cong}\pi^{\vee,-1}(U_{i})$

the corresponding isomorphisms for the dual line bundle $\pi^{\vee}\colon
L^{\vee}\to X$. Then, by construction, the restricton of
$\mathcal{O}_{L^{\vee,*}}^{-\beta}$ to $\pi^{\vee,-1}(U_{i})^{*}$ is
isomorphic to
$\widetilde{\psi}_{i,+}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U_{i}})$.

We denote by $j_{\mathds{C}}$ and $j_{\mathds{C}\times U_{i}}$ the open
embeddings $\mathds{C}^{*}$ into $\mathds{C}$ and of $\mathds{C}^{*}\times
U_{i}$ into $\mathds{C}\times U_{i}$, respectively. Then

$\displaystyle{\mathscr{M}_{|{\pi^{\vee,-1}(U_{i})}}}$
$\displaystyle\cong\operatorname{\textup{FL}}_{U_{i}}^{\pi^{-1}(U_{i})}\big{(}\psi_{i,+}j_{\mathds{C}\times
U_{i},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{i}})\big{)}$
$\displaystyle\cong\psi^{\vee}_{i,+}\operatorname{\textup{FL}}_{U_{i}}^{\mathds{C}\times
U_{i}}\big{(}j_{\mathds{C}\times
U_{i},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{i}})\big{)}$
$\displaystyle\cong\psi^{\vee}_{i,+}\big{(}\\!\operatorname{\textup{FL}}^{\mathds{C}}(j_{\mathds{C},+}\mathcal{O}_{\mathds{C}^{*}}^{\beta})\boxtimes\mathcal{O}_{U_{i}}\big{)},$

where the first isomorphism follows from 3.4, the second one from 3.3 and the
final one from combining 3.2 and 3.4. On the other hand, it follows from base
change that

$\displaystyle{\mathscr{N}_{|{\pi^{\vee,-1}(U_{i})}}}\cong\psi^{\vee}_{i,+}j_{\mathds{C}\times
U_{i},{\dagger}}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U_{i}})\cong\psi^{\vee}_{i,+}(j_{\mathds{C},{\dagger}}\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U_{i}}).$

There is an isomorphism
$\eta\colon\operatorname{\textup{FL}}^{\mathds{C}}(j_{\mathds{C},+}\mathcal{O}_{\mathds{C}^{*}}^{\beta})\xrightarrow{\cong}j_{\mathds{C},{\dagger}}\mathcal{O}_{\mathds{C}^{*}}^{-\beta}$.
It gives rise to

$\xi_{i}:=\psi^{\vee}_{i,+}(\eta\boxtimes\operatorname{\textup{id}}_{U_{i}})\colon{\mathscr{M}_{|{\pi^{\vee,-1}(U_{i})}}}\xrightarrow{\cong}{\mathscr{N}_{|{\pi^{\vee,-1}(U_{i})}}}.$

We need to check that $\xi_{i}$ and $\xi_{j}$ agree when restricted to
$\pi^{\vee,-1}(U_{ij})$. In terms of the transition maps

$\psi_{ij}=\psi_{s_{i}^{k}/s_{j}^{k}}\colon\mathds{C}\times
U_{ij}\xrightarrow{\cong}\mathds{C}\times
U_{ij}\quad\text{and}\quad\psi_{ij}^{\vee}=\psi_{s_{i}^{-k}/s_{j}^{-k}}\colon\mathds{C}\times
U_{ij}\xrightarrow{\cong}\mathds{C}\times U_{ij}$

of the line bundles $L$ and $L^{\vee}$ as in (3), or their restrictions
$\widetilde{\psi}_{ij}$ and $\widetilde{\psi}_{ij}^{\vee}$ to
$\mathds{C}^{*}\times U_{ij}$, this is equivalent to showing that the
following diagram commutes:

${\operatorname{\textup{FL}}_{U_{ij}}^{\mathds{C}\times
U_{ij}}\big{(}j_{\mathds{C}\times
U_{ij},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{ij}})\big{)}}$${j_{\mathds{C}\times
U_{ij},{\dagger}}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U_{ij}})}$${\operatorname{\textup{FL}}_{U_{ij}}^{\mathds{C}\times
U_{ij}}\big{(}j_{\mathds{C}\times
U_{ij},+}\widetilde{\psi}_{ij}^{+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{ij}})\big{)}}$${j_{\mathds{C}\times
U_{ij},{\dagger}}\widetilde{\psi}_{ij}^{\vee,+}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U_{ij}})}$${\psi_{ij}^{\vee,+}\Big{(}\\!\operatorname{\textup{FL}}_{U_{ij}}^{\mathds{C}\times
U_{ij}}\big{(}j_{\mathds{C}\times
U_{ij},+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\boxtimes\mathcal{O}_{U_{ij}})\big{)}\Big{)}}$${\psi_{ij}^{\vee,+}j_{\mathds{C}\times
U_{ij},{\dagger}}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\boxtimes\mathcal{O}_{U_{ij}}).}$$\scriptstyle{\eta\boxtimes\operatorname{\textup{id}}_{U_{ij}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\textstyle\cong}$$\scriptstyle{\textstyle\cong}$$\scriptstyle{\cong}$$\scriptstyle{\psi_{ij}^{\vee,+}(\eta\boxtimes\operatorname{\textup{id}}_{U_{ij}})}$

The vertical arrows are induced from the isomorphisms

$\widetilde{\psi}_{ij}^{+}(\mu_{ij})\colon\mathcal{O}_{\mathds{C}^{*}}^{\beta}\\!\boxtimes\\!\mathcal{O}_{U_{ij}}\xrightarrow{\cong}\widetilde{\psi}_{ij}^{+}(\mathcal{O}_{\mathds{C}^{*}}^{\beta}\\!\boxtimes\\!\mathcal{O}_{U_{ij}})\quad\text{and}\quad\widetilde{\psi}_{ij}^{\vee,+}(\mu_{ij}^{\vee})\colon\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\\!\boxtimes\\!\mathcal{O}_{U_{ij}}\xrightarrow{\cong}\widetilde{\psi}_{ij}^{\vee,+}(\mathcal{O}_{\mathds{C}^{*}}^{-\beta}\\!\boxtimes\\!\mathcal{O}_{U_{ij}}),$

from (4) in the construction of $\mathcal{O}_{L^{k,*}}^{\beta}$ and
$\mathcal{O}_{L^{-k,*}}^{-\beta}$, respectively.

First, we treat the case $\beta\notin\mathds{Z}$. Fixing a local coordinate
system $(t,x_{1},\dots,x_{n})$ on $V:=\mathds{C}\times U_{ij}$, we have
homomorphisms between cyclic $\mathscr{D}$-modules, which we may now write out
explicitly. Since $\beta\notin\mathds{Z}$, we have
$j_{\mathds{C},+}\mathcal{O}_{\mathds{C}^{*}}^{\beta}\cong
j_{\mathds{C},{\dagger}}\mathcal{O}_{\mathds{C}^{*}}^{\beta}=\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t+\beta)$.
Notice that

$\operatorname{\textup{FL}}^{\mathds{C}}(\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t+\beta))=\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-\beta-1)\xrightarrow{\cong}\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-\beta),$

where the last map is given by right-multiplication with $t$, which is an
isomorphism for $\beta\neq 0$. Under these identifications, the isomorphism
$\eta$ from above is simply given by the composition

$\eta\colon\operatorname{\textup{FL}}^{\mathds{C}}(j_{\mathds{C},+}\mathcal{O}_{\mathds{C}^{*}}^{\beta})\cong\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-\beta)\xrightarrow{\operatorname{\textup{id}}}j_{\mathds{C},{\dagger}}\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-\beta)\cong{}\mathcal{O}_{\mathds{C}^{*}}^{-\beta}.$

Using this as well as 2.2 and recalling the notation
$\alpha_{ij}=s_{i}/s_{j}\in\Gamma(U_{ij},\mathcal{O}_{U_{ij}}^{\times})$, the
diagram from above becomes:

${\mathscr{D}_{V}/\mathscr{D}_{V}\big{(}\partial_{t}t-\beta,\partial_{x_{\ell}}\mid\ell\big{)}}$${\mathscr{D}_{V}/\mathscr{D}_{V}\big{(}\partial_{t}t-\beta,\partial_{x_{\ell}}\mid\ell\big{)}}$${\mathscr{D}_{V}/\mathscr{D}_{V}\big{(}\partial_{t}t-\beta,\partial_{x_{\ell}}+k(\beta+1)\alpha_{ij}^{-1}\frac{\partial\alpha_{ij}}{\partial
x_{\ell}}\mid\ell\big{)}}$${\mathscr{D}_{V}/\mathscr{D}_{V}\big{(}\partial_{t}t-\beta,\partial_{x_{\ell}}+k(\beta-1)\alpha_{ij}^{-1}\frac{\partial\alpha_{ij}}{\partial
x_{\ell}}\mid\ell\big{)}}$${\mathscr{D}_{V}/\mathscr{D}_{V}\big{(}\partial_{t}t-\beta,\partial_{x_{\ell}}+k\beta\alpha_{ij}^{-1}\frac{\partial\alpha_{ij}}{\partial
x_{\ell}}\mid\ell\big{)}}$${\mathscr{D}_{V}/\mathscr{D}_{V}\big{(}\partial_{t}t-\beta,\partial_{x_{\ell}}+k(\beta-1)\alpha_{ij}^{-1}\frac{\partial\alpha_{ij}}{\partial
x_{\ell}}\mid\ell\big{)},}$$\scriptstyle{\cdot\alpha_{ij}^{k(\beta+1)}}$$\scriptstyle{\operatorname{\textup{id}}}$$\scriptstyle{\cdot\alpha_{ij}^{(-k)\cdot(-\beta+1)}}$$\scriptstyle{\cdot\alpha_{ij}^{-k}}$$\scriptstyle{\operatorname{\textup{id}}}$$\scriptstyle{\alpha_{ij}^{-k}}$

which commutes. On the other hand, if $\beta$ is an integer, then we have
$j_{\mathds{C},+}\mathcal{O}_{\mathds{C}^{*}}^{\beta}\cong\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t)$
and
$j_{\mathds{C},{\dagger}}\mathcal{O}_{\mathds{C}^{*}}^{\beta}\cong\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-1)$.
Then clearly

$\operatorname{\textup{FL}}^{\mathds{C}}(j_{\mathds{C},+}\mathcal{O}_{\mathds{C}^{*}}^{\beta})\cong\operatorname{\textup{FL}}^{\mathds{C}}(\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t))\cong\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-1),$

and under these identifications, the isomorphism $\eta$ is again given by the
identity map on
$\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}(\partial_{t}t-1)$. Hence,
we obtain the commutativity of the corresponding diagram by simply putting
$\beta$ to one in the above diagram. ∎

######  Corollary 3.7.

Let $k\in\mathds{Z}$ and let $\beta\in\mathds{R}$ with $k\beta\in\mathds{Z}$.
Then the Fourier–Laplace transform on $L$ of the $\mathscr{D}_{L}$-module
$j_{L,+}\mathcal{O}_{L^{*}}^{\beta}$ can be equipped with the structure of a
complex mixed Hodge module which is pure of weight $\dim(X)+1$ if
$\beta\notin{\mathds{Z}}$.

###### Proof.

We have just seen in the previous 3.6 that

$\operatorname{\textup{FL}}_{X}^{L}(j_{L,+}\mathcal{O}_{L^{*}}^{\beta})\cong
j_{L^{\vee},{\dagger}}\mathcal{O}_{L^{\vee,*}}^{-\beta}.$

On the other hand, we know by 2.9 that
$j_{L^{\vee},{\dagger}}\mathcal{O}_{L^{\vee,*}}^{-\beta}$ underlies the the
object

$j_{L^{\vee},!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{\vee,*}}\in\textup{MHM}(L^{\vee},{\mathds{C}}),$

and that it is pure if $\beta\notin{\mathds{Z}}$. ∎

###  3.3 Twisted cohomology of hyperplane sections

In this subsection, we describe a complex of $\mathscr{D}$-modules that
generically computes certain twisted cohomologies of hyperplane sections of
our variety $X$ (resp. the complement of those). We show that it underlies an
object in the derived category of mixed Hodge modules. In the more specific
situation studied later in Section 8, when $X$ arises a homogeneous space,
these $\mathscr{D}$-modules will appear as tautological systems.

With the notations from before, we fix a non-zero finite-dimensional subspace
$W$ of $\Gamma(X,\mathscr{L})$. Let $V:=W^{\vee}$ denote its dual vector
space. The linear system $W$ on $X$ defines a rational map $g\colon
X\dashrightarrow\mathds{P}V$. The natural evaluation morphism

$ev\colon W\times X\to L,$ (7)

is a morphism of vector bundles over $X$ and it induces a dual bundle morphism

$ev^{\vee}\colon L^{\vee}\to V\times X.$

The following diagram commutes:

${L^{\vee}}$${V\times X}$${X}$${\mathds{P}V\times
X}$$\scriptstyle{ev^{\vee}}$$\scriptstyle{\pi^{\vee}}$$\scriptstyle{g\times\operatorname{\textup{id}}_{X}}$

If the linear system $W$ is base-point-free, then $g\colon X\to\mathds{P}V$ is
a morphism and $ev^{\vee}$ restricts to a morphism

$\widetilde{ev}^{\vee}\colon L^{\vee,*}\to(V\setminus\\{0\\})\times X$

of complements of zero sections. In this case, we have the following
commutative diagram:

${L^{\vee}}$${V\times X}$${V}$${L^{\vee,*}}$${(V\setminus\\{0\\})\times
X}$${V\setminus\\{0\\}}$${X}$${\mathds{P}V\times
X}$${\mathds{P}V}$$\scriptstyle{ev^{\vee}}$$\scriptstyle{a_{V}}$${\times}$$\scriptstyle{\widetilde{ev}^{\vee}}$$\scriptstyle{j_{L^{\vee}}}$$\scriptstyle{j\times\operatorname{\textup{id}}_{X}}$${\times}$$\scriptstyle{j}$$\scriptstyle{g\times\operatorname{\textup{id}}_{X}}$

If, moreover, the linear system $W$ separates points and tangent directions
(in particular, $\mathscr{L}$ is very ample in this case), then $g\colon
X\to\mathds{P}V$ is a locally closed embedding. In this case, $L^{\vee,*}$ is
isomorphic to $\hat{X}\setminus\\{0\\}$, where $\hat{X}\subseteq V$ is the
affine cone over $g(X)\subseteq\mathds{P}V$, and $L^{\vee}$ is the blow-up of
$\hat{X}$ in the origin: $L^{\vee}\cong\operatorname{Bl}_{\\{0\\}}\hat{X}$.

We denote further by $\mathcal{Y}:=ev^{-1}(0)$ the inverse image of the zero
section of $L$, by $\mathcal{U}:=(W\times X)\backslash\mathcal{Y}$ its
complement, and we write $a_{\mathcal{Y}}:\mathcal{Y}\rightarrow W$ resp.
$a_{\mathcal{U}}:\mathcal{U}\rightarrow W$ for the restrictions of the
projection $a_{W}\colon W\times X\to W$ to $\mathcal{Y}$ resp. to
$\mathcal{U}$.

######  Proposition 3.8.

Assume $L$ to be very ample and let $W\subseteq H^{0}(X,\mathscr{L})$ be a
finite-dimensional linear system defining a locally closed embedding $g\colon
X\hookrightarrow\mathds{P}V$, where $V:=W^{\vee}$. Let $\hat{\iota}\colon
L^{\vee,*}\cong\hat{X}\setminus\\{0\\}\hookrightarrow V$ denote the locally
closed embedding of the punctured affine cone over $X$ into $V$. Then we have
the following.

  1. 1.

For all $\beta\in\mathds{C}$ with $k\beta\in\mathds{Z}$, the complexes of
$\mathscr{D}_{W}$-modules

$\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{\beta})\quad\quad\textup{and}\quad\quad\operatorname{\textup{FL}}^{V}(\hat{\iota}_{\dagger}\mathcal{O}_{L^{\vee,*}}^{-\beta})$

underly elements of $D^{b}\textup{MHM}(W,{\mathds{C}})$ that we denote by
${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}$ and by
${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$, respectively. We have

${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}\cong{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta+\ell}\quad\quad\textup{and}\quad\quad{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}\cong{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta+\ell}$

for any $\ell\in{\mathds{Z}}$.

  2. 2.

For $\beta\in{\mathds{Z}}$, the complexes
$\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{\beta})$
and
$\operatorname{\textup{FL}}^{V}(\hat{\iota}_{\dagger}\mathcal{O}_{L^{\vee,*}}^{-\beta})$
underly elements in $\textup{MHM}(W)$ that we denote unambiguously by
${{}^{H,*}\\!\\!}\mathcal{M}_{L}$ resp. by ${{}^{H,!}\\!\\!}\mathcal{M}_{L}$.

  3. 3.

For $\beta\notin{\mathds{Z}}$, we have an isomorphism
${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}\cong{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{\beta}$.
If $X$ is projective, then the cohomology modules
$H^{i}({{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta})$ are pure Hodge modules of
weight $\dim(X)+\dim(W)+i$.

  4. 4.

Let $\beta\in{\mathds{Z}}$ and assume again that $X$ is projective. Then for
any $k\in{\mathds{Z}}$, there exists morphisms in the abelian category of
mixed Hodge modules

$H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})\longrightarrow
H^{k}\mathopen{}\mathclose{{}\left({{}^{H,*}\\!\\!}\mathcal{M}_{L}}\right)\quad\text{resp.}\quad
H^{k}\mathopen{}\mathclose{{}\left({{}^{H,!}\\!\\!}\mathcal{M}_{L}}\right)\longrightarrow
H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})(-1)$

with constant kernel of weight $k+\dim X+\dim W-1$ resp. $k+\dim X+\dim W$ and
constant cokernel of weight $k+\dim X+\dim W$ resp. $k+\dim X+\dim W+1$. In
particular there are the following weight estimates for
${{}^{H,*}\\!\\!}\mathcal{M}_{L}$ and ${{}^{H,!}\\!\\!}\mathcal{M}_{L}$:

$\displaystyle\operatorname{\textup{Gr}}^{W}_{\ell}(H^{k}({{}^{H,*}\\!\\!}\mathcal{M}_{L}))=0$
$\displaystyle\quad\text{for}\quad\ell\neq k+\dim W+\dim X-1,k+\dim W+\dim X,$
$\displaystyle\operatorname{\textup{Gr}}^{W}_{\ell}(H^{k}({{}^{H,!}\\!\\!}\mathcal{M}_{L}))=0$
$\displaystyle\quad\text{for}\quad\ell\neq k+\dim W+\dim X,k+\dim W+\dim X+1.$

###### Proof.

  1. 1.

We start by showing the statement for
${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}$. For this purpose, we combine 3.5
and 3.6 to get a purely functorial description of this complex of
$\mathscr{D}_{W}$-modules not involving Fourier–Laplace transforms, namely

$\begin{array}[]{rcll}&&\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{\beta})\\\
&\cong&\operatorname{\textup{FL}}^{V}(a_{V,+}ev^{\vee}_{+}j_{{L^{\vee}},+}\mathcal{O}_{L^{\vee,*}}^{\beta})\\\
&\cong&a_{W,+}ev^{+}\operatorname{\textup{FL}}_{X}^{L^{\vee}}(j_{{L^{\vee}},+}\mathcal{O}_{L^{\vee,*}}^{\beta})&{\small\textup{\lx@cref{creftype~refnum}{prop:FLTransfer}}}\\\
&\cong&a_{W,+}ev^{+}j_{L,{\dagger}}\mathcal{O}_{L^{*}}^{-\beta}&{\small\textup{\lx@cref{creftype~refnum}{prop:FLofObeta}}}\\\
&\cong&a_{W,+}ev^{\dagger}j_{L,{\dagger}}\mathcal{O}_{L^{*}}^{-\beta}&{\small
ev\textup{ is smooth}}\\\ &=&a_{W,+}\,ev^{\dagger}[\dim L-\dim(W\times
X)]\,j_{L,{\dagger}}\mathcal{O}_{L^{*}}^{-\beta}[\dim W-1],\end{array}$ (8)

where the last equality is due to the obvious dimension count
$\dim(L)=\dim(X)+1$.

Recall from 2.8 that $\mathcal{O}_{L^{*}}^{-\beta}$ underlies the complex pure
Hodge module ${}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}$, therefore,
we obtain that

${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}:=a_{W,*}ev^{*}j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]\in D^{b}\textup{MHM}(W,{\mathds{C}}).$ (9)

Define

${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}:=\mathopen{}\mathclose{{}\left({\mathbb{D}}\,{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}}\right)(\dim(W\times
X))$

where ${\mathbb{D}}$ is the duality functor in $\textup{MHM}(V,{\mathds{C}})$
as recalled in the introduction. Clearly, the complex of
$\mathscr{D}_{W}$-modules that underlies
${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$ is then
${\mathbb{D}}\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}^{\beta}_{L^{\vee,*}})$,
where this time ${\mathbb{D}}$ is the holonomic duality functor on
$\mathscr{D}_{W}$-modules.

On the spaces $V$ and $L^{\vee,*}$, we consider the isomorphisms $c_{V}$ and
$c_{L^{\vee,*}}$ given by multiplication by $-1$ (in all variables for $c_{V}$
and fibrewise for $c_{L^{\vee,*}}$). Then since the Fourier transformation
$\operatorname{\textup{FL}}^{V}$ and the holonomic duality commute up to the
action of $c_{V}$ (i.e., since
${\mathbb{D}}\,\circ\operatorname{\textup{FL}}^{V}\cong\operatorname{\textup{FL}}^{V}\circ{\mathbb{D}}\circ
c_{V}^{+}$), we obtain the following isomorphisms in $D^{b}(\mathscr{D}_{W})$
for the complex of $\mathscr{D}_{W}$-modules underlying
${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$:

$\displaystyle{\mathbb{D}}\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}^{\beta}_{L^{\vee,*}})$
$\displaystyle\simeq\operatorname{\textup{FL}}^{V}{\mathbb{D}}\,c^{+}_{V}(\hat{\iota}_{+}\mathcal{O}^{\beta}_{L^{\vee,*}})$
$\displaystyle\simeq\operatorname{\textup{FL}}^{V}{\mathbb{D}}(\hat{\iota}_{+}c^{+}_{L^{\vee,*}}\mathcal{O}^{\beta}_{L^{\vee,*}})$
$\displaystyle{\small\textup{since }c_{V}\circ\hat{\iota}=\hat{\iota}\circ
c_{L^{\vee,*}}\textup{ by definition of }\hat{\iota}}$
$\displaystyle\simeq\operatorname{\textup{FL}}^{V}{\mathbb{D}}(\hat{\iota}_{+}\mathcal{O}^{\beta}_{L^{\vee,*}})$
$\displaystyle{\small\exists\textup{ isomorphism
}c^{+}_{L^{\vee,*}}\mathcal{O}^{\beta}_{L^{\vee,*}}\cong\mathcal{O}^{\beta}_{L^{\vee,*}}}$
$\displaystyle\simeq\operatorname{\textup{FL}}^{V}(\hat{\iota}_{\dagger}{\mathbb{D}}\,\mathcal{O}^{\beta}_{L^{\vee,*}})$
$\displaystyle{\small{\mathbb{D}}\,\hat{\iota}_{+}\cong\hat{\iota}_{\dagger}\,{\mathbb{D}}}$
$\displaystyle\simeq\operatorname{\textup{FL}}^{V}(\hat{\iota}_{\dagger}\mathcal{O}^{-\beta}_{L^{\vee,*}})$
$\displaystyle{\small{\mathbb{D}}\,\mathcal{O}^{\beta}_{L^{\vee,*}}\cong\mathcal{O}^{-\beta}_{L^{\vee,*}}\textup{
by \lx@cref{creftype~refnum}{lem:OLbetaDual}}}.$

This shows that the underlying complex of $\mathscr{D}_{W}$-modules of
${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$ is
$\operatorname{\textup{FL}}^{V}(\hat{\iota}_{\dagger}\mathcal{O}^{-\beta}_{L^{\vee,*}})$,
as claimed.

The second statement follows directly from 2.4.

  2. 2.

For $\beta\in{\mathds{Z}}$, we have
$\mathcal{O}_{L^{*}}^{-\beta}=\mathcal{O}_{L^{*}}$, which underlies an element
in $\textup{MHM}(L^{*})$, and by the above argument we get that
${{}^{H,*}\\!\\!}\mathcal{M}_{L},{{}^{H,!}\\!\\!}\mathcal{M}_{L}\in
D^{b}\textup{MHM}(W)$.

  3. 3.

Recall from Eq. 8 above that

$\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{-\beta})\cong
a_{W,+}ev^{\dagger}j_{L,{\dagger}}\mathcal{O}_{L^{*}}^{\beta}.$

Applying the holonomic duality functor yields

${\mathbb{D}}\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{-\beta})\cong
a_{W,+}ev^{\dagger}j_{L,+}{\mathbb{D}}\,\mathcal{O}_{L^{*}}^{\beta},\cong
a_{W,+}ev^{\dagger}j_{L,+}\mathcal{O}_{L^{*}}^{-\beta},$

since $a_{W,{\dagger}}\cong a_{W,+}$ ($a_{W}$ is proper) and since
$ev^{+}\cong ev^{\dagger}$ ($ev$ is smooth). Now if $\beta\notin{\mathds{Z}}$,
by using 2.9, we have $j_{L,+}\mathcal{O}_{L^{*}}^{-\beta}\cong
j_{L,{\dagger}}\mathcal{O}_{L^{*}}^{-\beta}$, and thus we obtain

${\mathbb{D}}\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{-\beta})\cong
a_{W,+}ev^{\dagger}j_{L,{\dagger}}\mathcal{O}_{L^{*}}^{-\beta}\cong\operatorname{\textup{FL}}^{V}(\hat{\iota}_{+}\mathcal{O}_{L^{\vee,*}}^{\beta}),$

from which we deduce an isomorphism

${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{\beta}\cong{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}$

in $D^{b}\textup{MHM}(W,{\mathds{C}})$.

Moreover, under the assumption that $\beta\notin{\mathds{Z}}$, we have seen in
3.7 that $j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}$ is pure
(of weight $\dim(X)+1$). Since the morphism $ev$ is smooth, and since $a_{W}$
is projective here, the second assertion thus follows from [Sai88, Théorème
1].

  4. 4.

Recall that we denoted by $j_{L}\colon L^{*}\rightarrow L$ the inclusion of
the complements of the zero section and denote by $i_{L}\colon X\rightarrow L$
the inclusion of the zero section of $L$. There is the following adjunction
triangle

$j_{L,!}\,j_{L}^{-1}\,{}^{H\\!}\underline{{\mathds{C}}}_{L}\longrightarrow\,^{H\\!}\underline{{\mathds{C}}}_{L}\longrightarrow
i_{L,!}i^{-1}_{L}\,{}^{H\\!}\underline{{\mathds{C}}}_{L}\overset{+1}{\longrightarrow}$

Since
$i^{-1}_{L}\,{}^{H\\!}\underline{{\mathds{C}}}_{L}=^{H\\!}\underline{{\mathds{C}}}_{X}[1]$
we get the triangle

$i_{L,!}\,^{H\\!}\underline{{\mathds{C}}}_{X}\longrightarrow
j_{L,!}\,j_{L}^{-1}\,{}^{H\\!}\underline{{\mathds{C}}}_{L}\longrightarrow\,^{H\\!}\underline{{\mathds{C}}}_{L}\overset{+1}{\longrightarrow}$
(10)

Since the map $j_{L}$ is affine, the functor $j_{L,!}$ from
$\textup{MHM}(L^{*})$ to $\textup{MHM}(L)$ is exact and
$H^{0}(j_{L_{!}}{{}^{H\\!}}\underline{{\mathds{C}}}_{L^{*}})$ is the only non-
zero cohomology. Therefore we get the short exact sequence

$0\longrightarrow i_{L,!}\,^{H\\!}\underline{{\mathds{C}}}_{X}\longrightarrow
H^{0}(j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}_{L^{*}})\longrightarrow\,^{H\\!}\underline{{\mathds{C}}}_{L}\longrightarrow
0$

We have the following diagram with cartesian squares

$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\mathcal{Y}}}$$\textstyle{X\times
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\textstyle{\mathcal{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L}$$\textstyle{L^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Applying the exact functor $ev^{*}[\dim W-1]$ to the short exact sequence (10)
we get the short exact sequence

$0\longrightarrow
i_{\mathcal{Y},!}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}}\longrightarrow
H^{0}(ev^{*}j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}_{L^{*}}[\dim
W-1])\longrightarrow\,^{H\\!}\underline{{\mathds{C}}}_{X\times
W}\longrightarrow 0$ (11)

Notice that $i_{\mathcal{Y},!}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}}$
is pure of weight $\dim X+\dim W-1$ and that
${}^{H\\!}\underline{{\mathds{C}}}_{X\times W}$ is pure of weight $\dim X+\dim
W$. We apply the functor $a_{W,*}$ to (11) and get

$H^{k-1}(a_{W,*}\,^{H\\!}\underline{{\mathds{C}}}_{X\times W})\rightarrow
H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})\rightarrow
H^{k}\mathopen{}\mathclose{{}\left({{}^{H,*}\\!\\!}\mathcal{M}_{L}}\right)\rightarrow
H^{k}(a_{W,*}\,^{H\\!}\underline{{\mathds{C}}}_{X\times W}).$ (12)

Since
$H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})$ is
pure of weight $k+\dim X+\dim W-1$ and the constant mixed Hodge module
$H^{k}(a_{W,*}^{H\\!}\underline{{\mathds{C}}}_{X\times W})$ is pure of weight
$k+\dim X+\dim W$ we conclude that

$\operatorname{\textup{Gr}}^{W}_{\ell}(H^{k}({{}^{H,*}\\!\\!}\mathcal{M}_{L}))=0\quad\text{for}\quad\ell\neq
k+\dim W+\dim X-1,k+\dim W+\dim X.$

and there exists a map
$H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})\rightarrow
H^{k}\mathopen{}\mathclose{{}\left({{}^{H,*}\\!\\!}\mathcal{M}_{L}}\right)$
with constant kernel and cokernel. Applying $\mathbb{D}$ to the sequence (12)
and doing a Tate-twist by $-(\dim X\times W)$ we get for $m=-k$

$H^{m}(a_{W,*}\,^{H\\!}\underline{{\mathds{C}}}_{X\times W})\rightarrow
H^{m}\mathopen{}\mathclose{{}\left({{}^{H,!}\\!\\!}\mathcal{M}_{L}}\right)\rightarrow
H^{m}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})(-1)\rightarrow
H^{m+1}(a_{W,*}\,^{H\\!}\underline{{\mathds{C}}}_{X\times W})$

Since
$H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})$ is
pure of weight $k+\dim X+\dim W+1$ we conclude that

$\operatorname{\textup{Gr}}^{W}_{\ell}(H^{m}({{}^{H,!}\\!\\!}\mathcal{M}_{L}))=0\quad\text{for}\quad\ell\neq
m+\dim W+\dim X,m+\dim W+\dim X+1.$

and there exists a map
$H^{k}\mathopen{}\mathclose{{}\left({{}^{H,!}\\!\\!}\mathcal{M}_{L}}\right)\rightarrow
H^{k}(a_{\mathcal{Y},*}\,^{H\\!}\underline{{\mathds{C}}}_{\mathcal{Y}})(-1)$
with constant kernel and cokernel.

∎

We will discuss next a natural geometric interpretation of the complex of
mixed Hodge modules ${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}$ resp.
${{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$.

For this purpose, fix some value $\lambda\in W$. Then, by definition, we have
$\lambda\in\Gamma(X,\mathscr{L})$, and interpreting this global section as a
morphism $\lambda\colon X\hookrightarrow L$, we can consider the image
$L_{\lambda}:=\operatorname{\textup{im}}(\lambda)\subseteq L$. We identify the
zero section of the projection $\pi\colon L\twoheadrightarrow X$ inside $L$
with $X$ and recall that $L^{*}:=L\setminus X$ denotes the complement of the
zero section. We denote by $H_{\lambda}:=L_{\lambda}\cap X\subseteq X$ the
zero locus of the section $\lambda$ (which was called $Z(\lambda)$ in 1.2) and
by $U_{\lambda}:=X\setminus H_{\lambda}$ its complement in $X$.

Notice that the full family of zero loci of sections of $L$ is given by
$\mathcal{Y}:=ev^{-1}(0)\to W$, $(s,\lambda)\mapsto\lambda$, i.e. the fibre of
this map over a point $\lambda\in W$ is exactly the hypersurface
$H_{\lambda}$. Similarly, we have $\mathcal{U}=(W\times
X)\setminus\mathcal{Y}=\bigcup_{\lambda\in W}U_{\lambda}$, the evaluation
morphism $ev$ from Formula (7) then restricts to a morphism

$ev_{|\mathcal{U}}\colon\mathcal{U}\longrightarrow L^{*}.$

Recall that we defined the constant (complex) pure Hodge module
${}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{L^{*}}$ in 2.8. We then put

${}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\mathcal{U}}:={ev_{|\mathcal{U}}^{*}}\,{{}^{H\\!}}\underline{{\mathds{C}}}^{\beta}_{L^{*}}[\dim
W-1]\in\textup{HM}(\mathcal{U},{\mathds{C}}).$

Moreover, for $\lambda\in W$ as above, consider the restriction
$\lambda_{|U_{\lambda}}\colon U_{\lambda}\hookrightarrow L^{*}$. We put, for
$\beta\in{\mathds{Q}}$,

${}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\lambda}:=\lambda^{*}_{|U_{\lambda}}\,{}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{L^{*}}[-1]\in\textup{HM}(U_{\lambda},{\mathds{C}})$

######  Proposition 3.9.

We continue with the setup of 3.8 and additionally assume that $X$ is
projective. Let $\beta\in\mathds{C}$ with $k\beta\in\mathds{Z}$. Then the
following statements hold true:

  1. 1.

Let $a_{\mathcal{U}}\colon\mathcal{U}\rightarrow W$ be the restriction of the
projection $a_{W}\colon W\times X\rightarrow W$. Then we have an isomorphisms

$a_{\mathcal{U},!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\mathcal{U}}\cong{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}\quad\quad\textup{and}\quad\quad
a_{\mathcal{U},*}\,^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\mathcal{U}}(2\dim
W+2\dim X)\cong{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$

in $D^{b}\textup{MHM}(W,{\mathds{C}})$.

  2. 2.

For any $m\in{\mathds{N}}$, and any $\lambda\in W$ we have isomorphisms of
(complex) mixed Hodge structures

$\begin{array}[]{rcl}H^{m}(i_{\lambda}^{*}\,{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}[-\dim
W])&\cong&H_{c}^{\dim(X)+m}(U_{\lambda},\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\lambda}),\\\
\\\ H^{m}(i_{\lambda}^{!}\,{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}[\dim
W])&\cong&H^{\dim(X)+m}(U_{\lambda},\,^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\lambda})(\dim
W+2\dim X).\\\ \\\ \end{array}$

###### Proof.

In the course of the proof, we will make repeatedly use of the base change
property for algebraic mixed Hodge modules, as stated in [Sai90, Section
4.4.3].

  1. 1.

This is almost immediate by considering the following cartesian diagram

${\mathcal{U}}$${L^{*}}$${W}$${X\times
W}$${L}$$\scriptstyle{ev_{|\mathcal{U}}}$$\scriptstyle{j_{\mathcal{U}}}$$\scriptstyle{a_{\mathcal{U}}}$$\scriptstyle{j_{L}}$$\scriptstyle{ev}$$\scriptstyle{a_{W}}$

which yields (using Eq. 9)

$\begin{array}[]{rcl}{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}&\stackrel{{\scriptstyle(*)}}{{\cong}}&a_{W,!}\,ev^{*}\,{j_{L,!}}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]\\\
&\stackrel{{\scriptstyle(**)}}{{\cong}}&a_{W,!}\,j_{\mathcal{U},!}\,{ev_{|\mathcal{U}}^{*}}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]\\\\[5.0pt]
&=&a_{\mathcal{U},!}\,{ev_{|\mathcal{U}}^{*}}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]\\\
&=&{a_{\mathcal{U},!}}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\mathcal{U}},\end{array}$

where the isomorphism $(*)$ holds because $a_{W}$ is proper (since $X$ is
projective) and $ev$ is smooth, and where $(**)$ follows by base change. We
then apply the duality functor ${\mathbb{D}}$ on
$D^{b}\textup{MHM}(W,{\mathds{C}})$ on both sides of
$a_{\mathcal{U},!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\mathcal{U}}\cong{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}$
to obtain that
$a_{\mathcal{U},*}\,^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\mathcal{U}}(2\dim
X+2\dim W)\cong{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}$.

  2. 2.

Write

${{}^{H,*}\\!\\!}\mathcal{N}_{L}^{\beta}:=ev^{*}j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]\in\textup{MHM}(X\times W,{\mathds{C}}),$

then by the proof of the previous 3.8 we have that

${{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}\cong
a_{W,!}\,{{}^{H,*}\\!\\!}\mathcal{N}_{L}^{\beta}.$

Now consider the cartesian diagram

${X\times\mathopen{}\mathclose{{}\left\\{\lambda}\right\\}}$${X\times
W}$${\mathopen{}\mathclose{{}\left\\{\lambda}\right\\}}$${W.}$$\scriptstyle{i_{\lambda}^{X}}$$\scriptstyle{a^{X}}$$\scriptstyle{a_{W}}$$\scriptstyle{i_{\lambda}}$

Then

$\begin{array}[]{rclcl}i_{\lambda}^{*}\,{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}&\cong&i_{\lambda}^{*}{a_{W,!}}\,{{}^{H,*}\\!\\!}\mathcal{N}_{L}^{\beta}\\\
\\\
&\cong&a^{X}_{!}i^{X,*}_{\lambda}\,{{}^{H,*}\\!\\!}\mathcal{N}_{L}^{\beta}=a^{X}_{!}\,i^{X,*}_{\lambda}ev^{*}j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]&&(\textup{base change})\\\ \\\
&\cong&a_{!}^{X}\,\lambda^{*}j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]&&(ev\circ i^{X}_{\lambda}=\lambda)\\\ \\\
&\cong&a_{*}^{X}\,\lambda^{*}j_{L,!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]&&(a^{X}\textup{ proper}).\end{array}$

Now we consider the diagram

${X}$${L}$${U_{\lambda}}$${L^{*},}$$\scriptstyle{\lambda}$$\scriptstyle{j}$$\scriptstyle{\lambda_{|U_{\lambda}}}$$\scriptstyle{j_{L}}$

then base change yields $\lambda^{*}\,j_{L,!}\cong
j_{!}\,\lambda_{|U_{\lambda}}^{*}$, so we get an isomorphism of objects in
$D^{b}\textup{MHM}(\\{\lambda\\},{\mathds{C}})$ (which we identify with the
derived category of complex mixed Hodge structures).

$\begin{array}[]{rl}i_{\lambda}^{*}\,{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}&\cong
a_{*}^{X}\,j_{!}\,\lambda_{|U_{\lambda}}^{*}\,{}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{L^{*}}[\dim
W-1]\cong
a_{*}^{X}\,j_{!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\lambda}[\dim
W]\\\ &\cong
a_{!}^{X}\,j_{!}\,^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\lambda}[\dim
W]=a^{U_{\lambda}}_{!}\,{}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\lambda}[\dim
W],\end{array}$

where $a^{U_{\lambda}}\colon U_{\lambda}\twoheadrightarrow\\{\lambda\\}$ and
where we have used $a^{X}_{*}=a^{X}_{!}$ since $X$ is projective. We apply
$H^{m}(-)$ to both sides to obtain an isomorphism of complex mixed Hodge
structures

$H^{m}(i_{\lambda}^{*}\,{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}[-\dim W])\cong
H^{m}(a^{U_{\lambda}}_{!}\,{}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\lambda})=H^{m+\dim
X}_{c}(U_{\lambda},{{}^{H\\!}}\underline{{\mathds{C}}}^{-\beta}_{\lambda}).$

recall that we use the convention
${}^{H\\!}\underline{{\mathds{C}}}_{X}:=a_{X}^{*}{{}^{H}\\!}{\mathds{C}}_{pt}[\dim(X)]$.

For the second statement, we apply the duality functor ${\mathbb{D}}$ in
$D^{b}\textup{MHM}(W,{\mathds{C}})$ to the isomorphism
$i_{\lambda}^{*}\,{{}^{H,*}\\!\\!}\mathcal{M}_{L}^{\beta}\cong
a^{U_{\lambda}}_{!}\,{}^{H\\!}\underline{{\mathds{C}}}^{-\beta}_{\lambda}[\dim
W]$ just proved, which gives

$i_{\lambda}^{!}\,{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}\cong
a^{U_{\lambda}}_{*}\,{}^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\lambda}(\dim
W+2\dim X)[-\dim W],$

and then by taking cohomology again we find that

$H^{m}(i_{\lambda}^{!}\,{{}^{H,!}\\!\\!}\mathcal{M}_{L}^{-\beta}[\dim W])\cong
H^{\dim(X)+m}(U_{\lambda},\,^{H\\!}\underline{{\mathds{C}}}^{\beta}_{\lambda})(\dim
W+2\dim X),$

as required. ∎

In the subsequent sections of this article, we will investigate to which
extent tautological systems for homogeneous spaces $X$ are examples of the
$\mathscr{D}$-modules underlying ${}^{H\\!}\mathcal{M}_{L}^{\beta}$ for
particular line bundles $L$ and values $\beta$.

##  4 Equivariant constructions and tautological systems

In this section, we consider varieties with group actions and introduce
particular $\mathscr{D}$-modules that arise naturally from global vector
fields induced by the group action.

_Notations about vector fields:_ For a smooth variety $X$, we denote its
tangent sheaf by $\Theta_{X}$, which coincides with the sheaf
$\operatorname{\mathscr{D}\kern-1.0pt\textit{er}\kern 1.0pt}(\mathcal{O}_{X})$
of derivations on $X$, i.e.,
$\Gamma(U,\Theta_{X})=\operatorname{\textup{Der}}(\mathcal{O}_{U})$ for
$U\subseteq X$. A vector field on $X$ is a global section of $\Theta_{X}$. A
morphism $f\colon X\to Y$ between smooth varieties induces a push-forward
homomorphism of $\mathcal{O}_{X}$-modules
$\operatorname{d\\!}f\colon\Theta_{X}\to f^{*}\Theta_{Y}$. Under the natural
isomorphism

$\displaystyle f^{*}\Theta_{Y}$ $\displaystyle\cong
f^{*}\\!\operatorname{\mathscr{H}\kern-3.0pt\textit{om}\kern
1.0pt}_{\mathcal{O}_{Y}}(\Omega_{Y},\mathcal{O}_{Y})\cong\operatorname{\mathscr{H}\kern-3.0pt\textit{om}\kern
1.0pt}_{\mathcal{O}_{X}}(f^{*}\Omega_{Y},\mathcal{O}_{X})\cong\operatorname{\mathscr{H}\kern-3.0pt\textit{om}\kern
1.0pt}_{\mathcal{O}_{X}}(\mathcal{O}_{X}\otimes_{f^{-1}\mathcal{O}_{Y}}\Omega_{f^{-1}\mathcal{O}_{Y}},\mathcal{O}_{X})$
$\displaystyle\cong\operatorname{\mathscr{H}\kern-3.0pt\textit{om}\kern
1.0pt}_{f^{-1}\mathcal{O}_{Y}}(\Omega_{f^{-1}\mathcal{O}_{Y}},\mathcal{O}_{X})\cong\operatorname{\mathscr{D}\kern-1.0pt\textit{er}\kern
1.0pt}(f^{-1}\mathcal{O}_{Y},\mathcal{O}_{X}),$

the push-forward homomorphism $\operatorname{d\\!}f$ is given by pre-composing
derivations on (an open subset of) $X$ with the homomorphism
$f^{-1}\mathcal{O}_{Y}\to\mathcal{O}_{X}$ induced by $f$. We identify the
fiber of $\Theta_{X}$ at a point $p\in X$ with the tangent space $T_{p}X$. The
push-forward homomorphism $\operatorname{d\\!}f$ induces on the level of
fibers the differential map $\operatorname{d\\!}f_{p}\colon T_{p}X\to
T_{f(p)}Y$ between tangent spaces. For a tangent vector $v\in T_{p}X$, we will
simply write $\operatorname{d\\!}f(v)$ instead of
$\operatorname{d\\!}f_{p}(v)$ whenever convenient.

###  4.1 Vector fields from group actions

In this section we consider the action of a connected linear algebraic group
$G^{\prime}$ on a variety $Y$. The main case of interest arises when we are
given an action of an algebraic group $G$ on a variety $X$, and an equivariant
line bundle $\mathscr{L}$ on $X$. Denoting $G^{\prime}$ the group
${\mathds{C}}^{*}\times G$, we let $G^{\prime}$ act (in various ways depending
on the choice of a character) on $Y$, which we take to be the total space $L$
(or the complement $L^{*}$ of the zero section) of $\mathscr{L}$.

We begin with recalling some facts concerning actions of algebraic groups on
smooth varieties. The proofs are rather elementary and therefore omitted.

######  Lemma 4.1.

Let $G^{\prime}$ be an algebraic group acting on a smooth variety $Y$. Then
there is a unique Lie algebra homomorphism

$Z_{Y}\colon\mathfrak{g}^{\prime}\to\Gamma(Y,\Theta_{Y})$

associating to every element $\xi$ of the Lie algebra $\mathfrak{g}^{\prime}$
of $G^{\prime}$ a vector field $Z_{Y}(\xi)$ on $Y$ with the following point-
wise description: At a point $y\in Y$, the tangent vector of the vector field
$Z_{Y}(\xi)$ is given by $-\operatorname{d\\!}\varphi^{y}(\xi)$, where
$\varphi^{y}\colon G^{\prime}\to Y$, $g\mapsto g\cdot y$, and $\xi$ is
understood as a tangent vector to $G^{\prime}$ at the point $1\in G^{\prime}$.

In the complex analytic category, the vector field $Z_{Y}(\xi)$ may be defined
as the derivation

$Z_{Y}(\xi)(f)=\frac{d}{dt}{f\mathopen{}\mathclose{{}\left(\exp(t\xi)^{-1}\cdot{(-)}}\right)_{|{t=0}}}.$

If the $G^{\prime}$-variety $Y$ considered is clear from the context, we will
drop the index and just write $Z(\xi)$. In the literature, the vector field
$Z(\xi)$ is sometimes denoted by $L_{\xi}$, see e.g. [Hot98, II.2].

######  Example 4.2.

Consider the action of $G^{\prime}$ on itself by left-multiplication (i.e.,
$Y=G^{\prime}$). Then $-Z_{G^{\prime}}(\xi)$ is the right-invariant vector
field associated to $\xi\in\mathfrak{g}^{\prime}$. If, for example,
$G^{\prime}=(\mathds{C}^{*})^{d}$ and
$\xi\in\mathds{C}^{d}=\mathfrak{g}^{\prime}$, then

$Z_{(\mathds{C}^{*})^{d}}(\xi)=-\sum_{i=1}^{d}\xi_{i}\,t_{i}\partial_{t_{i}},$

where $(t_{1},\dots,t_{d})$ are the standard coordinates on
$(\mathds{C}^{*})^{d}$.

For group actions on finite-dimensional vector spaces, we also have the
following description:

######  Lemma 4.3.

Let $\rho\colon G^{\prime}\to\operatorname{\textup{GL}}(V)$ be a finite-
dimensional rational representation of an algebraic group $G^{\prime}$. The
induced left action of $G^{\prime}$ on $\mathds{C}[V]=\bigoplus_{d\geq
0}\operatorname{\textup{Sym}}^{d}V^{\vee}$ describes a morphism of algebraic
groups $G^{\prime}\to\operatorname{\textup{GL}}_{\mathds{C}}(\mathds{C}[V])$
whose induced Lie algebra homomorphism
$\mathfrak{g}^{\prime}\to\operatorname{\textup{End}}_{\mathds{C}}(\mathds{C}[V])$
makes the following diagram commute:

${\operatorname{\textup{Lie}}(G^{\prime})}$${\operatorname{\textup{End}}_{\mathds{C}}(\mathds{C}[V])}$${\operatorname{\textup{Der}}(\mathds{C}[V])}$.$\scriptstyle{Z_{V}}$

Explicitly, if we fix coordinates $x_{1},\ldots,x_{n}$ on $V$ and consider the
associated Lie algebra representation
$\operatorname{d\\!}\rho\colon\mathfrak{g}^{\prime}\to\mathfrak{gl}(V)=\mathfrak{gl}(n,\mathds{C})=\mathds{C}^{n\times
n}$, then

$Z_{V}(\xi)=-\sum_{i,j=1}^{n}\operatorname{d\\!}\rho(\xi)_{ji}\,x_{i}\partial_{x_{j}}$

for all $\xi\in\operatorname{\textup{Lie}}(G^{\prime})$.

######  Example 4.4.

Let $G^{\prime}=(\mathds{C}^{*})^{d}$ be a $d$-dimensional torus acting
linearly on an $n$-dimensional vector space $V$. We identify $V$ with
$\mathds{C}^{n}$ by picking a basis that diagonalizes the action, i.e.,
$t=(t_{1},\dots,t_{d})\in(\mathds{C}^{*})^{d}$ acts on
$x=(x_{1},\dots,x_{n})\in\mathds{C}^{n}$ by

$t\cdot x=(t^{\alpha_{1}}x_{1},\dots,t^{\alpha_{n}}x_{n})\qquad\text{with
}\alpha_{1},\ldots,\alpha_{n}\in\mathds{Z}^{d}.$

If
$\xi\in\mathds{Z}^{d}=\operatorname{\textup{Lie}}\mathopen{}\mathclose{{}\left((\mathds{C}^{*})^{d}}\right)$
is the $i$-th standard basis vector $e_{i}$, we get the vector field

$Z_{V}(e_{i})=-\sum_{j=1}^{n}(\alpha_{j})_{i}x_{j}\partial_{x_{j}}$

on $V$. These are the vector fields showing up in GKZ-systems associated to
the given torus action.

######  Lemma 4.5.

Let $G^{\prime}$ be an algebraic group acting transitively on a smooth variety
$Y$. Fix $y\in Y$ and consider the morphism $\varphi^{y}\colon G^{\prime}\to
Y$, $g\mapsto g\cdot y$. Let $\xi\in\mathfrak{g}^{\prime}$. Then $Z_{Y}(\xi)$
is the unique vector field on $Y$ lifting the push-forward under $\varphi^{y}$
of the right-invariant vector field on $G^{\prime}$ associated to $-\xi$.

###### Proof.

Let $\theta\in\Gamma(G^{\prime},\Theta_{G^{\prime}})$ be the right-invariant
vector field associated to $-\xi\in\mathfrak{g}^{\prime}=T_{1}G^{\prime}$.
Point-wise, it associates to a point $g\in G^{\prime}$ the tangent vector
$\operatorname{d\\!}R_{g}(-\xi)$, where $R_{g}\colon G^{\prime}\to G^{\prime}$
is given by right-multiplication with $g$. Its push-forward under
$\varphi^{y}$ associates to $g\in G^{\prime}$ the tangent vector
$\operatorname{d\\!}\varphi^{y}(\operatorname{d\\!}R_{g}(-\xi))=-\operatorname{d\\!}\varphi^{g\cdot
y}(\xi)$ to $Y$ at $g\cdot y$. By 4.1, $Z_{Y}(\xi)$ is a vector field on $Y$
lifting
$\operatorname{d\\!}\varphi^{y}(\theta)\in\Gamma(G^{\prime},\varphi^{y,*}\Theta_{Y})$.
On the other hand, since $G^{\prime}$ acts transitively on $Y$, the morphism
$\varphi^{y}$ is surjective, hence the lift of
$\operatorname{d\\!}\varphi^{y}(\theta)$ is unique. ∎

######  Example 4.6.

Let $X\subseteq\mathds{P}^{k}$ be the rational normal curve of degree $k$,
i.e., the image of

$\mathds{P}^{1}\xrightarrow{|\mathcal{O}(k)|}\mathds{P}^{k},\qquad[x_{0}:x_{1}]\mapsto\big{[}{\textstyle\binom{k}{i}\,x_{0}^{k-i}x_{1}^{i}\mid
i=0,\dots,k}\big{]},$

and let $Y:=\hat{X}\setminus\\{0\\}$ be the punctured affine cone over $X$ in
$V:=\mathds{C}^{k+1}$. The group $\operatorname{\textup{SL}}(2)$ acts on on
$V=H^{0}(\mathds{P}^{1},\mathcal{O}(k))^{\vee}=\operatorname{\textup{Sym}}^{k}(\mathds{C}^{2})$,
the $k$-th symmetric power of the standard
$\operatorname{\textup{SL}}(2)$-representation, and we extend this to an
action of $G^{\prime}:=\operatorname{\textup{SL}}(2)\times\mathds{C}^{*}$ by
letting the $\mathds{C}^{*}$-factor act by scaling on $V$. The Lie algebra
$\mathfrak{g}^{\prime}$ is generated by
$E_{12},E_{21},E_{11}-E_{22}\in\mathfrak{sl}(2)$ and the generator
$\mathbf{e}$ of $\operatorname{\textup{Lie}}(\mathds{C}^{*})\cong\mathds{C}$.
The induced vector fields on $V$ are

$\begin{array}[]{c@{\hspace{3em}}c}\begin{aligned}
Z_{V}(E_{12})&=-\sum_{i=1}^{k}i\,z_{i}\partial_{z_{i-1}},\\\
Z_{V}(E_{11}-E_{22})&=-\sum_{i=0}^{k}(k-2i)\,z_{i}\partial_{z_{i}},\end{aligned}\hfil\hskip
30.00005pt&\begin{aligned}
Z_{V}(E_{21})&=-\sum_{i=1}^{k}(k-i+1)\,z_{i-1}\partial_{z_{i}},\\\
Z_{V}(\mathbf{e})&=-\sum_{i=0}^{k}z_{i}\partial_{z_{i}},\end{aligned}\end{array}$

where $z_{0},\dots,z_{k}$ denote the coordinates on $V=\mathds{C}^{k+1}$. Note
that the minus signs appear because we differentiate the _contragredient_
action on the coordinate ring of $V$.

On the $G^{\prime}$-invariant subset $Y$, these vector fields restrict to the
vector fields $Z_{Y}(\xi)$. In local charts, these can be expressed as
follows: We may cover $Y$ by the two open subsets $U_{0}$ and $U_{1}$ given by
the non-vanishing of $x_{0}^{k}\in V^{\vee}$ and $x_{1}^{k}\in V^{\vee}$,
respectively. Identifying

$\displaystyle U_{0}$
$\displaystyle\cong\mathds{C}^{*}\times\mathds{C},\qquad\lambda\cdot{\textstyle(1,ks,\binom{k}{2}s^{2},\dots,ks^{k-1},s^{k})}\quad\mapsfrom\
(\lambda,s),$ $\displaystyle U_{1}$
$\displaystyle\cong\mathds{C}^{*}\times\mathds{C},\qquad\mu\cdot{\textstyle(t^{k},kt^{k-1},\binom{k}{2}t^{k-2},\dots,kt,1)}\
\,\mapsfrom\ (\mu,t),$

the vector fields induced from the $G^{\prime}$-action on $Y$ are:

$\begin{array}[]{c@{\hspace{3em}}c}\begin{aligned}
{Z_{Y}(E_{12})_{|{U_{0}}}}&=-ks\lambda\partial_{\lambda}+s^{2}\partial_{s},\\\
{Z_{Y}(E_{21})_{|{U_{0}}}}&=-\partial_{s},\\\
{Z_{Y}(E_{11}-E_{22})_{|{U_{0}}}}&=-k\lambda\partial_{\lambda}+2s\partial_{s},\\\
{Z_{Y}(\mathbf{e})_{|{U_{0}}}}&=-\lambda\partial_{\lambda},\end{aligned}\hfil\hskip
30.00005pt&\begin{aligned} {Z_{Y}(E_{12})_{|{U_{1}}}}&=-\partial_{t},\\\
{Z_{Y}(E_{21})_{|{U_{1}}}}&=-kt\mu\partial_{\mu}+t^{2}\partial_{t},\\\
{Z_{Y}(E_{11}-E_{22})_{|{U_{1}}}}&=k\mu\partial_{\mu}-2t\partial_{t},\\\
{Z_{Y}(\mathbf{e})_{|{U_{1}}}}&=-\mu\partial_{\mu}.\end{aligned}\end{array}$

Note that these local expressions coincide on the intersection $U_{0}\cap
U_{1}$ under the gluing
$\mathds{C}^{*}\times\mathds{C}^{*}\xrightarrow{\cong}\mathds{C}^{*}\times\mathds{C}^{*}$,
$(\lambda,s)\mapsto(\lambda s^{k},s^{-1})=(\mu,t)$.

###  4.2 $\mathscr{D}$-modules from group actions

Using the vector fields defined in the previous section, we introduce the
following $\mathscr{D}$-modules:

######  Definition 4.7.

Let $G^{\prime}$ be an algebraic group acting on a smooth variety $Y$ and let
$\beta\colon\mathfrak{g}^{\prime}\to\mathds{C}$ be a Lie algebra homomorphism.
Then we define the left $\mathscr{D}_{Y}$-module

$\mathscr{N}_{Y}^{\beta}:=\omega_{Y}^{\vee}\otimes_{\mathcal{O}_{Y}}\mathscr{D}_{Y}/(Z_{Y}(\xi)-\beta(\xi)\mid\xi\in\mathfrak{g}^{\prime})\mathscr{D}_{Y}.$

######  Example 4.8.

Let $G^{\prime}=T=(\mathds{C}^{*})^{d}$ be a $d$-dimensional torus acting on
itself. We identify Lie algebra homomorphisms
$\beta\colon\mathds{C}^{d}=\mathfrak{g}^{\prime}\to\mathds{C}$ with vectors
$\beta\in\mathds{C}^{d}$. Then

$\mathscr{N}_{T}^{\beta}=\omega_{T}^{\vee}\otimes_{\mathcal{O}_{T}}\mathscr{D}_{T}/(-t_{i}\partial_{t_{i}}-\beta_{i}\mid
i=1,\dots,d)\mathscr{D}_{T}\cong\mathscr{D}_{T}/\mathscr{D}_{T}(\partial_{t_{i}}t_{i}-\beta_{i}\mid
i=1,\dots,d).$

This $\mathscr{D}_{T}$-module was called $\mathcal{O}_{T}^{-\beta}$ in [RS20].

######  Example 4.9.

We reconsider the action of
$G^{\prime}=\operatorname{\textup{SL}}(2)\times\mathds{C}^{*}$ on the
punctured affine cone $Y$ over the rational normal curve of degree $k$ from
example 4.6 and use the notations from before. Every Lie algebra homomorphism
$\beta\colon\mathfrak{g}^{\prime}\to\mathds{C}$ is given by
${\beta_{|{\mathfrak{sl}(2)}}}\equiv 0$ and
$\beta(\mathbf{e})=\beta_{0}\in\mathds{C}$. By the computations in example
4.6, in the local chart $U_{0}\cong\mathds{C}^{*}\times\mathds{C}\subseteq Y$,
the $\mathscr{D}_{Y}$-module $\mathscr{N}_{Y}^{\beta}$ can be expressed as

$\displaystyle{(\mathscr{N}_{Y}^{\beta})_{|{U_{0}}}}$
$\displaystyle\cong\omega_{U_{0}}^{\vee}\otimes_{\mathcal{O}_{U_{0}}}\mathscr{D}_{U_{0}}/(-ks\lambda\partial_{\lambda}+s^{2}\partial_{s},\;-\partial_{s},\;-k\lambda\partial_{\lambda}+2s\partial_{s},\;-\lambda\partial_{\lambda}-\beta_{0})\mathscr{D}_{U_{0}}$
$\displaystyle\cong\mathscr{D}_{U_{0}}/\mathscr{D}_{U_{0}}(ks\partial_{\lambda}\lambda-\partial_{s}s^{2},\;\partial_{s},\;k\partial_{\lambda}\lambda-2\partial_{s}s,\;\partial_{\lambda}\lambda-\beta_{0})$
$\displaystyle\cong\mathscr{D}_{U_{0}}/\mathscr{D}_{U_{0}}(ks\lambda\partial_{\lambda}-s^{2}\partial_{s}+(k-2)s,\;\partial_{s},\;k\lambda\partial_{\lambda}-2s\partial_{s}+(k-2),\;\lambda\partial_{\lambda}+1-\beta_{0})$
$\displaystyle\cong\mathscr{D}_{U_{0}}/\mathscr{D}_{U_{0}}(\partial_{s},\;\lambda\partial_{\lambda}+1-\beta_{0},\;k(-1+\beta_{0})+(k-2))$
$\displaystyle\cong\begin{cases}\mathscr{D}_{\mathds{C}^{*}}/\mathscr{D}_{\mathds{C}^{*}}(\partial_{\lambda}\lambda-\beta_{0})\boxtimes\mathscr{D}_{\mathds{C}}/\mathscr{D}_{\mathds{C}}\cdot\partial_{s}\quad&\text{if
}\beta_{0}=2/k,\\\ 0&\text{otherwise}\end{cases}$

and similarly for the other local chart $U_{1}$ of $Y$. In particular, for one
specific value for $\beta(\mathbf{e})$, we obtain a non-zero
$\mathscr{D}_{Y}$-module that will be of interest to us.

Note that in contrast, if we define the cyclic _left_ module

$\tilde{\mathscr{N}}_{Y}^{\beta}:=\mathscr{D}_{Y}/\mathscr{D}_{Y}(Z_{Y}(\xi)-\beta(\xi)\mid\xi\in\mathfrak{g}^{\prime}),$

then, in this example, we get

$\displaystyle{(\tilde{\mathscr{N}}_{Y}^{\beta})_{|{U_{0}}}}$
$\displaystyle\cong\mathscr{D}_{U_{0}}/\mathscr{D}_{U_{0}}(-ks\lambda\partial_{\lambda}+s^{2}\partial_{s},\;-\partial_{s},\;-k\lambda\partial_{\lambda}+2s\partial_{s},\;-\lambda\partial_{\lambda}-\beta_{0})$
$\displaystyle\cong\mathscr{D}_{U_{0}}/\mathscr{D}_{U_{0}}(\partial_{s},\;\lambda\partial_{\lambda}+\beta_{0},\;k\beta_{0})\cong\begin{cases}\mathcal{O}_{U_{0}}&\text{if
}\beta_{0}=0,\\\ 0&\text{otherwise}.\end{cases}$

For $k=2$, we have
$\mathscr{N}_{Y}^{(\beta_{0}=1)}=\tilde{\mathscr{N}}_{Y}^{(\beta_{0}=0)}$, but
in general they do not agree with each other. In fact, one can show that
although $\mathscr{N}_{Y}^{\beta}$ is _locally_ a cyclic left
$\mathscr{D}_{Y}$-module, it does not admit a _global_ description as a cyclic
left $\mathscr{D}_{Y}$-module for $k\geq 3$.

The main reason we wish to consider the $\mathscr{D}_{Y}$-module
$\mathscr{N}_{Y}^{\beta}$ defined via the right-left-transformation of a
cyclic right-module is the following behavior under equivariant closed
embeddings:

######  Proposition 4.10.

Let $G^{\prime}$ be an algebraic group and let $i\colon Y_{1}\hookrightarrow
Y_{2}$ be a $G^{\prime}$-equivariant closed embedding between smooth
$G^{\prime}$-varieties $Y_{1},Y_{2}$. Then, for all Lie algebra homomorphisms
$\beta\colon\mathfrak{g}^{\prime}\to\mathds{C}$, we have

$i_{+}\mathscr{N}_{Y_{1}}^{\beta}\cong\omega_{Y_{2}}^{\vee}\otimes_{\mathcal{O}_{Y_{2}}}\mathscr{D}_{Y_{2}}/\big{(}\mathcal{I}+(Z_{Y_{2}}(\xi)-\beta(\xi)\mid\xi\in\mathfrak{g}^{\prime})\big{)}\mathscr{D}_{Y_{2}}$

where $\mathcal{I}\subseteq\mathcal{O}_{Y_{2}}$ is the ideal sheaf of $Y_{1}$
in $Y_{2}$.

###### Proof.

Since $i\colon Y_{1}\hookrightarrow Y_{2}$ is a closed embedding, the functor
$i_{*}$ is exact and the transfer module $\mathscr{D}_{Y_{1}\to Y_{2}}$ is a
flat $\mathscr{D}_{Y_{1}}$-module. Therefore, the direct image of
$\mathscr{N}_{Y_{1}}^{\beta}$ under $i$ is given by

$i_{+}\mathscr{N}_{Y_{1}}^{\beta}\cong\omega_{Y_{2}}^{\vee}\otimes_{\mathcal{O}_{Y_{2}}}i_{*}\big{(}\mathscr{D}_{Y_{1}}/(Z_{Y_{1}}(\xi)-\beta(\xi)\mid\xi\in\mathfrak{g}^{\prime})\mathscr{D}_{Y_{1}}\otimes_{\mathscr{D}_{Y_{1}}}\mathscr{D}_{Y_{1}\to
Y_{2}}\big{)}.$

Hence, the claim is that

$\displaystyle\mathscr{D}_{Y_{1}}/(Z_{Y_{1}}(\xi)-\beta(\xi)\mid\xi\in\mathfrak{g}^{\prime})\mathscr{D}_{Y_{1}}\otimes_{\mathscr{D}_{Y_{1}}}\mathscr{D}_{Y_{1}\to
Y_{2}}$
$\displaystyle\qquad\stackrel{{\scriptstyle!}}{{\cong}}i^{-1}\big{(}\mathscr{D}_{Y_{2}}/((Z_{Y_{2}}(\xi)-\beta(\xi)\mid\xi\in\mathfrak{g}^{\prime})\mathscr{D}_{Y_{2}}+\mathcal{I}\mathscr{D}_{Y_{2}})\big{)}.$

as right $i^{-1}\mathscr{D}_{Y_{2}}$-modules. Note that $\mathscr{D}_{Y_{1}\to
Y_{2}}\cong i^{-1}(\mathscr{D}_{Y_{2}}/\mathcal{I}\mathscr{D}_{Y_{2}})$ as
right $i^{-1}\mathscr{D}_{Y_{2}}$-modules, since $i$ is a closed embedding.
Under the left $\mathscr{D}_{Y_{1}}$-module structure on
$\mathscr{D}_{Y_{1}\to Y_{2}}$, vector fields on $Y_{1}$ act via the push-
forward homomorphism

$\operatorname{d\\!}i\colon\Theta_{Y_{1}}\to
i^{*}\Theta_{Y_{2}}=\mathcal{O}_{Y_{1}}\otimes_{i^{-1}\mathcal{O}_{Y_{2}}}i^{-1}\Theta_{Y_{2}}\cong
i^{-1}(\mathcal{O}_{Y_{2}}/\mathcal{I}\otimes_{\mathcal{O}_{Y_{2}}}\Theta_{Y_{2}}).$

We note that the push-forward of the vector field $Z_{Y_{1}}(\xi)$ on $Y_{1}$
agrees with the restriction of the vector field $Z_{Y_{2}}(\xi)$ on $Y_{2}$ to
$Y_{1}$, i.e., $\operatorname{d\\!}i(Z_{Y_{1}}(\xi))=1\otimes Z_{Y_{2}}(\xi)$.
Indeed, this follows from the construction of $Z_{Y_{1}}(\xi)$ and
$Z_{Y_{2}}(\xi)$, using the commutativity of

${G^{\prime}\times Y_{1}}$${Y_{1}}$${G^{\prime}\times
Y_{2}}$${Y_{2},}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\operatorname{\textup{id}}_{G^{\prime}}\times
i}$$\scriptstyle{i}$$\scriptstyle{\varphi_{2}}$

where $\varphi_{1},\varphi_{2}$ are the morphisms given by the
$G^{\prime}$-actions.

This shows that
$Z_{Y_{1}}(\xi)\in\operatorname{\textup{Der}}(\mathcal{O}_{Y_{1}})$ acts on
the right $i^{-1}\mathscr{D}_{Y_{2}}$-module $\mathscr{D}_{Y_{1}\to
Y_{2}}\cong i^{-1}(\mathscr{D}_{Y_{2}}/\mathcal{I}\mathscr{D}_{Y_{2}})$ by
left-multiplication with $Z_{Y_{2}}(\xi)$. This implies the claimed
description as a cyclic right $i^{-1}\mathscr{D}_{Y_{2}}$-module of
$\mathscr{D}_{Y_{1}}/(Z_{Y_{1}}(\xi)-\beta(\xi)\mid\xi)\mathscr{D}_{Y_{1}}\otimes_{\mathscr{D}_{Y_{1}}}\mathscr{D}_{Y_{1}\to
Y_{2}}$, concluding the proof. ∎

The $\mathscr{D}$-modules in 4.10 look similar to the $\beta$-twistedly
equivariant $\mathscr{D}$-modules considered in [Hot98, II.2], yet they are
different: Instead of considering a cyclic _left_ module obtained by
quotienting out a $G^{\prime}$-stable ideal and the vector fields induced by
the group action (twisted with $\beta$), we instead consider the _right_
module constructed in the same way and apply a right-left transformation to
obtain a left $\mathscr{D}$-module. The behavior under direct images of closed
embeddings in 4.10 is the reason why for our purposes we work with the
definition via right modules in 4.7.

We next consider the situation where $Y$ is an orbit of a rational
representation $\rho$ of our group $G^{\prime}$ in a given vector space $V$.
Recall from our basic 1.1 that under this hypothesis, we can define, for any
Lie algebra homomorphism
$\beta\colon\mathfrak{g}^{\prime}\rightarrow{\mathds{C}}$, the
$\mathscr{D}_{V}$-module $\hat{\tau}(\rho,\overline{Y},\beta)$ (as well as its
Fourier-Laplace transform $\tau(\rho,\overline{Y},\beta)$ which was called
tautological system in 1.1). The next result tells us about a technically easy
but imporant relation of this $\hat{\tau}(\rho,\overline{Y},\beta)$ to the
$\mathscr{D}_{Y}$-module $\mathscr{N}_{Y}^{\beta}$ considered above.

######  Corollary 4.11.

Let $\rho\colon G^{\prime}\to\operatorname{\textup{GL}}(V)$ be a finite-
dimensional rational representation of an algebraic group and let
$\beta\colon\mathfrak{g}^{\prime}\to\mathds{C}$ be a Lie algebra homomorphism.
Let $Y\subseteq V$ be a $G^{\prime}$-orbit, let $\overline{Y}$ be its closure
and let $\partial Y:=\overline{Y}\setminus Y$. Then

$j^{+}\hat{\tau}(\rho,\overline{Y},\beta)\cong i_{+}\mathscr{N}_{Y}^{\beta},$

where $Y=\overline{Y}\setminus\partial
Y\xhookrightarrow{i}U:=V\setminus\partial Y\xhookrightarrow{j}V$.

In particular, if $\hat{\tau}(\rho,\overline{Y},\beta)$ is localized at
$\partial Y$ (meaning
$j_{+}j^{+}\hat{\tau}(\rho,\overline{Y},\beta)\cong\hat{\tau}(\rho,\overline{Y},\beta)$),
then it is the direct image of $\mathscr{N}_{Y}^{\beta}$ under the locally
closed embedding $Y\hookrightarrow V$.

###### Proof.

We apply 4.10 to the $G^{\prime}$-spaces $Y_{1}:=Y$,
$Y_{2}:=V\setminus\partial Y$ and the closed embedding $i\colon
Y_{1}\hookrightarrow Y_{2}$ to see that

$i_{+}\mathscr{N}_{Y}^{\beta}\cong\omega_{U}^{\vee}\otimes_{\mathcal{O}_{U}}\mathscr{D}_{U}/\big{(}\mathcal{I}+\\{Z_{U}(\xi)-\beta(\xi)\\}\big{)}\mathscr{D}_{U}.$

Choosing coordinates $x_{1},\dots,x_{n}$ on $V$, we may by 4.3 express the
vector field $Z_{U}(\xi)$ as the derivation
$-\sum_{i,j=1}^{n}\operatorname{d\\!}\rho(\xi)_{ji}\,x_{i}\partial_{x_{j}}$.
The right-left transformation
$\omega_{U}^{\vee}\otimes_{\mathcal{O}_{U}}(\cdot)$ is then explicitly given
by transposing operators:

$i_{+}\mathscr{N}_{Y}^{\beta}\cong\mathscr{D}_{U}/\mathscr{D}_{U}\big{(}\mathcal{I}+\\{Z_{U}(\xi)^{T}-\beta(\xi)\\}\big{)}.$

An explicit computation of the transposed vector fields yields:

$Z_{V}(\xi)^{T}=\sum_{i,j=1}^{n}\operatorname{d\\!}\rho(\xi)_{ji}\,\partial_{x_{j}}x_{i}=\sum_{i,j=1}^{n}\operatorname{d\\!}\rho(\xi)_{ji}\,x_{i}\partial_{x_{j}}+\sum_{i=1}^{n}\operatorname{d\\!}\rho(\xi)_{ii}=-Z_{V}(\xi)+\operatorname{trace}(\operatorname{d\\!}\rho(\xi)),$

hence (using $Z_{V}(\xi)_{|U}=Z_{U}(\xi)$) we have that
$i_{+}\mathscr{N}_{Y}^{\beta}\cong j^{+}\hat{\tau}(\rho,\overline{Y},\beta)$.
∎

######  Example 4.12 (GKZ-systems).

Consider a torus representation
$\rho\colon(\mathds{C}^{*})^{n}\to\operatorname{\textup{GL}}(n,\mathds{C})$
that is given by
$\rho(t_{1},\dots,t_{d})=\operatorname{\textup{diag}}(t^{\alpha_{1}},\dots,t^{\alpha_{n}})$
with $\alpha_{i}\in\mathds{Z}^{d}$. Let $\overline{Y}\subseteq\mathds{C}^{n}$
be the orbit closure of the point $(1,\dots,1)\in\mathds{C}^{n}$; this is a
(not necessarily normal) affine toric variety. The
$\mathscr{D}_{\mathds{C}^{n}}$-module $\hat{\tau}(\rho,\overline{Y},\beta)$ is
the Fourier–Laplace transform $\widehat{\mathscr{M}}_{A}(-\beta)$ of the GKZ-
system $\mathscr{M}_{A}(-\beta)$ (see, e.g. [RSSW21] for an overview and for
the notation used here), where $A$ is the $d\times n$-matrix whose $i$-th
column is $\alpha_{i}$ and
$\beta\colon\operatorname{\textup{Lie}}\mathopen{}\mathclose{{}\left((\mathds{C}^{*})^{d}}\right)=\mathds{Z}^{d}\to\mathds{C}$
is identified with the vector
$\mathopen{}\mathclose{{}\left(\beta(e_{i})}\right)_{i=1,\dots,d}\in\mathds{C}^{d}$.

In this case, 4.11 applied to $Y=\overline{Y}\cap(\mathds{C}^{*})^{n}$ says
that $\widehat{\mathscr{M}}_{A}(-\beta)$ is the direct image of
$\mathscr{N}_{(\mathds{C}^{*})^{d}}^{\beta}=\mathcal{O}_{(\mathds{C}^{*})^{d}}^{-\beta}$
under the locally closed embedding $(\mathds{C}^{*})^{d}\cong
Y\hookrightarrow\mathds{C}^{n}$, whenever $\mathscr{M}_{A}(-\beta)$ is
localized at the intersection of $Y$ with the union of coordinate hyperplanes
of $\mathds{C}^{n}$. This was observed in [SW09], where an explicit
combinatorial characterization of the localization property in terms of $A$
and $\beta$ was proved using Euler–Koszul complexes.

######  Example 4.13.