The Künneth Formula for the Twisted de Rham and Higgs Cohomologies

The Künneth Formula for the Twisted de Rham
and Higgs Cohomologies††This paper is a contribution to the Special Issue on
Modular Forms and String Theory in honor of Noriko Yui. The full collection is
available at http://www.emis.de/journals/SIGMA/modular-forms.html

Kai-Chieh CHEN † and Jeng-Daw YU ‡

K.-C. Chen and J.-D. Yu

† Department of Mathematics, University of California, Berkeley, Berkeley, CA,
USA<EMAIL_ADDRESS>https://math.berkeley.edu/~kaichieh/

‡ Department of Mathematics, National Taiwan University, Taipei, Taiwan
<EMAIL_ADDRESS>http://homepage.ntu.edu.tw/~jdyu/

Received February 23, 2018, in final form June 02, 2018; Published online June
12, 2018

We prove the Künneth formula for the irregular Hodge filtrations on the
exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-
projective complex varieties. The method involves a careful comparison of the
underlying chain complexes under a certain elimination of indeterminacy.

de Rham complex; Hodge filtration; Künneth formula

14F40; 18F20; 14C30

##  1 The main result

Let $U$ be a smooth quasi-projective variety over the field $\mathbb{C}$ of
complex numbers, and $f\in\Gamma(U,\mathscr{O}_{U})$ a regular function.
Attached to such a pair $(U,f)$ and a non-negative integer $k$, one has the
$k$-th de Rham cohomology $\mathrm{H}_{\mathrm{dR}}^{k}(U,f)$ and Higgs
cohomology $\mathrm{H}_{\mathrm{Hig}}^{k}(U,f)$, defined in Section 2. The two
spaces
$\mathrm{H}_{\diamond}^{k}(U,f),\diamond\in\\{\mathrm{dR},\mathrm{Hig}\\}$,
are of the same finite dimension over $\mathbb{C}$ (see [1, Remark 1.3.3]),
and are equipped with the decreasing irregular Hodge filtrations

$\displaystyle
F^{\lambda}\mathrm{H}_{\diamond}^{k}(U,f),\qquad\lambda\in\mathbb{Q},$

indexed by the ordered set $\mathbb{Q}$ of rational numbers with finitely many
jumps. In the following, we omit the adjective and just call them the Hodge
filtrations. For the motivations and the basic properties of the Hodge
filtration, including in particular the degeneration of the Hodge to de Rham
spectral sequence, see [1, 3] and the references therein. We recall the
construction in Section 2.

Now consider two such pairs $(U_{i},f_{i})$, $i=1,2$. On the product
$U:=U_{1}\times U_{2}$, consider the regular function $f$ defined by
$f\colon(x_{1},x_{2})\mapsto f_{1}(x_{1})+f_{2}(x_{2})$. We call the pair
$(U,f)$ the product of the two $(U_{i},f_{i})$. For
$\diamond\in\\{\mathrm{dR},\mathrm{Hig}\\}$, there is the canonical map

$\displaystyle\bigoplus_{i+j=k}\mathrm{H}_{\diamond}^{i}(U_{1},f_{1})\otimes\mathrm{H}_{\diamond}^{j}(U_{2},f_{2})\to\mathrm{H}^{k}_{\diamond}(U,f)$
(1)

induced by cup product. We equip the space on the left hand side with the
product filtration, i.e., the $\lambda$-th filtration for
$\lambda\in\mathbb{Q}$ is given by the subspace

$\displaystyle\bigoplus_{i+j=k}\left(\sum_{a+b=\lambda}F^{a}\mathrm{H}^{i}_{\diamond}(U_{1},f_{1})\otimes
F^{b}\mathrm{H}^{j}_{\diamond}(U_{2},f_{2})\right),$

where the inner sum is taken inside
$\mathrm{H}_{\diamond}^{i}(U_{1},f_{1})\otimes\mathrm{H}_{\diamond}^{j}(U_{2},f_{2})$.
In this article, we prove the following Künneth formula.

######  Theorem 1.1.

With notations as above, the map (1) is an isomorphism of filtered spaces.

In particular, denoting $\operatorname{Gr}_{F}^{\lambda}V$ the $\lambda$-th
graded piece of a filtered space $(V,F)$, one has the identification

$\displaystyle\operatorname{Gr}_{F}^{\lambda}\mathrm{H}_{\diamond}^{k}(U,f)=\bigoplus_{r,s}\operatorname{Gr}_{F}^{r}\mathrm{H}_{\diamond}^{s}(U_{1},f_{1})\otimes\operatorname{Gr}_{F}^{\lambda-r}\mathrm{H}_{\diamond}^{k-s}(U_{2},f_{2}).$

Recently different proofs of the Künneth formula in a more general setting
appear in [8, Theorem 3.39] where the involved coefficients in the cohomology
are allowed to be exponential twists of complex Hodge modules. One of the main
ideas in that work is to enrich the de Rham and Higgs cohomologies into the
Brieskorn lattice (see the last section Section 4) or a twistor structure and
treat the irregular Hodge filtration as a byproduct of the enrichment. The
notion of a $V$-adapted trivializing lattice for a meromorphic connection on
$\mathbb{P}^{1}$ of special type is introduced [8, Section 3.2.b] in order to
obtain the Künneth formula for irregular Hodge filtrations. The methods depend
on the deep theory of twistor $\mathscr{D}$-modules mainly developed by Sabbah
and Mochizuki (see [6]). On the other hand, our approach is much more
elementary. We believe that the concrete filtered de Rham complex used here
would be suitable for computations in some interesting examples of irregular
Hodge structures in the future work.

The rest of the article begins in Section 2 with a brief of the construction
of the Hodge filtration. We follow the approach of [11] by putting a
filtration on the de Rham complex or the Higgs complex via a certain
compactification of the pair $(U,f)$. Here we introduce the notion of a non-
degenerate compactification, which is weaker than that of a good
compactification used in [11], but appears naturally in the later section (see
also [7, Section 4], [9, Section 7.3], [5]). In order to compare the
cohomologies with the filtrations of the summands $(U_{i},f_{i})$ and of their
product $(U,f)$, we construct a particular compactification of $(U,f)$ from
the fixed ones of $(U_{i},f_{i})$ in Section 3. The proof of the Künneth
formula is obtained by a careful investigation of the relations between the
involved complexes on the compactifications. In the last Section 4, we remark
that one can interpolate the two spaces $\mathrm{H}_{\mathrm{dR}}^{k}(U,f)$
and $\mathrm{H}_{\mathrm{Hig}}^{k}(U,f)$ as the fibers of the Kontsevich
bundle on the projective line $\mathbb{P}^{1}$ over $1$ and $0$, respectively.
In fact, the fiber over $c\in\mathbb{C}\setminus\\{0\\}$ of the bundle is
equal to $\mathrm{H}_{\mathrm{dR}}^{k}(U,f/c)$ and hence the Künneth formula
also holds true. However at $\infty$ the situation is more complicated in this
regard and the direct analogue of the Künneth formula does not hold in
general.

##  2 The filtrations

In this section, we fix a pair $(U,f)$ consisting of a smooth quasi-projective
variety $U$ over $\mathbb{C}$ and a regular function
$f\in\Gamma(U,\mathscr{O}_{U})$.

###  2.1 The compactification

Let $X$ be a projective variety over $\mathbb{C}$ containing $U$ such that the
reduced subvariety $S:=X\setminus U$ is a normal crossing divisor. Regard
$f\in\Gamma(X,\mathscr{O}_{X}(*S))$ as a rational function on $X$. Let $P$ and
$Z$ be the pole divisor and the zero divisor of $f$ on $X$, respectively, and
let ${P}_{\mathrm{red}}$ be the support of $P$.

  1. (i)

We call $X$ a non-degenerate compactification of $(U,f)$ if there exists a
neighborhood $V\subset X$ of $P$ such that $Z\cap V$ is smooth and $Z+S$ forms
a reduced normal crossing divisor on $V$.

  2. (ii)

We call $X$ a good compactification of $(U,f)$ if $f$ indeed defines a
morphism $f\colon X\to\mathbb{P}^{1}$.

If $X$ is non-degenerate, analytically locally at a point of $P$, there exists
a coordinate system

$\displaystyle\\{x_{1},\dots,x_{l},y_{1},\dots,y_{m},z_{1},\dots,z_{r}\\}$

such that

$\displaystyle S=(xy)\qquad\text{and}\qquad
f=\frac{1}{x^{e}}\qquad\text{or}\qquad f=\frac{z_{1}}{x^{e}}$ (2)

for some $e\in\mathbb{Z}_{>0}^{l}$. (Here and afterwards, we use the standard
multi-index convention.) If $X$ is good, the second case of $f$ in the above
expression does not occur.

For example, consider the case where $U$ is a complex torus and $f$ a Laurent
polynomial. Assume that $f$ is non-degenerate with respect to its Newton
polytope (for the definition, see [11, Section 4]). Then any toric smooth
compactification compatible with the Newton polytope is a non-degenerate
compactification of $(U,f)$ by [11, Proposition 4.3] or [5, Lemma 6.6]. The
non-degenerate compactification also appears in the considerations of
rescaling from a good compactification [9, Section 7.3], and of Fourier
transform [7, Section 4]. It is discussed in [5] where in this situation, the
author calls the meromorphic function $f$ on $X$ non-degenerate along $S$ [5,
Definition 2.6]. In this case, the author investigates the structures of the
twistor $\mathscr{D}$-module associated with the meromorphic connection
$(\mathscr{O}_{X}(*S),\mathrm{d}+\mathrm{d}f)$; it is shown in particular that
if $S$ equals the pole divisor of $f$, the resulting twistor
$\mathscr{D}$-module is pure [5, Lemma 2.10 and Corollary 3.12].

###  2.2 The filtered complexes

Fix a non-degenerate compactification $X$ of $(U,f)$ with boundary $S$. Regard
$f$ as a rational function on $X$ and let $P=f^{*}(\infty)$ be the pole
divisor with multiplicities. We have
$\mathrm{d}f\in\Gamma\big{(}X,\Omega_{X}^{1}(\log S)(P)\big{)}$.

  1. (i)

The twisted de Rham complex and the Higgs complex are the complexes

$\displaystyle(\Omega_{X}^{\bullet}(\log
S)(*P),\Theta)=\big{[}\mathscr{O}_{X}(*P)\\!\xrightarrow{\Theta}\\!\cdots\\!\xrightarrow{\Theta}\\!\Omega_{X}^{i}(\log
S)(*P)\\!\xrightarrow{\Theta}\\!\Omega_{X}^{i+1}(\log
S)(*P)\xrightarrow{\Theta}\\!\cdots\big{]},$

where $\Theta=\mathrm{d}+\mathrm{d}f$ and $\mathrm{d}f$, respectively. Here
$\mathrm{d}f$ sends a local section $\omega$ to $\mathrm{d}f\wedge\omega$.

  2. (ii)

We call the associated hypercohomology groups
$\mathbb{H}^{k}(X,(\Omega_{X}^{\bullet}(\log S)(*P),\Theta))$ the de Rham
cohomology and the Higgs cohomology of $(U,f)$, and denote respectively by

$\displaystyle\mathrm{H}_{\mathrm{dR}}^{k}(U,f)=\mathrm{H}^{k}(U,\mathrm{d}+\mathrm{d}f)\qquad\text{and}\qquad\mathrm{H}_{\mathrm{Hig}}^{k}(U,f)=\mathrm{H}^{k}(U,\mathrm{d}f).$

  3. (iii)

For an effective divisor $D$ on $X$ and $\mu\in\mathbb{Q}$, let

$\displaystyle\Omega_{X}^{i}(\log S)(\lfloor\mu
D\rfloor)_{+}=\begin{cases}\Omega_{X}^{i}(\log S)(\lfloor\mu
D\rfloor),&\text{if $\mu\geq 0$},\\\ 0,&\text{otherwise}.\end{cases}$

For $\Theta\in\\{\mathrm{d}+\mathrm{d}f,\mathrm{d}f\\},\lambda\in\mathbb{Q}$,
let

$\displaystyle
F_{X}^{\lambda}(\Theta)=F^{\lambda}(\Theta)=\big{[}\mathscr{O}_{X}(\lfloor-\lambda
P\rfloor)_{+}\xrightarrow{\Theta}\cdots\xrightarrow{\Theta}\Omega_{X}^{i}(\log
S)(\lfloor(i-\lambda)P\rfloor)_{+}\xrightarrow{\Theta}\cdots\big{]}.$

The subindex $X$ in $F_{X}^{\lambda}(\Theta)$ will be omitted if the base
variety is clear. The Hodge filtrations of the de Rham and the Higgs
cohomologies are

$\displaystyle
F^{\lambda}\mathrm{H}^{k}(U,\Theta)=\mathrm{Image}\big{\\{}\mathbb{H}^{k}(X,F^{\lambda}(\Theta))\to\mathrm{H}^{k}(U,\Theta)\big{\\}}$
(3)

induced by the inclusions of complexes.

  4. (iv)

For $\alpha\in\mathbb{Q}$, the Kontsevich sheaf of differential $p$-forms is
the coherent subsheaf of $\Omega_{X}^{p}(\log S)(*P)$

$\displaystyle\Omega_{f}^{p}(\alpha)=\ker\big{\\{}\Theta\colon\Omega_{X}^{p}(\log
S)(\lfloor\alpha P\rfloor)\to\Omega_{X}^{p+1}(*S)/\Omega_{X}^{p+1}(\log
S)(\lfloor\alpha P\rfloor)\big{\\}},$

and it forms the Kontsevich complex $(\Omega_{f}^{\bullet}(\alpha),\Theta)$
equipped with the filtration
$(\Omega_{f}^{\bullet}(\alpha),\Theta)_{\bullet\geq p}$ by direct truncation.
We simply write $\Omega_{f}^{p}$ for $\Omega_{f}^{p}(0)$.

######  Proposition 2.1.

  1. $\mathrm{(i)}$

For $\alpha\in\mathbb{Q}$ and $p\in\mathbb{Z}$, the $\mathscr{O}_{X}$-module
$\Omega_{f}^{p}(\alpha)$ is locally free of rank $\binom{\dim X}{p}$ with

$\displaystyle\Omega_{f}^{p}(\alpha)=\Omega_{f}^{p}\otimes_{\mathscr{O}_{X}}\mathscr{O}_{X}(\lfloor\alpha
P\rfloor).$

  2. $\mathrm{(ii)}$

For $\Theta\in\\{\mathrm{d}+\mathrm{d}f,\mathrm{d}f\\}$ the three inclusions

$\displaystyle F^{0}(\Theta)\to F^{-\alpha}(\Theta),\qquad 0\leq\alpha,$
$\displaystyle(\Omega_{f}^{\bullet}(\alpha),\Theta)\to
F^{-\alpha}(\Theta),\qquad 0\leq\alpha,$
$\displaystyle(\Omega_{f}^{\bullet}(\alpha),\Theta)_{\bullet\geq p}\to
F^{-\alpha+p}(\Theta),\qquad 0\leq\alpha<1$

are quasi-isomorphisms.

######  Proof 2.2.

Both statements are local properties for coherent sheaves on $X$ and we can
restrict to the coordinates such that (2) holds.

(i) In case $f=\frac{1}{x^{e}}$, the local freeness and the quasi-isomorphisms
are obtained in [1, equation (1.3.1)], [3, Lemma 2.12(a)] and [11, Proposition
1.3.], [1, Proposition 1.4.2], respectively. In fact, the methods are similar
to the arguments below.

Consider the second case $f=\frac{z_{1}}{x^{e}}$ so that

$\displaystyle
z_{1}\frac{\mathrm{d}f}{f}=x^{e}\mathrm{d}f=\mathrm{d}z_{1}-z_{1}\sum
e_{i}\frac{\mathrm{d}x_{i}}{x_{i}}.$

In this chart, the $\mathscr{O}_{X}$-module $\Omega_{X}^{p}(\log S)$ is freely
generated by

$\displaystyle
z_{1}\frac{\mathrm{d}f}{f}\wedge\bigwedge_{i=1}^{p-1}\xi_{i},\qquad\\{\xi_{i}\\}_{i=1}^{p-1}\subset\left\\{\mathrm{d}z_{2},\dots,\mathrm{d}z_{r},\frac{\mathrm{d}x_{1}}{x_{1}},\dots,\frac{\mathrm{d}x_{l}}{x_{l}},\frac{\mathrm{d}y_{1}}{y_{1}},\dots,\frac{\mathrm{d}y_{m}}{y_{m}}\right\\}$
(4)

and

$\displaystyle\bigwedge_{i=1}^{p}\eta_{i},\qquad\\{\eta_{i}\\}_{i=1}^{p}\subset\left\\{\mathrm{d}z_{2},\dots,\mathrm{d}z_{r},\frac{\mathrm{d}x_{1}}{x_{1}},\dots,\frac{\mathrm{d}x_{l}}{x_{l}},\frac{\mathrm{d}y_{1}}{y_{1}},\dots,\frac{\mathrm{d}y_{m}}{y_{m}}\right\\}.$
(5)

The $\mathscr{O}_{X}$-module $\Omega_{f}^{p}(\alpha)$ is indeed freely
generated by

$\displaystyle x^{-\lfloor\alpha e\rfloor}\omega_{1}\qquad\text{and}\qquad
x^{e-\lfloor\alpha e\rfloor}\omega_{2},$ (6)

where $\omega_{1}$ and $\omega_{2}$ run through elements in (4) and (5),
respectively.

(ii) Let $D$ be a divisor supported on ${P}_{\mathrm{red}}$, and $E$ an
irreducible component of ${P}_{\mathrm{red}}$. Let

$\displaystyle A^{p}_{k}(D)=\Omega^{p}(\log S)(D+pP+kE),\qquad
p,k\in\mathbb{Z},\alpha\in\mathbb{Q}.$

To obtain the quasi-isomorphisms, it suffices to show that the inclusion of
complexes

$\displaystyle(A_{k-1}^{\bullet}(D),\Theta)\to(A_{k}^{\bullet}(D),\Theta)$ (7)

is a quasi-isomorphism for any $k,\alpha$ and all choices of $D$ (cf. [11,
Proposition 1.2]). In fact, the first desired quasi-isomorphism then follows
immediately. The last two quasi-isomorphisms can be derived by a decreasing
induction (see the analogous statement [1, Proposition 1.4.2] in the case of
good compactification and its proof, which works for both
$\Theta=\mathrm{d}+\mathrm{d}f,\mathrm{d}f$).

To check the quasi-isomorphism (7), we may assume that $E=(x_{1})$ in the
local model. Consider local sections $h\in\mathscr{O}_{X}(D)$, and
$\omega_{1}$ and $\omega_{2}$ in (4) and (5), respectively. Then
$\big{\\{}h\cdot x^{-pe}x_{1}^{-k}\omega_{i}\big{\\}}$ generates
$A_{k}^{p}(D)$ and

$\displaystyle\Theta\colon\ $ $\displaystyle A_{k}^{p}(D)/A_{k-1}^{p}(D)\to
A_{k}^{p+1}(D)/A_{k-1}^{p+1}(D),$
$\displaystyle\frac{h}{x^{pe}x_{1}^{k}}\cdot\begin{cases}\omega_{1},\\\
\omega_{2}\end{cases}\mapsto\begin{cases}0,\\\
\frac{h}{x^{(p+1)e}x_{1}^{k}}(x^{e}\mathrm{d}f)\wedge\omega_{2}.\end{cases}$

Therefore the quotient
$\big{(}A_{k}^{p}(D)/A_{k-1}^{p}(D),\Theta\big{)}_{p\in\mathbb{Z}}$ of (7) is
exact.

In a more functorial way, one can consider the $\mathscr{D}$-module
$\mathscr{M}$ on $X$ attached to the meromorphic connection
$(\mathscr{O}_{X}(*S),\mathrm{d}+\mathrm{d}f)$ and define the irregular Hodge
filtration on $\mathscr{M}$ as given in [9, Definition 5.1]. By [9, Lemma
9.17], the filtered complex $F^{\lambda}(\mathrm{d}+\mathrm{d}f)$ defined
above is quasi-isomorphic to the filtered de Rham complex associated with the
filtered $\mathscr{D}$-module $\mathscr{M}$. When $X$ is a good
compactification, this is obtained in [1, Proposition 1.7.4]. Forgetting the
filtrations, the quasi-isomorphisms of various de Rham complexes of
$\mathscr{M}$ are also derived in [5, Lemmas 2.13 and 2.15] in the case where
$X$ is non-degenerate, including the compactly supported counterpart.

###  2.3 The independence

For $\diamond\in\\{\mathrm{dR},\mathrm{Hig}\\}$, the space
$\mathrm{H}_{\diamond}^{k}(U,f)$ with the Hodge filtration is independent of
the choice of the non-degenerate compactification of $(U,f)$. More precisely,
if $\pi\colon X^{\prime}\to X$ is a morphism between non-degenerate
compactifications extending the identity on $U$, then there is a natural
quasi-isomorphism

$\displaystyle\mathrm{R}\pi_{*}F^{\lambda}_{X^{\prime}}(\Theta)\simeq
F^{\lambda}_{X}(\Theta),\qquad\lambda\in\mathbb{Q}.$ (8)

######  Proof 2.3.

The assertion for good compactifications is proved in [11, Theorem 1.7], by
comparing the degree $p$ components $F_{X}^{\lambda}(\Theta)^{p}$ of
$F_{X}^{\lambda}(\Theta)$ on various $X$ for a fixed $p$ (and hence the proof
works for both $\Theta=\mathrm{d}+\mathrm{d}f$ and $\mathrm{d}f$).

In the following we show that by performing successively certain blowups, one
can replace a non-degenerate compactification $X$ (thus in the second case of
(2)) by a good one and compare the involved chain complexes (cf. [11, Section
4($b$)]). Let $\varpi\colon\widetilde{X}\to X$ be the blowup along the
intersection $\Xi$ of $Z=(z_{1})$ and the irreducible component $(x_{1})$ of
$P$ with multiplicity $e_{1}$. Let $E$ be the exceptional divisor,
$\widetilde{S}=\widetilde{X}\setminus U$ and $\widetilde{P}$ the pole divisor
of $\tilde{f}:=\varpi^{*}f$. We want to establish that
$\mathrm{R}\varpi_{*}F_{\widetilde{X}}^{\lambda}(\Theta)$ and
$F_{X}^{\lambda}(\Theta)$ are canonically quasi-isomorphic.

In case $e_{1}>1$, it can be proved similarly as [11, Proposition 4.4]. In
more details, write $\lambda=-\alpha+p$ where $0\leq\alpha<1$ and
$p\in\mathbb{Z}$. On $\widetilde{X}$ define the complex

$\displaystyle
R^{\lambda}(\Theta)=\big{(}\Omega_{\widetilde{X}}^{\bullet}\big{(}\log\widetilde{S}\big{)}(\lfloor(\alpha-p+\bullet)\big{(}\widetilde{P}+E)\rfloor\big{)}_{+},\Theta\big{)}.$

We have

$\displaystyle\varpi^{*}F_{X}^{\lambda}(\Theta),F_{\widetilde{X}}^{\lambda}(\Theta)\subset
R^{\lambda}(\Theta).$

By [11, Proposition 4.4(i)], each component of the complex
$R^{\lambda}(\Theta)/\varpi^{*}F_{X}^{\lambda}(\Theta)$ is a direct sum of
copies of $\mathscr{O}_{E/\Xi}(-1)$. Hence the adjunction
$F_{X}^{\lambda}(\Theta)\to\mathrm{R}\varpi_{*}R^{\lambda}(\Theta)$ is a
quasi-isomorphism. On the other hand, there are increasing complexes
$R^{\lambda}_{q}(\Theta)$ on $\widetilde{X}$ (those denoted by
$R_{\lambda}(q)$ in [11, equation (27)], which is a complex under either
$\mathrm{d}+\mathrm{d}f$ or $\mathrm{d}f$) with
$R^{\lambda}_{-1}(\Theta)=F^{\lambda}_{\widetilde{X}}(\Theta)$ and
$R^{\lambda}_{\dim\widetilde{X}-p}(\Theta)=R^{\lambda}(\Theta)$ such that
$R^{\lambda}_{q}(\Theta)/R^{\lambda}_{q-1}(\Theta)$ is quasi-isomorphic to the
complex consisting of a direct sum of copies of $\mathscr{O}_{E/\Xi}(-1)$
concentrated at degree $(p+q)$. In fact, in [11, Lemma 4.5], we further
introduce the complexes $(K_{\rho,\eta,\xi}^{\bullet},\mathrm{d}+\mathrm{d}f)$
and prove that the inclusion
$(K_{\rho,\eta,\xi}^{\bullet},\mathrm{d}+\mathrm{d}f)\subset(K_{\rho,\eta+1,\xi}^{\bullet},\mathrm{d}+\mathrm{d}f)$
is a quasi-isomorphism by showing the quotient complex is exact. Now one
notices that $K_{\rho,\eta,\xi}^{\bullet}$ is indeed also stable under
$\mathrm{d}f$; on the quotient
$K_{\rho,\eta+1,\xi}^{\bullet}/K_{\rho,\eta,\xi}^{\bullet}$, one has the
equality $\mathrm{d}+\mathrm{d}f=\mathrm{d}f$ of the differential maps under
the condition $e_{1}>1$. Therefore the proof of [11, Proposition 4.4(ii)]
describing $R^{\lambda}_{q}(\Theta)/R^{\lambda}_{q-1}(\Theta)$ goes through in
both cases $\Theta=\mathrm{d}+\mathrm{d}f$ and $\Theta=\mathrm{d}f$. Hence one
concludes that the inclusion $F_{\widetilde{X}}^{\lambda}(\Theta)\subset
R^{\lambda}(\Theta)$ induces a quasi-isomorphism
$\mathrm{R}\varpi_{*}F_{\widetilde{X}}^{\lambda}(\Theta)\to\mathrm{R}\varpi_{*}R^{\lambda}(\Theta)$
under the assumption $e_{1}>1$.

The following arguments work for any $e_{1}\geq 1$ and simplify those in [11,
Section 4($b$)]. (Cf., [3, Lemma 2.12(b)].) By Proposition 2.1, it suffices to
show that for any $0\leq\alpha<1$, $p\in\mathbb{Z}$, we have the inclusion
relation
$\varpi^{*}\Omega_{f}^{p}(\alpha)\subset\Omega_{\tilde{f}}^{p}(\alpha)$ on
$\widetilde{X}$ and it induces an isomorphism
$\Omega_{f}^{p}(\alpha)\to\mathrm{R}\varpi_{*}\Omega_{\tilde{f}}^{p}(\alpha)$
on $X$.

We first compute $\Omega_{\tilde{f}}^{p}(\alpha)$. Explicitly the blowup
$\widetilde{X}$ is defined by the equation

$\displaystyle\det\begin{pmatrix}x_{1}&z_{1}\\\ u&v\end{pmatrix}=0$

in $X\times\mathbb{P}^{1}$ where $[u:v]$ is the homogeneous coordinate on
$\mathbb{P}^{1}$. On the chart $v\neq 0$ with local coordinates

$\displaystyle\left\\{\bar{u}=\frac{u}{v},x_{2},\dots,x_{l},y_{1},\dots,y_{m},z_{1},\dots,z_{r}\right\\},$

the $\mathscr{O}_{\widetilde{X}}$-module $\Omega_{\tilde{f}}^{p}(\alpha)$ is
generated by the basis

$\displaystyle\delta\frac{\mathrm{d}\tilde{f}}{\tilde{f}}\wedge\bigwedge_{i=1}^{p-1}\xi_{i},\qquad\begin{array}[]{@{}l}\delta=z_{1}^{-\lfloor\alpha(e_{1}-1)\rfloor}\bar{u}^{-\lfloor\alpha
e_{1}\rfloor}x_{2}^{-\lfloor\alpha e_{2}\rfloor}\cdots x_{l}^{-\lfloor\alpha
e_{l}\rfloor},\\\
\displaystyle\\{\xi_{i}\\}_{i=1}^{p-1}\subset\left\\{\frac{\mathrm{d}z_{1}}{z_{1}}+\frac{\mathrm{d}\bar{u}}{\bar{u}},\frac{\mathrm{d}x_{2}}{x_{2}},\dots,\frac{\mathrm{d}x_{l}}{x_{l}},\frac{\mathrm{d}y_{1}}{y_{1}},\dots,\frac{\mathrm{d}y_{m}}{y_{m}},\mathrm{d}z_{2},\dots,\mathrm{d}z_{r}\right\\},\end{array}$
(11)

and

$\displaystyle\delta\frac{1}{\tilde{f}}\bigwedge_{i=1}^{p}\eta_{i},\qquad\\{\eta_{i}\\}_{i=1}^{p}\subset\left\\{\frac{\mathrm{d}z_{1}}{z_{1}}+\frac{\mathrm{d}\bar{u}}{\bar{u}},\frac{\mathrm{d}x_{2}}{x_{2}},\dots,\frac{\mathrm{d}x_{l}}{x_{l}},\frac{\mathrm{d}y_{1}}{y_{1}},\dots,\frac{\mathrm{d}y_{m}}{y_{m}},\mathrm{d}z_{2},\dots,\mathrm{d}z_{r}\right\\}.$
(12)

On the chart $u\neq 0$ with local coordinates

$\displaystyle\left\\{x_{1},\dots,x_{l},y_{1},\dots,y_{m},\bar{v}=\frac{v}{u},z_{2},\dots,z_{r}\right\\},$

the sheaf $\Omega_{\tilde{f}}^{p}(\alpha)$ is generated by the basis

$\displaystyle\varepsilon\bar{v}\frac{\mathrm{d}\tilde{f}}{\tilde{f}}\wedge\bigwedge_{i=1}^{p-1}\xi_{i},\qquad\begin{array}[]{@{}l}\varepsilon=x_{1}^{-\lfloor\alpha(e_{1}-1)\rfloor}x_{2}^{-\lfloor\alpha
e_{2}\rfloor}\cdots x_{l}^{-\lfloor\alpha e_{l}\rfloor},\\\
\displaystyle\\{\xi_{i}\\}_{i=1}^{p-1}\subset\left\\{\frac{\mathrm{d}x_{1}}{x_{1}},\dots,\frac{\mathrm{d}x_{l}}{x_{l}},\frac{\mathrm{d}y_{1}}{y_{1}},\dots,\frac{\mathrm{d}y_{m}}{y_{m}},\mathrm{d}z_{2},\dots,\mathrm{d}z_{r}\right\\},\end{array}$
(15)

and

$\displaystyle\varepsilon\frac{\bar{v}}{\tilde{f}}\bigwedge_{i=1}^{p}\eta_{i},\qquad\\{\eta_{i}\\}_{i=1}^{p}\subset\left\\{\frac{\mathrm{d}x_{1}}{x_{1}},\dots,\frac{\mathrm{d}x_{l}}{x_{l}},\frac{\mathrm{d}y_{1}}{y_{1}},\dots,\frac{\mathrm{d}y_{m}}{y_{m}},\mathrm{d}z_{2},\dots,\mathrm{d}z_{r}\right\\}.$
(16)

On the intersection $u,v\neq 0$, one has
$\frac{\mathrm{d}x_{1}}{x_{1}}=\frac{\mathrm{d}z_{1}}{z_{1}}+\frac{\mathrm{d}\bar{u}}{\bar{u}}$.

Using the basis (6) of $\Omega_{f}^{p}(\alpha)$, a direct computation reveals
that $\varpi^{*}\Omega_{f}^{p}(\alpha)$ is contained in
$\Omega_{\tilde{f}}^{p}(\alpha)$. Moreover, the $\mathscr{O}_{E}$-module
$\Omega_{\tilde{f}}^{p}(\alpha)/\varpi^{*}\Omega_{f}^{p}(\alpha)$ equals
either zero if $\lfloor\alpha(e_{1}-1)\rfloor\neq\lfloor\alpha e_{1}\rfloor$
or otherwise $\binom{l+m+r}{p}$ copies of $\mathscr{O}_{E/\Xi}(-1)$ generated
by (11), (12) and (15), (16) on the two charts, respectively. Therefore we
obtain that $F_{X}^{\lambda}(\Theta)$ and
$\mathrm{R}\varpi_{*}F_{\widetilde{X}}^{\lambda}(\Theta)$ are naturally quasi-
isomorphic.

One then iteratively takes the blowups along the intersections of irreducible
components of the zero and the pole divisors as in [11, Section 4$(b)$] (the
diagram (26) therein) to obtain a good compactification $X^{\prime}$ of
$(U,f)$ from the non-degenerate $X$ with the canonical quasi-isomorphism (8).

  1. (i)

By the $E_{1}$-degeneration [1, Theorem 1.2.2], [4, Theorem 1.1] on a good
compactification (see also [3, Theorem 2.18]), the above proposition implies
that the arrow in (3) is injective for any non-degenerate $X$ and indices
$k,\lambda$.

  2. (ii)

For $X$ non-degenerate, we have
$\mathrm{H}_{\mathrm{dR}}^{k}(U,f)=\mathbb{H}^{k}(X,(\Omega_{X}^{\bullet}(*S),\mathrm{d}+\mathrm{d}f))$
by the arguments of [11, Corollary 1.4]. On the other hand,
$\mathrm{H}_{\mathrm{Hig}}^{k}(U,f)\neq\mathbb{H}^{k}(X,(\Omega_{X}^{\bullet}(*S),\mathrm{d}f))$
in general which can be seen by considering the case $U$ affine and $f=0$.

##  3 The proof of the main result

We begin with two pairs $(U_{i},f_{i})$, $i=1,2$, and their product $(U,f)$.
Fix good compactifications $X_{i}$ of $(U_{i},f_{i})$ such that
$S_{i}:=X_{i}\setminus U_{i}$ are strict normal crossing divisors. The proof
of Theorem 1.1 consists of two steps. In the first step Section 3.1, we
construct explicitly a non-degenerate compactification $X$ of $(U,f)$ from
$X_{1}\times X_{2}$ by successive blowups. In step two Section 3.2, we compare
the filtered de Rham or the Higgs complex on $X$, which gives the Hodge
filtration on $\mathrm{H}_{\diamond}^{k}(U,f)$, with a certain filtered
complex on $X_{1}\times X_{2}$ that gives the product filtration using the
explicit construction of $X$.

###  3.1 An explicit elimination

For each $i=1,2$, take an open covering of $X_{i}$ with a system of local
coordinates

$\displaystyle\\{x_{i,1},\dots,x_{i,l_{i}},x_{i,l_{i}+1},\dots,x_{i,l_{i}+m_{i}},x_{i,l_{i}+m_{i}+1},\dots,x_{i,l_{i}+m_{i}+r_{i}}\\}$

such that

$\displaystyle f_{i}=\frac{1}{x_{i,1}^{e_{i,1}}\cdots
x_{i,l_{i}}^{e_{i,l_{i}}}},\quad e_{i,j}>0,\qquad S_{i}=(x_{i,1}\cdots
x_{i,l_{i}+m_{i}}).$

As for the initial data in the inductive construction, we consider the
compactification $X_{1}\times X_{2}$ of $U$ with the systems of local
coordinates $\\{x_{i,j}\\}$.

Suppose we have constructed a compactification $Y$ of $U$ and its systems of
local coordinates

$\displaystyle\\{y_{1},\dots,y_{l},y_{l+1},\dots,y_{l+m},y_{l+m+1},\dots,y_{l+m+r}\\},$
(17)

together with a birational map $\pi\colon Y\to X_{1}\times X_{2}$ such that

$\displaystyle\pi^{*}f_{1}=\frac{1}{y_{1}^{a_{1}}\cdots
y_{l}^{a_{l}}},\qquad\pi^{*}f_{2}=\frac{1}{y_{1}^{b_{1}}\cdots
y_{l}^{b_{l}}},\qquad Y\setminus U=(y_{1}\cdots y_{l+m})$ (18)

for some $a_{i},b_{i}\geq 0$ with $b_{i}>0$ if $a_{i}=0$. Let $T$ be the
boundary divisor $Y\setminus U$. For a pair of irreducible components $D_{1}$,
$D_{2}$ of $T$, set

$\displaystyle\Delta_{Y}(D_{1},D_{2})=\begin{cases}\displaystyle\prod_{i=1}^{2}\big{(}\operatorname{ord}_{D_{i}}(\pi^{*}f_{1})-\operatorname{ord}_{D_{i}}(\pi^{*}f_{2})\big{)},&\text{if
$D_{1}\neq D_{2},D_{1}\cap D_{2}\neq\varnothing$},\\\
0,&\text{otherwise}.\end{cases}$

Notice that if $\Delta_{Y}(D_{1},D_{2})\geq 0$ for any pair $(D_{1},D_{2})$,
i.e., $a\geq b$ or $b\geq a$ in terms of the systems of local coordinates as
above, then the zero divisor of $\pi^{*}f$ is smooth in a neighborhood of $T$
and intersects $T$ transversally. That is, $Y$ is a non-degenerate
compactification of $(U,f)$ in this case.

Otherwise, pick a pair $(D_{1},D_{2})$ such that $\Delta_{Y}(D_{1},D_{2})<0$
and is the smallest among all possible values of $\Delta_{Y}$. Let
$\widetilde{Y}$ be the blowup of $Y$ along $D_{1}\cap D_{2}$ and
$\tilde{\pi}\colon\widetilde{Y}\to X_{1}\times X_{2}$ the induced map. Then
$\widetilde{T}:=\widetilde{Y}\setminus U$ consists of the exceptional divisor
$E$ and $\\{\widetilde{D}\\}$ where $\widetilde{D}$ denotes the proper
transform of an irreducible component $D$ of $T$. To construct the explicit
systems of local coordinates of $\widetilde{Y}$, we may assume that
$D_{i}=(y_{i})$ for $i=1,2$ with $a_{1}>b_{1}$ and $a_{2}<b_{2}$ after
rearrangement. The blowup is defined by the equation $y_{1}v=y_{2}u$ where
$[u:v]$ is the homogeneous coordinate of $\mathbb{P}^{1}$. Over this chart of
coordinates of $Y$, we add two charts to $\widetilde{Y}$ (and away from the
blowup center, we pass the charts of $Y$ to $\widetilde{Y}$). In the chart
$v\neq 0$ of $\mathbb{P}^{1}$, we consider the local coordinates

$\displaystyle\left\\{\bar{u}:=\frac{u}{v},y_{2},y_{3},\dots,y_{l+m+r}\right\\}.$
(19)

Then

$\displaystyle\tilde{\pi}^{*}f_{1}=\frac{1}{\bar{u}^{a_{1}}y_{2}^{a_{1}+a_{2}}y_{3}^{a_{3}}\cdots
y_{l}^{a_{l}}},\qquad\tilde{\pi}^{*}f_{2}=\frac{1}{\bar{u}^{b_{1}}y_{2}^{b_{1}+b_{2}}y_{3}^{b_{3}}\cdots
y_{l}^{b_{l}}},$ (20) $\displaystyle\widetilde{T}=(\bar{u}y_{2}\cdots
y_{l+m}),\qquad\widetilde{D}_{1}=(\bar{u}),\qquad E=(y_{2}).$

In the chart $u\neq 0$ of $\mathbb{P}^{1}$, we consider the local coordinates

$\displaystyle\left\\{\bar{v}:=\frac{v}{u},y_{1},y_{3},\dots,y_{l+m+r}\right\\}.$
(21)

Then

$\displaystyle\tilde{\pi}^{*}f_{1}=\frac{1}{\bar{v}^{a_{2}}y_{1}^{a_{1}+a_{2}}y_{3}^{a_{3}}\cdots
y_{l}^{a_{l}}},\qquad\tilde{\pi}^{*}f_{2}=\frac{1}{\bar{v}^{b_{2}}y_{1}^{b_{1}+b_{2}}y_{3}^{b_{3}}\cdots
y_{l}^{b_{l}}},$ (22) $\displaystyle\widetilde{T}=(\bar{v}y_{1}y_{3}\cdots
y_{l+m}),\qquad\widetilde{D}_{2}=(\bar{v}),\qquad E=(y_{1}).$

One has $\widetilde{D}_{1}\cap\widetilde{D}_{2}=\varnothing$ and

$\displaystyle\Delta_{\widetilde{Y}}(\widetilde{D},\widetilde{D}^{\prime})=\Delta_{Y}(D,D^{\prime})\qquad\text{for
$(D,D^{\prime})\neq(D_{1},D_{2}),(D_{2},D_{1})$},$
$\displaystyle\Delta_{\widetilde{Y}}(\widetilde{D}_{i},E)=\Delta_{Y}(D_{1},D_{2})+\big{(}\operatorname{ord}_{D_{i}}(\pi^{*}f_{1})-\operatorname{ord}_{D_{i}}(\pi^{*}f_{2})\big{)}^{2}>\Delta_{Y}(D_{1},D_{2}),$
$\displaystyle\Delta_{\widetilde{Y}}(\widetilde{D},E)=\Delta_{Y}(D,D_{1})+\Delta_{Y}(D,D_{2}).$

Notice that

$\displaystyle\Delta_{Y}(D,D_{i})=\left(\operatorname{ord}_{D}(\pi^{*}f_{1})-\operatorname{ord}_{D}(\pi^{*}f_{2})\right)\cdot(-a_{i}+b_{i})$

and consequently

$\displaystyle\Delta_{\widetilde{Y}}(\widetilde{D},E)\
\begin{cases}>\min\\{\Delta_{Y}(D,D_{1}),\Delta_{Y}(D,D_{2})\\}&\text{if
$\operatorname{ord}_{D}(\pi^{*}f_{1})\neq\operatorname{ord}_{D}(\pi^{*}f_{2})$},\\\
=0&\text{if
$\operatorname{ord}_{D}(\pi^{*}f_{1})=\operatorname{ord}_{D}(\pi^{*}f_{2})$}.\end{cases}$

We then replace $Y$ with its systems of local coordinates by $\widetilde{Y}$
with the coordinates constructed above.

Observe that after a finite number of blowups in this procedure, the smallest
possible value of the function $\Delta$, if it is negative on $Y$, strictly
increases. Hence repeating this construction, it produces a non-degenerate
compactification $X$ of $(U,f)$ obtained by a sequence $X\to\cdots\to
X_{1}\times X_{2}$ of explicit blowups.

###  3.2 Relations between complexes

We shall put a filtered complex on each step of the sequence of blowups
constructed in Section 3.1 and compare them under push-forwards.

Consider a birational map $\pi\colon Y\to X_{1}\times X_{2}$ and let
$P_{Y,f_{j}}$ be the pole divisor of the pullback of $f_{j}$ on $Y$. For any
$\lambda\in\mathbb{Q}$, consider the subsheaf $\mathscr{O}_{Y}^{(\lambda)}$ of
$\mathscr{O}_{Y}(*(P_{Y,f_{1}}+P_{Y,f_{2}}))$ given by

$\displaystyle\mathscr{O}_{Y}^{(\lambda)}=\sum_{0\leq\theta\leq
1}\mathscr{O}_{Y}\left(\lfloor\lambda((1-\theta)P_{Y,f_{1}}+\theta
P_{Y,f_{2}})\rfloor\right),$

which is coherent but not locally free in general. Let
$T=Y\setminus\pi^{-1}(U)$ and

$\displaystyle\Omega_{Y}^{p,(\lambda)}=\mathscr{O}_{Y}^{(\lambda)}\otimes_{\mathscr{O}_{Y}}\Omega_{Y}^{p}(\log
T).$

For $\Theta\in\\{\mathrm{d}+\mathrm{d}f,\mathrm{d}f\\}$, consider the complex
on $Y$

$\displaystyle
F_{Y}^{(\lambda)}(\Theta)=\big{[}\mathscr{O}_{Y}^{(-\lambda)_{+}}\xrightarrow{\Theta}\Omega_{Y}^{1,(1-\lambda)_{+}}\xrightarrow{\Theta}\Omega_{Y}^{2,(2-\lambda)_{+}}\xrightarrow{\Theta}\cdots\big{]},$

where

$\displaystyle\Omega_{Y}^{p,(p-\lambda)_{+}}=\begin{cases}\Omega_{Y}^{p,(p-\lambda)},&\text{if
$p-\lambda\geq 0$},\\\ 0,&\text{otherwise}.\end{cases}$

(It is indeed a sub-complex of $(\Omega_{Y}^{\bullet}(*T),\Theta)$.)

If $Y$ is a non-degenerate compactification of $(U,f)$, e.g., $Y$ equals the
iterated blowup $X$ constructed in the previous subsection, then
$F_{Y}^{(\lambda)}(\Theta)$ is indeed the filtration defining the desired
Hodge filtration. The following two lemmas describe the situations in the
initial step $Y=X_{1}\times X_{2}$ and in each blowup
$\varpi\colon\widetilde{Y}\to Y$ appeared in the sequence occurred in Section
3.1, respectively.

Consider $Y=X_{1}\times X_{2}$ and let $F_{\boxtimes}^{\lambda}(\Theta)$ be
the product filtration of $F_{X_{i}}^{\lambda}(\Theta_{i})$ whose $p$-th
component is

$\displaystyle
F_{\boxtimes}^{\lambda}(\Theta)^{p}:=\sum_{a\in\mathbb{Q},q\in\mathbb{Z}}F_{X_{1}}^{a}(\Theta_{1})^{q}\boxtimes
F_{X_{2}}^{\lambda-a}(\Theta_{2})^{p-q}$

inside $\Omega_{Y}^{p}(*T)$. $($Here $\Theta_{i}=\mathrm{d}+\mathrm{d}f_{i}$
and $\mathrm{d}f_{i}$ if $\Theta=\mathrm{d}+\mathrm{d}f$ and $\mathrm{d}f$,
respectively.$)$

  1. $\mathrm{(i)}$

The filtration $F_{Y}^{(\lambda)}(\Theta)$ coincides with
$F_{\boxtimes}^{\lambda}(\Theta)$.

  2. $\mathrm{(ii)}$

For all $\lambda\geq\rho$, the induced map

$\displaystyle\mathbb{H}^{k}\big{(}Y,F_{\boxtimes}^{\lambda}(\Theta)\big{)}\to\mathbb{H}^{k}\big{(}Y,F_{\boxtimes}^{\rho}(\Theta)\big{)}$

is injective; it is an isomorphism if $\lambda\leq 0$. One has

$\displaystyle\mathbb{H}^{k}\big{(}Y,F_{\boxtimes}^{\lambda}(\Theta)\big{)}=\bigoplus_{i+j=k}\left(\sum_{a+b=\lambda}\mathbb{H}^{i}\big{(}X_{1},F_{X_{1}}^{a}(\Theta_{1})\big{)}\otimes\mathbb{H}^{j}\big{(}X_{2},F^{b}_{X_{2}}(\Theta_{2})\big{)}\right)$
(23)

for any $\lambda$ where the inner sum is taken inside the vector space
$\mathrm{H}^{i}(U_{1},\Theta_{1})\otimes\mathrm{H}^{j}(U_{2},\Theta_{2})$.

######  Proof 3.1.

(i) Indeed the inclusion $F_{\boxtimes}^{\lambda}(\Theta)\subset
F_{Y}^{(\lambda)}(\Theta)$ is clear. Conversely suppose that $\mu:=p-\lambda$
$\geq 0$. The $p$-th degree component $\Omega_{Y}^{p,(\mu)}$ of
$F_{Y}^{(\lambda)}(\Theta)$ is generated by products of elements in

$\displaystyle\Omega_{X_{1}}^{q}(\log S_{1})(\lfloor(1-\theta)\mu
P_{X_{1}}\rfloor)\qquad\text{and}\qquad\Omega_{X_{2}}^{p-q}(\log
S_{2})(\lfloor\theta\mu P_{X_{2}}\rfloor),$

which are the $q$-th and the $(p-q)$-th components of
$F_{X_{1}}^{q-(1-\theta)\mu}(\Theta_{1})$ and
$F_{X_{2}}^{p-q-\theta\mu}(\Theta_{2})$, respectively. The latter two
contribute to $F_{\boxtimes}^{\lambda}(\Theta)$.

(ii) We have the natural external products

$\displaystyle F_{X_{1}}^{a}(\Theta_{1})\boxtimes F_{X_{2}}^{b}(\Theta_{2})\to
F_{\boxtimes}^{\lambda}(\Theta)$

for all $a+b=\lambda$. Using Čech resolution or representatives in smooth
forms, one obtains the cup product

$\displaystyle\mathbb{H}^{i}\big{(}X_{1},F_{X_{1}}^{a}(\Theta_{1})\big{)}\otimes\mathbb{H}^{j}\big{(}X_{2},F^{b}_{X_{2}}(\Theta_{2})\big{)}\to\mathbb{H}^{i+j}\big{(}Y,F_{X_{1}}^{a}(\Theta_{1})\boxtimes
F_{X_{2}}^{b}(\Theta_{2})\big{)}.$

On the other hand, one has from the definition that

$\displaystyle\operatorname{Gr}_{F_{\boxtimes}(\Theta)}^{\lambda}=\bigoplus_{a+b=\lambda}\operatorname{Gr}_{F_{X_{1}}(\Theta_{1})}^{a}\boxtimes\operatorname{Gr}_{F_{X_{2}}(\Theta_{2})}^{b}.$

Again there is the cup product

$\displaystyle\bigoplus_{i+j=k}\mathbb{H}^{i}\big{(}X_{1},\operatorname{Gr}_{F_{X_{1}}(\Theta_{1})}^{a}\big{)}\otimes\mathbb{H}^{j}\big{(}X_{2},\operatorname{Gr}_{F_{X_{2}}(\Theta_{2})}^{b}\big{)}\to\mathbb{H}^{k}\big{(}Y,\operatorname{Gr}_{F_{X_{1}}(\Theta_{1})}^{a}\boxtimes\operatorname{Gr}_{F_{X_{2}}(\Theta_{2})}^{b}\big{)}.$
(24)

To complete the assertions, it suffices to show that the arrow above is an
isomorphism. Indeed, denote the sum inside the big round brackets in the right
side of (23) by $\Phi(i,j,\lambda)$. The $E_{1}$-degeneration of the spectral
sequence attached to each filtered complex $F_{X_{i}}(\Theta_{i})$ implies
that there is the natural exact sequence

$\displaystyle
0\to\bigcup_{\mu>\lambda}\Phi(i,j,\mu)\to\Phi(i,j,\lambda)\to\bigoplus_{a+b=\lambda}\mathbb{H}^{i}\big{(}X_{1},\operatorname{Gr}_{F_{X_{1}}(\Theta_{1})}^{a}\big{)}\otimes\mathbb{H}^{j}\big{(}X_{2},\operatorname{Gr}_{F_{X_{2}}(\Theta_{2})}^{b}\big{)}\to
0.$

Together with the isomorphism of (24), a decreasing induction on the index
$\lambda$ in (23) then gives the desired statements. Finally the isomorphism
of (24) can be obtained by directly truncating the involved complexes and
inductively using the Künneth formula for coherent sheaves [10, Theorem 1].

Let $\varpi\colon\widetilde{Y}\to Y$ be the blowup constructed in Section 3.1.
Then the pullback $\varpi^{*}F_{Y}^{(\lambda)}(\Theta)$ is a sub-complex of
$F_{\widetilde{Y}}^{(\lambda)}(\Theta)$ and the following hold.

  1. $\mathrm{(i)}$

Each component
$F_{\widetilde{Y}}^{(\lambda)}(\Theta)^{p}/\varpi^{*}F_{Y}^{(\lambda)}(\Theta)^{p}$
of the quotient is supported on the exceptional divisor $E$ and is a direct
sum of copies of relative $\mathscr{O}(-1)$ of the $\mathbb{P}^{1}$-bundle $E$
over the center of the blowup $\varpi$.

  2. $\mathrm{(ii)}$

We have the canonical quasi-isomorphism
$F_{Y}^{(\lambda)}(\Theta)\to\mathrm{R}\varpi_{*}F_{\widetilde{Y}}^{(\lambda)}(\Theta)$.

######  Proof 3.2.

(i) Let $\widetilde{T}=\widetilde{Y}\setminus U$. As in the proof of the
previous lemma, assume that $\mu:=p-\lambda\geq 0$. Since
$\varpi^{*}\Omega_{Y}^{p}(\log
T)=\Omega_{\widetilde{Y}}^{p}(\log\widetilde{T})$, we only need to consider
the difference between $\mathscr{O}_{\widetilde{Y}}^{(\mu)}$ and
$\varpi^{*}\mathscr{O}_{Y}^{(\mu)}$. We use the local coordinates (17) with
the properties (18). Then $\mathscr{O}_{Y}^{(\mu)}$ is generated by various

$\displaystyle g:=\frac{1}{y^{\lfloor c\rfloor}},$ (25)

where $c=(1-\theta)\mu a+\theta\mu b$ for $0\leq\theta\leq 1$. On the other
hand, $\mathscr{O}_{\widetilde{Y}}^{(\mu)}$ is generated by various

$\displaystyle h_{\bar{u}}:=\frac{1}{\bar{u}^{\lfloor
c_{1}\rfloor}y_{2}^{\lfloor c_{1}+c_{2}\rfloor}y_{3}^{\lfloor
c_{3}\rfloor}\cdots y_{l}^{\lfloor c_{l}\rfloor}}\qquad\text{and}\qquad
h_{\bar{v}}:=\frac{1}{\bar{v}^{\lfloor c_{2}\rfloor}y_{1}^{\lfloor
c_{1}+c_{2}\rfloor}y_{3}^{\lfloor c_{3}\rfloor}\cdots y_{l}^{\lfloor
c_{l}\rfloor}}$

on the two charts (19) and (21) satisfying (20) and (22), respectively. We
have

$\displaystyle\left\langle
h_{\bar{u}}\right\rangle_{\mathscr{O}_{\widetilde{Y}}}/\varpi^{*}\left\langle
g\right\rangle_{\mathscr{O}_{Y}}=\begin{cases}\left\langle
h_{\bar{u}}\right\rangle_{\mathscr{O}_{E}},&\text{if $\lfloor
c_{1}+c_{2}\rfloor=\lfloor c_{1}\rfloor+\lfloor c_{2}\rfloor+1$},\\\
0,&\text{if $\lfloor c_{1}+c_{2}\rfloor=\lfloor c_{1}\rfloor+\lfloor
c_{2}\rfloor$},\end{cases}$

and similarly on the chart $u\neq 0$. Observe on the intersection that
$h_{\bar{u}}=\bar{v}^{-1}h_{\bar{v}}$ if $\lfloor
c_{1}+c_{2}\rfloor\neq\lfloor c_{1}\rfloor+\lfloor c_{2}\rfloor$. Thus
$\mathscr{O}_{\widetilde{Y}}^{(\mu)}/\varpi^{*}\mathscr{O}_{Y}^{(\mu)}$ is a
direct sum of copies of the relative $\mathscr{O}(-1)$.

(ii) There are only finitely many $y^{-\lfloor c\rfloor}$ occurred in the
generators in (25); call the appeared (distinct) monomials
$\xi_{1},\dots,\xi_{k}$ ordered by the increment of the corresponding
parameter $\theta$. Consider locally the filtration
$M_{i}=\left\langle\xi_{1},\dots,\xi_{i}\right\rangle$ of
$\mathscr{O}_{Y}$-submodules of $\mathscr{O}_{Y}^{(\mu)}$. On this chart one
has the short exact sequence

$\displaystyle 0\to N_{i+1}\to M_{i}\oplus\mathscr{O}_{Y}\cdot\xi_{i+1}$
$\displaystyle\to M_{i+1}\to 0,$ $\displaystyle(\omega,\eta)$
$\displaystyle\mapsto\omega-\eta.$

Here if $\xi_{i}=\prod y_{j}^{-c_{i,j}}$, then $c_{i,j}$ is monotone as a
function of $i$ and $N_{i+1}=M_{i}\cap\mathscr{O}_{Y}\xi_{i+1}$ is the
invertible sheaf generated by

$\displaystyle\prod y_{j}^{-\min\\{c_{i,j},c_{i+1,j}\\}}$

(with diagonal embedding to $M_{i}\oplus\mathscr{O}_{Y}\xi_{i+1}$). Applying
inductively the projection formula (e.g., [2, Exercise III.8.3]) to the
locally free $\mathscr{O}_{Y}\xi_{i+1}$ and $N_{i}$, one obtains that

$\displaystyle\mathrm{R}\varpi_{*}\varpi^{*}M_{i}=\mathrm{R}\varpi_{*}\big{(}\mathscr{O}_{\widetilde{Y}}\otimes\varpi^{*}M_{i}\big{)}=\mathrm{R}\varpi_{*}\mathscr{O}_{\widetilde{Y}}\otimes
M_{i}=M_{i},$

since $\mathrm{R}\varpi_{*}\mathscr{O}_{\widetilde{Y}}=\mathscr{O}_{Y}$ for
the birational morphism $\varpi$. Together with the computation in (i), the
assertion follows.

The proof of the main Theorem 1.1 is now completed by the above two lemmas.

##  4 The Brieskorn lattice

We indicate that the Künneth formula for $\mathrm{H}_{\mathrm{dR}}^{k}(U,f)$
and $\mathrm{H}_{\mathrm{Hig}}^{k}(U,f)$ can be put together into a family
version (cf. [9, Sections 1.3 ans 6.2], [3, Section 3.2] and [4]).

Consider the affine line $\mathbb{A}^{1}_{u}$ with a fixed coordinate $u$.

Fix a pair $(U,f)$ and a non-degenerate compactification $X$. Let $\pi\colon
X\times\mathbb{A}^{1}_{u}\to\mathbb{A}^{1}_{u}$ be the projection. The $k$-th
Brieskorn lattice of $(U,f)$ (cf. [9, Section 6.1]) is the coherent sheaf on
$\mathbb{A}^{1}_{u}$

$\displaystyle\mathscr{G}^{k}(U,f):=\mathrm{R}^{k}\pi_{*}\big{(}\Omega_{f}^{q}(\alpha)\boxtimes\mathscr{O}_{\mathbb{A}^{1}_{u}},u\cdot\mathrm{d}_{X}+\mathrm{d}f\big{)}_{q\geq
0}.$

Here $\mathrm{d}_{X}$ is the derivative with respective to the component $X$
only.

Let

$\displaystyle
F^{-\alpha+p}\mathscr{G}^{k}(U,f)=\mathrm{R}^{k}\pi_{*}\big{(}\Omega_{f}^{q}(\alpha)\boxtimes\mathscr{O}_{\mathbb{A}^{1}_{u}},u\cdot\mathrm{d}_{X}+\mathrm{d}f\big{)}_{q\geq
p},\qquad 0\leq\alpha<1.$

According to the quasi-isomorphisms in Proposition 2.1(ii) and the
$E_{1}$-degeneration

$\displaystyle\dim\mathrm{H}^{k}(U,\Theta)=\sum_{q}\dim\mathrm{H}^{k-q}\big{(}X,\Omega_{f}^{q}(\alpha)\big{)},\qquad\Theta\in\\{\mathrm{d}+\mathrm{d}f,\mathrm{d}f\\},$

we have that

  * •

the $\mathscr{O}_{\mathbb{A}^{1}_{u}}$-module $\mathscr{G}^{k}(U,f)$ is free
and independent of the choice of $\alpha$, and

  * •

the canonical maps $F^{-\alpha+p}\mathscr{G}^{k}(U,f)\to\mathscr{G}^{k}(U,f)$
indeed define a filtration by free subsheaves of $\mathscr{G}^{k}(U,f)$ with
free quotients whose fibers are

$\displaystyle
F^{-\alpha+p}\mathscr{G}^{k}(U,f)_{|{u=c}}=\begin{cases}F^{-\alpha+p}\mathrm{H}_{\mathrm{Hig}}^{k}(U,f),&c=0,\\\
F^{-\alpha+p}\mathrm{H}_{\mathrm{dR}}^{k}(U,f/c),&c\neq 0\end{cases}$ (26)

under the base-change map (cf. the arguments in the proof of [1, Proposition
1.5.1]).

Now consider as in Section 3 two pairs $(U_{i},f_{i})$ and their product
$(U,f)$. We again have the natural map

$\displaystyle\bigoplus_{0\leq i\leq
k}\mathscr{G}^{i}(U_{1},f_{1})\otimes_{\mathscr{O}_{\mathbb{A}^{1}_{u}}}\mathscr{G}^{k-i}(U_{2},f_{2})\to\mathscr{G}^{k}(U,f)$

obtained by cup product. Similar to the proof of [1, Proposition 1.5.1], the
fiber-wise Künneth formula Theorem 1.1 shows that the above map is an
isomorphism strictly compatible with the filtrations.

On the other hand, fix $0\leq\alpha<1$. One can naturally complete
$\big{(}\mathscr{G}^{k}(U,f),F^{-\alpha+p}\big{)}_{p\in\mathbb{Z}}$ into a
filtered bundle on $\mathbb{P}^{1}$ by adding the filtered cohomology space

$\displaystyle
F^{p}\mathrm{H}_{\mathrm{d},\alpha}^{k}(U,f):=\mathrm{Image}\big{\\{}\mathbb{H}^{k}\big{(}X,(\Omega_{f}^{\bullet}(\alpha),\mathrm{d})_{\bullet\geq
p}\big{)}\to\mathbb{H}^{k}\big{(}X,(\Omega_{f}^{\bullet}(\alpha),\mathrm{d})\big{)}\big{\\}}$

as the fiber over $u=\infty$. The resulting sheaf on $\mathbb{P}^{1}$ is
called the Kontsevich bundle and denoted by $\mathscr{K}_{\alpha}^{k}(U,f)$.
The filtered space $\mathrm{H}_{\mathrm{d},\alpha}^{k}(U,f)$ depends on
$\alpha$ but does not depend on the choice of the non-degenerate
compactification $X$ since in fact ([9, Theorem 1.11(a), Section 1.3], [4,
Theorem 1.2(i,ii)])

  1. $\mathrm{(i)}$

the bundle $\mathscr{K}_{\alpha}^{k}(U,f)$ can be obtained (as a Deligne
extension) by using a natural algebraic connection on $\mathscr{G}^{k}(U,f)$
(see [9, Lemma 6.2] with the aid of Proposition 2.1(ii) in the case of non-
degenerate compactification $X$) with regular singularity at $u=\infty$, and

  2. $\mathrm{(ii)}$

under the base-change, one has

$\displaystyle\big{(}\mathrm{H}_{\mathrm{d},\alpha}^{k}(U,f),F^{\bullet}\big{)}=\big{(}\mathscr{K}_{\alpha}^{k}(U,f),\mathrm{HN}^{\bullet}\big{)}_{|{u=\infty}},$
(27)

where $\mathrm{HN}^{p}\mathscr{K}^{k}_{\alpha}(U,f)$ is the Harder–Narasimhan
filtration on the locally free sheaf $\mathscr{K}^{k}_{\alpha}(U,f)$
normalized with $\operatorname{Gr}_{\mathrm{HN}}^{p}$ isomorphic to a direct
sum of copies of $\mathscr{O}(p)$ on $\mathbb{P}^{1}$.

In fact, we also have

$\displaystyle
F^{-\alpha+\bullet}\mathscr{G}^{k}(U,f)=\big{(}\mathscr{K}_{\alpha}^{k}(U,f),\mathrm{HN}^{\bullet}\big{)}_{|{u\neq\infty}}.$
(28)

However, one does not have the direct analogue of the Künneth formula for
$\mathrm{H}_{\mathrm{d},\alpha}^{\bullet}$ in general. For example, in the
case $(U_{i},f_{i})=(\mathbb{A}^{1},x_{i}^{2})$ where $x_{i}$ is a global
coordinate on the affine line $\mathbb{A}^{1}$, one has

$\displaystyle\dim\mathrm{H}_{\mathrm{d},0}^{k}\big{(}\mathbb{A}^{1},x_{i}^{2}\big{)}=\begin{cases}1,&\text{if
$k=1$},\\\ 0,&\text{otherwise},\end{cases}$

and the cup product

$\displaystyle\mathrm{H}_{\mathrm{d},0}^{1}\big{(}\mathbb{A}^{1},x_{1}^{2}\big{)}\otimes\mathrm{H}_{\mathrm{d},0}^{1}\big{(}\mathbb{A}^{1},x_{2}^{2}\big{)}\to\mathrm{H}_{\mathrm{d},0}^{2}\big{(}\mathbb{A}^{2},x_{1}^{2}+x_{2}^{2}\big{)}$
(29)

is zero. The vanishing can be obtained by a careful cohomology calculation or
manipulating Kontsevich bundles as follows.

Indeed, consider the complex
$\big{[}\Omega_{f_{i}}^{0}\xrightarrow{\mathrm{d}}\Omega_{f_{i}}^{1}\big{]}$
on $\mathbb{P}^{1}$, whose hypercohomology gives
$\mathrm{H}_{\mathrm{d},0}^{k}\big{(}\mathbb{A}^{1},x_{i}^{2}\big{)}$. Let
$\pi\colon\widetilde{X}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ be the blowup at
$(\infty,\infty)$. Then $\widetilde{X}$ is a non-degenerate compactification
of $\big{(}\mathbb{A}^{2},f=f_{1}+f_{2}\big{)}$. On $\widetilde{X}$ we have
the inclusion

$\displaystyle
A:=\pi^{*}\big{(}\boxtimes_{i=1}^{2}(\Omega_{f_{i}}^{\bullet},\mathrm{d})\big{)}\to
B:=(\Omega_{f}^{\bullet},\mathrm{d})$

and the arrow (29) factors through the induced map
$\mathbb{H}^{2}\big{(}\widetilde{X},A\big{)}\to\mathbb{H}^{2}\big{(}\widetilde{X},B\big{)}$.
One checks that the last map is zero.

On the other hand, for $\Theta=\mathrm{d}+\mathrm{d}x_{i}^{2}$ or
$\mathrm{d}x_{i}^{2}$, we have

$\displaystyle\mathrm{H}^{1}\big{(}\mathbb{A}^{1},\Theta\big{)}=\operatorname{Gr}_{F}^{1/2}\mathrm{H}^{1}\big{(}\mathbb{A}^{1},\Theta\big{)}=\mathrm{H}^{0}\big{(}\mathbb{P}^{1},\Omega^{1}(2[\infty])\big{)},$

which is generated by the class $\mathrm{d}x_{i}$. By (26), (27), (28), one
obtains that

$\displaystyle\mathscr{K}^{1}_{\alpha}\big{(}\mathbb{A}^{1},x_{i}^{2}\big{)}=\begin{cases}\mathscr{O}_{\mathbb{P}^{1}}\cdot\mathrm{d}x_{i},&0\leq\alpha<\frac{1}{2},\\\
\mathscr{O}_{\mathbb{P}^{1}}([\infty])\cdot\mathrm{d}x_{i},&\frac{1}{2}\leq\alpha<1,\end{cases}$

and in particular,
$\mathrm{H}_{\mathrm{d},0}^{1}\big{(}\mathbb{A}^{1},x_{i}^{2}\big{)}$ is
generated by $\mathrm{d}x_{i}$ as a fiber of
$\mathscr{K}^{1}_{0}\big{(}\mathbb{A}^{1},x_{i}^{2}\big{)}$. Similar argument
shows that

$\displaystyle\mathscr{K}^{2}_{\alpha}\big{(}\mathbb{A}^{2},x_{1}^{2}+x_{2}^{2}\big{)}=\mathscr{O}_{\mathbb{P}^{1}}([\infty])\cdot\mathrm{d}x_{1}\mathrm{d}x_{2},\qquad
0\leq\alpha<1.$

We conclude that the map (29) sends the generator
$\mathrm{d}x_{1}\otimes\mathrm{d}x_{2}$ to $\mathrm{d}x_{1}\mathrm{d}x_{2}$,
which represents zero in
$\mathrm{H}_{\mathrm{d},0}^{2}\big{(}\mathbb{A}^{2},x_{1}^{2}+x_{2}^{2}\big{)}$
as a fiber of
$\mathscr{K}^{2}_{0}\big{(}\mathbb{A}^{2},x_{1}^{2}+x_{2}^{2}\big{)}$.

### Acknowledgements

The second author thanks Noriko Yui for providing him the wonderful
opportunity to work with her during the years 2006 to 2008 and the happy lunch
time at Queen’s University. We thank the referee for the careful reading,
helpful comments and in particular pointing out the insufficiency of the proof
of Proposition 2.1 in an earlier version. This work was partially supported by
the MoST and the NCTS, Taiwan.

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