# A characterization of irreducible Hermitian symmetric spaces of tube type by
$\displaystyle{\mathbb{C}}^{*}$-actions

Yingqi Liu  Yingqi Liu, AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun
East Road, Beijing, 100190, China and University of Chinese Academy of
Sciences, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

A $\displaystyle{\mathbb{C}}^{*}$-action on a projective variety
$\displaystyle X$ is said to be of Euler type at a nonsingular fixed point
$\displaystyle x$ if the isotropy action of $\displaystyle{\mathbb{C}}^{*}$ on
$\displaystyle T_{x}X$ is by scalar multiplication. In this paper, it’s proven
that a smooth projective variety of Picard number one $\displaystyle X$ is
isomorphic to an irreducible Hermitian symmetric space of tube type if and
only if for a general pair of points $\displaystyle x,y$ on $\displaystyle X$,
there exists a $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$
which is of Euler type at $\displaystyle x$ and its inverse action is of Euler
type at $\displaystyle y$.

##  1\. Introduction

###  1.1. Main result

The study of complex torus actions on algebraic varieties is a classical topic
in algebraic geometry. It is an interesting problem to classify algebraic
varieties with special $\displaystyle{\mathbb{C}}^{*}$-actions. Let
$\displaystyle X$ be a smooth projective variety. A
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ is said to be
equalized at a fixed point $\displaystyle x$ if any weight of the isotropy
action on $\displaystyle T_{x}X$ equals to 0 or $\displaystyle\pm 1$. We call
the action equalized if it’s equalized at each fixed point. Denote by
$\displaystyle X^{{\mathbb{C}}^{*}}$ the set of fixed points of the
$\displaystyle{\mathbb{C}}^{*}$-action. An irreducible component of
$\displaystyle X^{{\mathbb{C}}^{*}}$ is called extremal if it intersects with
general $\displaystyle{\mathbb{C}}^{*}$-orbit closures. In the series of works
[RW20][ORSW22][ORSW21], the authors study equalized
$\displaystyle{\mathbb{C}}^{*}$-actions on projective manifolds with isolated
extremal fixed components. For rational homogenous spaces they proved the
following:

######  Theorem 1.1.

[ORSW21] Let $\displaystyle X=G/P$ be a rational homogenous space of Picard
number one, then $\displaystyle X$ admits an equalized
$\displaystyle{\mathbb{C}}^{*}$-action with two isolated extremal fixed points
if and only if X is isomorphic to one of the followings:

(i) a smooth hyperquadric $\displaystyle{\mathbb{Q}}^{n}$.

(ii) the Grassmaninan variety $\displaystyle Gr(n,2n)$.

(iii) the Lagrangian Grassmanian variety $\displaystyle Lag(n,2n)$.

(iv) the spinor variety $\displaystyle{\mathbb{S}}_{2n}$.

(v) the 27 dimensional $\displaystyle E_{7}$-variety $\displaystyle
E_{7}/P_{7}$.

The varieties classified above are exactly irreducible Hermitian symmetric
spaces (IHSS for short) of tube type. Recall that an IHSS is called of tube
type if its dual, as a bounded symmetric domain, is holomorphically equivalent
to a tube domain over a self dual cone. In [Mok02] Mok showed that an IHSS is
of tube type if and only if for a point $\displaystyle o\in X$ the complement
of the vectors of maximal rank in $\displaystyle T_{o}(X)$ is a hypersurface.
Translating in the language of $\displaystyle{\mathbb{C}}^{*}$-actions, one
can regard Theorem 1.1 as a generalization of Mok’s result to rational
homogenous spaces.

It is then a natural problem to characterize IHSS of tube type by
$\displaystyle{\mathbb{C}}^{*}$-actions in a more general context. A
$\displaystyle{\mathbb{C}}^{*}$-action on a projective manifold $\displaystyle
X$ is said to be of Euler type at a fixed point $\displaystyle x$ if the
isotropy action of $\displaystyle{\mathbb{C}}^{*}$ on $\displaystyle T_{x}X$
is by scalar multiplication, or equivalently if the
$\displaystyle{\mathbb{C}}^{*}$-action is equalized at $\displaystyle x$ and
$\displaystyle x$ is an isolated extremal fixed point. In Theorem 1.1, by
taking certain conjugates of $\displaystyle{\mathbb{C}}^{*}$ in $\displaystyle
Aut^{0}(X)$ one can show that for a general pair of points $\displaystyle x,y$
on $\displaystyle X$ there exists a $\displaystyle{\mathbb{C}}^{*}$-action
which is of Euler type at $\displaystyle x$ and its inverse action is of Euler
type at $\displaystyle y$ (see Section 4 for more details). Here a general
pair of points on $\displaystyle X$ is defined as a pair of points lying in a
Zariski open dense subset of $\displaystyle X\times X$. Our main result proves
the converse:

######  Theorem 1.2.

Let $\displaystyle X$ be a smooth projective variety of Picard number one,
then $\displaystyle X$ is isomorphic to an IHSS of tube type if and only if
for a general pair of points $\displaystyle x,y$ on $\displaystyle X$, there
exists a $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ which is
of Euler type at $\displaystyle x$ and its inverse action is of Euler type at
$\displaystyle y$.

###  1.2. Outline of the proof

The main ingredient of the proof is the VMRT theory developed by Hwang and
Mok. Let $\displaystyle X$ be a Fano manifold of Picard number one and let
$\displaystyle{\mathcal{K}}$ be a fixed irreducible dominant family of minimal
rational curves on $\displaystyle X$. The variety of minimal rational tangents
(VMRT for short) at a general point $\displaystyle x\in X$ is the closed
subvariety $\displaystyle{\mathcal{C}}_{x}\subseteq{\mathbb{P}}T_{x}X$
consisting of all tangent directions at $\displaystyle x$ of curves in
$\displaystyle{\mathcal{K}}$ passing through $\displaystyle x$. A large part
of the global geometry of the manifold is controlled by the VMRT
$\displaystyle{\mathcal{C}}_{x}\subseteq{\mathbb{P}}T_{x}X$ at a general point
$\displaystyle x$. The first step is to show that $\displaystyle X$ in Theorem
1.2 is an equivariant compactification of vector group hence $\displaystyle X$
can be recovered from its VMRT by Cartan-Fubini type theorem [HM01, Theorem
1.2]. Then we follow the methods developed in [HM05] [FH12] to classify the
projective subvariety
$\displaystyle{\mathcal{C}}_{x}\subseteq{\mathbb{P}}T_{x}X$ by studying its
prolongaion of infinitesimal linear automorphisms.

######  Definition 1.3.

(1) Let $\displaystyle\mathfrak{g}\subseteq\mathfrak{g}l(V)$ be a Lie
subalgebra, then the k-th prolongation of $\displaystyle\mathfrak{g}$ is the
space of symmetric multi-linear homomorphisms $\displaystyle
A:Sym^{k+1}V\rightarrow V$ such that for any fixed $\displaystyle
v_{1},...,v_{k}\in V$, the homomorphism $\displaystyle
A_{v_{1},...,v_{k}}:V\rightarrow V$ defined by:

$v\in V\rightarrow A(v,v_{1},...,v_{k})\in V$

is in $\displaystyle\mathfrak{g}$.

(2) Let $\displaystyle S\subseteq{\mathbb{P}}V$ be a projective subvariety.
Let $\displaystyle\hat{S}\subseteq V$ be its affine cone and $\displaystyle
T_{\alpha}(\hat{S})$ the tangent space at a smooth point
$\displaystyle\alpha\in\hat{S}$. The Lie algebra of infinitesimal linear
automorphisms of $\displaystyle\hat{S}$ is

$\mathfrak{a}ut(\hat{S})=\\{g\in End(V)|g(\alpha)\in
T_{\alpha}(\hat{S})\,\text{for any smooth point
$\displaystyle\alpha\in\hat{S}$}\\}=\\{g\in
End(V)|exp(tg)\cdot\hat{S}\subset\hat{S},\,t\in{\mathbb{C}}\\}.$

Its k-th prolongation $\displaystyle\mathfrak{a}ut(\hat{S})^{(k)}$ will be
called the k-th prolongation of $\displaystyle S\subseteq{\mathbb{P}}V$.

A crucial step of our proof is the following result.

######  Theorem 1.4.

Let $\displaystyle X$ be a smooth projective variety of Picard number one.
Assume for a general pair of points $\displaystyle(x,y)\in X\times X$ , there
is a $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ which is of
Euler type at $\displaystyle x$ and its inverse action is of Euler type at
$\displaystyle y$, then for the VMRT
$\displaystyle{\mathcal{C}}_{x}\subseteq{\mathbb{P}}T_{x}X$ we have:

(1.1) $dim(\mathfrak{a}ut(\hat{{\mathcal{C}}_{x}})^{(1)})=dim(X)$

The identity (1.1) is known to be valid for any IHSS. Let $\displaystyle
X=G/P$ be an IHSS defined by a semi-simple algebraic group $\displaystyle G$
and a maximal parabolic subgroup $\displaystyle P$. Denote by $\displaystyle
P^{-}$ the opposite group of $\displaystyle P$ and let $\displaystyle x=eP$,
$\displaystyle y=\dot{w}_{0}x$ where $\displaystyle w_{0}$ is the longest
element in the Weyl group. Then there is an equalized
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ such that
$\displaystyle x$ is an isolated extremal fixed point. And
$\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})^{(1)}$ can be
identified with the Lie algebra of $\displaystyle R_{u}(P)$, induced by the
$\displaystyle{\mathbb{Z}}/2{\mathbb{Z}}$-grading of the Lie algebra
$\displaystyle\mathfrak{g}=Lie(G)$. When X is of tube type, then
$\displaystyle y$ is also an isolated extremal fixed point of the
$\displaystyle{\mathbb{C}}^{*}$-action and X is an equivariant
compactification of $\displaystyle R_{u}(P)$ with open orbit $\displaystyle
R_{u}(P)\cdot y$. Thus in this case the nonzero prolongation of
$\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})$ comes from the vector
group compactification on $\displaystyle X$ with origin $\displaystyle y$.

Generally assume that $\displaystyle X$ satisfies the conditions of Theorem
1.2. Then the vector groups action of $\displaystyle T_{y}X$ and
$\displaystyle T_{x}X$ on themselves can be extended to equivariant
compactifications of vector groups on $\displaystyle X$. Choosing a suitable
non-degenrate projective embedding of $\displaystyle X\subseteq{\mathbb{P}}V$.
The actions of $\displaystyle T_{y}X$ and $\displaystyle T_{x}X$ on
$\displaystyle X$ can be lifted to linear actions on $\displaystyle V$.
Identifying $\displaystyle Lie(T_{y}X),Lie(T_{x}X)$ with their images in
$\displaystyle{\mathfrak{g}}l(V)$, the adjoint actions of the two Lie
subalgebras inside $\displaystyle{\mathfrak{g}}l(V)$ will induce the
identification: $\displaystyle
Lie(T_{y}X)\cong\mathfrak{a}ut(\hat{{\mathcal{C}}_{x}})^{(1)}$.

In [FH12] Fu and Hwang classified irreducible non-degenrate nonsingular
projective subvariety $\displaystyle S\subseteq{\mathbb{P}}V$ with nonzero
prolongations. Most of them are the VMRT of an IHSS or a nonsingular linear
section of some IHSS. Using a case by case calculation of $\displaystyle
dim(\mathfrak{a}ut(\hat{S})^{(1)})$, we show that
$\displaystyle{\mathcal{C}}_{x}\subseteq{\mathbb{P}}T_{x}X$ is projectively
isomorphic to the VMRT of an IHSS. Then by a
$\displaystyle{\mathbb{C}}^{*}$-equivariant Cartan-Fubini extension theorem,
we conclude that $\displaystyle X$ is
$\displaystyle{\mathbb{C}}^{*}$-isomorphic to an IHSS. The
$\displaystyle{\mathbb{C}}^{*}$-action on IHSS was studied in detail in
[ORSW21], as a corollary it has two isolated extremal fixed points if and only
if it is isomorphic to an IHSS of tube type.

The article is organized as follows. In Section 2 we first review the approach
in [FH20] to associate $\displaystyle{\mathbb{C}}^{*}$-actions with vector
group actions, then we apply it to the case when the
$\displaystyle{\mathbb{C}}^{*}$-action has two isolated extremal fixed points.
In Section 3 we study the prolongations of projective varieties, we first
prove Theorem 1.4 then we calculate the dimension of prolongation for certain
projective subvariety with nonzero prolongations. In Section 4 we finish the
proof of our main result.

### Notations

Throughout this article we work over the field of complex numbers. Given a
line bundle $\displaystyle{\mathcal{L}}$ on a variety $\displaystyle X$, the
principal open subset of a section $\displaystyle s\in H^{0}(X,{\mathcal{L}})$
is denoted by $\displaystyle D_{+}(s)=\\{x\in X:s(x)\not=0\\}$, and the cycle-
theoretic zero locus of $\displaystyle s$ is denoted by $\displaystyle Z(s)$.
For a vector space $\displaystyle V$ of dimension $\displaystyle n$, we
identify the regular functions on $\displaystyle V$ with the total space of
symmetric multilinear forms:
$\displaystyle{\mathbb{C}}[V]\cong\oplus_{k\geqslant 0}Sym^{k}(V^{*})$ by
assigning a function $\displaystyle f$ on $\displaystyle V$ to $\displaystyle
P_{f}=\sum_{k\geqslant 0}P_{f,k}$ such that $\displaystyle
f(v)=\sum_{k\geqslant 0}P_{f,k}(v,v...,v)$.

##  2\. From $\displaystyle{\mathbb{C}}^{*}$-action to vector group action

First we recall some basic facts and notions on
$\displaystyle{\mathbb{C}}^{*}$-actions. Let $\displaystyle X$ be a smooth
projective variety with a $\displaystyle{\mathbb{C}}^{*}$-action. For a
nontrival $\displaystyle{\mathbb{C}}^{*}$-orbit of a point $\displaystyle z$
on $\displaystyle X$, the orbit map
$\displaystyle\psi_{z}:{\mathbb{C}}^{*}\rightarrow X$ extends uniquely to a
morphism $\displaystyle\Psi_{z}:{\mathbb{P}}^{1}\rightarrow X$. We denote the
source (resp. sink) of the orbit to be $\displaystyle\lim_{t\rightarrow
0}tz=\Psi_{z}(0)$ (resp.
$\displaystyle\lim_{t\rightarrow\infty}tz=\Psi_{z}(\infty)$). Denote by
$\displaystyle{\mathcal{Y}}$ the set of irreducible components of
$\displaystyle X^{{\mathbb{C}}^{*}}$. Then for each $\displaystyle
Y\in{\mathcal{Y}}$, $\displaystyle Y$ is a smooth closed subvariety. The
isotropy action of $\displaystyle{\mathbb{C}}^{*}$ on $\displaystyle TX|_{Y}$
gives a decomposition of $\displaystyle TX|_{Y}=T^{+}(Y)\oplus T^{-}(Y)\oplus
TY$, where $\displaystyle T^{+}(Y),T^{-}(Y)$ are the subbundles of
$\displaystyle TX|_{Y}$ on which $\displaystyle{\mathbb{C}}^{*}$ acts with
positive, negative weights. Denote $\displaystyle C^{\pm}(Y)=\\{x\in
X:\lim_{t^{\pm 1}\rightarrow 0}tx\in Y\\}$. We recall the following theorem of
Białynicki-Birula [BB73].

######  Theorem 2.1.

Assume that $\displaystyle X$ is a smooth projective variety with a
$\displaystyle{\mathbb{C}}^{*}$-action, then:

(1) For each $\displaystyle Y\in{\mathcal{Y}}$, $\displaystyle C^{\pm}(Y)$ are
locally closed subsets and there are decompositions:

$X=\bigcup\limits_{Y\in{\mathcal{Y}}}C^{+}(Y)=\bigcup\limits_{Y\in{\mathcal{Y}}}C^{-}(Y)$

(2) For each $\displaystyle Y\in{\mathcal{Y}}$, there are
$\displaystyle{\mathbb{C}}^{*}$-isomorphisms $\displaystyle C^{+}(Y)\cong
T^{+}(Y)$ and $\displaystyle C^{-}(Y)\cong T^{-}(Y)$ lifting the natural map
$\displaystyle C^{\pm}(Y)\rightarrow Y$.The map $\displaystyle
C^{\pm}(Y)\rightarrow Y$ is algebraic and is a
$\displaystyle{\mathbb{C}}^{v^{\pm}(Y)}$-fibration, where we set
$\displaystyle v^{\pm}(Y)=rank(T^{\pm}(Y))$.

We call the unique $\displaystyle Y$ such that $\displaystyle C^{+}(Y)$ (resp.
$\displaystyle C^{-}(Y)$) is dense the source (resp. sink) of the action. For
any two components $\displaystyle Y,Y^{\prime}\in{\mathcal{Y}}$, we call
$\displaystyle Y\prec Y^{\prime}$ if there exists a point $\displaystyle x\in
X$ such that $\displaystyle lim_{t\rightarrow 0}t\cdot x\in Y$ and
$\displaystyle lim_{t\rightarrow\infty}t\cdot x\in Y^{\prime}$.

######  Proposition 2.2.

[ORSW22, Lemma 2.8 ] and [RW20, Lemma 3.5]
Let $\displaystyle X$ be a smooth projective variety of Picard number one.
Assume that there is a $\displaystyle{\mathbb{C}}^{*}$-action on
$\displaystyle X$ which is of Euler-type at $\displaystyle x$, then there is a
unique $\displaystyle Y\in{\mathcal{Y}}$ such that

(2.1) $\\{Y^{\prime}\in{\mathcal{Y}}:Y^{\prime}\prec Y\\}=\\{x\\}.$

Moreover $\displaystyle C^{-}(Y)$ is a line bundle over $\displaystyle Y$,
i.e., $\displaystyle v^{-}(Y)=1$.

To associate $\displaystyle{\mathbb{C}}^{*}$-actions with vector group actions
we recall Euler-symmetric varieties defined by Fu and Hwang in [FH20].

######  Definition 2.3.

Let $\displaystyle Z\subseteq{\mathbb{P}}V$ be a projective subvariety. A
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle Z$ is called of
Euler-type at a nonsingular point $\displaystyle x$ if the isotropic action on
the tangent space $\displaystyle T_{x}Z$ is by scalar multiplication. We say
$\displaystyle Z\subseteq{\mathbb{P}}V$ is an Euler-symmetric variety if for a
general point $\displaystyle x\in Z$, there exists a
$\displaystyle{\mathbb{C}}^{*}$-action which is of Euler type at x, where the
$\displaystyle{\mathbb{C}}^{*}$-action comes from a multiplicative subgroup of
$\displaystyle GL(V)$.

From [FH20, Theorem 3.7] any Euler-symmetric variety is an equivariant
compactification of vector group. Moreover we have the following by [FH20,
Theorem 3.7].

######  Proposition 2.4.

Assume $\displaystyle X$ to be a normal projective variety and let
$\displaystyle{\mathcal{L}}$ be a very ample line bundle on $\displaystyle X$,
then the followings are equivalent:

(1) For a general point $\displaystyle x$ on $\displaystyle X$, there is a
$\displaystyle{\mathbb{C}}^{*}$-action which of Euler type at $\displaystyle
x$.

(2) $\displaystyle X$ is an equivariant compactification of vector group and
the scalar multiplication of $\displaystyle{\mathbb{C}}^{*}$ on the vector
group can be extended to a $\displaystyle{\mathbb{C}}^{*}$-action on
$\displaystyle X$.

(3) The projective subvariety $\displaystyle
X\subset{\mathbb{P}}H^{0}(X,{\mathcal{L}})^{\vee}$ is Euler-symmetric.

In the following we always assume that $\displaystyle X$ is a smooth
projective variety of Picard number one. We study
$\displaystyle{\mathbb{C}}^{*}$-actions on $\displaystyle X$ from the
perspective of local projective differential geometry following [LM03] and
FH20,

######  Definition 2.5.

(1) Let $\displaystyle x\in X\subseteq{\mathbb{P}}V$ be a nonsingular point of
a nondegenerate projective variety, and let
$\displaystyle{\mathcal{L}}={\mathcal{O}}_{{\mathbb{P}}V}(1)|_{X}$ be the line
bundle on $\displaystyle X$. For each nonnegative integer $\displaystyle k$,
$\displaystyle\mathfrak{m}_{x,X}^{k}$ be the $\displaystyle k$-th power of the
maximal ideal $\displaystyle\mathfrak{m}_{x,X}$. For a section $\displaystyle
s\in H^{0}(X,{\mathcal{L}})$, let $\displaystyle j_{x}^{k}(s)$ be the
$\displaystyle k$-jet of $\displaystyle s$ at $\displaystyle x$ such that
$\displaystyle j_{x}^{0}=s_{x}\in{\mathcal{L}}_{x}$. The induced homomorphism
:

$(V^{*}\cap Ker(j_{x}^{k-1}))/(V^{*}\cap
Ker(j_{x}^{k}))\rightarrow{\mathcal{L}}_{x}\otimes Sym^{k}T_{x}^{*}X$

is injective. For each $\displaystyle k\geqslant 2$, the subspace
$\displaystyle{\mathbb{F}}_{x}^{k}\subseteq Sym^{k}T_{x}^{*}X$ defined by the
image of this homomorphism is called the $\displaystyle k$-th fundamental form
of $\displaystyle X$ at $\displaystyle x$. Set $\displaystyle
F_{x}^{0}={\mathbb{C}}$ and $\displaystyle F_{x}^{1}=T_{x}^{*}X$, The
collection of subspaces $\displaystyle{\mathbb{F}}_{x}=\oplus_{k\geqslant
0}{\mathbb{F}}_{x}^{k}\subset\oplus_{k\geqslant 0}Sym^{k}T_{x}^{*}X$ is called
the system of fundamental forms of $\displaystyle X$ at $\displaystyle x$.

(2) Let $\displaystyle W$ be a vector space. For $\displaystyle w\in W$, the
contraction homomorphism $\displaystyle\iota_{w}:Sym^{k+1}W^{*}\rightarrow
Sym^{k}W^{*}$ sending $\displaystyle\phi\in Sym^{k+1}W^{*}$ to
$\displaystyle\iota_{w}\phi\in Sym^{k}W^{*}$ is defined by:

$\iota_{w}\phi(w_{1},...,w_{k})=\phi(w,w_{1},...,w_{k})$

for any $\displaystyle w_{1},...,w_{k}\in W$. By convention we define
$\displaystyle\iota_{w}(Sym^{0}W^{*})=0$.

(3) A subspace $\displaystyle{\mathbb{F}}=\oplus_{k\geqslant
0}F^{k}\subset\oplus_{k\geqslant 0}Sym^{k}W^{*}$ with $\displaystyle
F^{0}={\mathbb{C}},F^{1}=W^{*},F^{r}\not=0$, and $\displaystyle F^{r+i}=0$ for
all $\displaystyle i\geqslant 1$ is called a symbol system of rank r if
$\displaystyle\iota_{w}F^{k+1}\subseteq F^{k}$ for any $\displaystyle w\in W$
and any $\displaystyle k\geqslant 0$.

We recall the following theorem of Cartan (see for example [LM03, Section
2.1]).

######  Theorem 2.6.

Let $\displaystyle X\subseteq{\mathbb{P}}V$ be a nondegenerate subvariety, and
let $\displaystyle x\in X$ be a general point. Then the system of fundamental
forms $\displaystyle{\mathbb{F}}_{x}\subset\oplus_{k\geqslant
0}Sym^{k}T_{x}^{*}X$ is a symbol system.

Now assume that $\displaystyle X$ admits a
$\displaystyle{\mathbb{C}}^{*}$-action which is of Euler type at a fixed point
$\displaystyle x$. Denote by $\displaystyle{\mathcal{L}}$ a very ample line
bundle on $\displaystyle X$ and $\displaystyle
V=H^{0}(X,{\mathcal{L}})^{\vee}$. Choose a
$\displaystyle{\mathbb{C}}^{*}$-linearization on $\displaystyle{\mathcal{L}}$
such that if we write $\displaystyle
H^{0}(X,{\mathcal{L}})=\oplus_{k=0}^{r}H^{0}(X,{\mathcal{L}})_{w_{k}}$ as the
sum of nonzero weight subspaces with respected to the
$\displaystyle{\mathbb{C}}^{*}$-action, then we have $\displaystyle
0=w_{0}>w_{1}>w_{2}>...>w_{r}$. The fundamental forms of $\displaystyle
X\subset{\mathbb{P}}V$ at $\displaystyle x$ can be identified as follows.

######  Lemma 2.7.

(1) The subspace $\displaystyle\\{s\in
H^{0}(X,{\mathcal{L}})\,|\,D_{+}(s)=C^{+}(x)\\}$ is of dimension one and it
equals to $\displaystyle H^{0}(X,{\mathcal{L}})_{0}$. Take a nonzero section
$\displaystyle s_{0}$ with $\displaystyle D_{+}(s_{0})=C^{+}(x)$ and consider
the linear map:

$\displaystyle\displaystyle\eta:H^{0}(X,{\mathcal{L}})$
$\displaystyle\displaystyle\rightarrow{\mathbb{C}}[C^{+}(x)]$
$\displaystyle\displaystyle s$ $\displaystyle\displaystyle\rightarrow\eta(s)$

defined by: $\displaystyle s|_{C^{+}(x)}=\eta(s)s_{0}|_{C^{+}(x)}$, for any
$\displaystyle s\in H^{0}(X,{\mathcal{L}})$. Then:

(2) $\displaystyle\eta$ is an injective
$\displaystyle{\mathbb{C}}^{*}$-equivariant linear map, where the
$\displaystyle{\mathbb{C}}^{*}$-action on
$\displaystyle{\mathbb{C}}[C^{+}(x)]$ is given by $\displaystyle z\cdot
f(u)=f(z^{-1}\cdot u)$, for any $\displaystyle z\in{\mathbb{C}}^{*}$,
$\displaystyle u\in C^{+}(x)$ and $\displaystyle f\in{\mathbb{C}}[C^{+}(x)]$.

(3) Identifying
$\displaystyle{\mathbb{C}}[C^{+}(x)]\cong{\mathbb{C}}[T_{x}X]\cong\oplus_{k\geqslant
0}Sym^{k}T_{x}^{*}X$, then the image of $\displaystyle\eta$ is identified with
$\displaystyle{\mathbb{F}}_{x}$, under which
$\displaystyle\eta(H^{0}(X,{\mathcal{L}})_{w_{k}})={\mathbb{F}}_{x}^{-w_{k}}$.
Furthermore $\displaystyle w_{1}=-1$ and
$\displaystyle\eta|_{H^{0}(X,{\mathcal{L}})_{w_{1}}}$ is an isomorphism.

###### Proof.

As $\displaystyle C^{+}(x)$ is isomorphic to the affine space $\displaystyle
T_{x}X$, $\displaystyle D_{x}:=X\backslash C^{+}(x)$ is a divisor by Hartgo’s
theorem and $\displaystyle Cl(X)$ is freely generated by the irreducible
components of $\displaystyle D_{x}$. Then by our assumption that
$\displaystyle X$ is of Picard number one, $\displaystyle D_{x}$ is the prime
generator of $\displaystyle Cl(X)$. Thus we can write
$\displaystyle{\mathcal{L}}\cong{\mathcal{O}}_{X}(r_{0}D_{x})$ for some
positive integer $\displaystyle r_{0}$. For any nonzero section $\displaystyle
s$ with $\displaystyle D_{+}(s)=C^{+}(x)$, write $\displaystyle Z(s)=rD_{x}$
for some positive integer $\displaystyle r$. Then we have $\displaystyle
rD_{x}\sim r_{0}D_{x}$, implying that $\displaystyle r=r_{0}$ and thus the
subspace $\displaystyle\\{s\in H^{0}(X,{\mathcal{L}}):D_{+}(s)=B^{+}(x)\\}$ is
of dimension one. The subspace is $\displaystyle{\mathbb{C}}^{*}$-invariant
and we denote its weight to be $\displaystyle w^{\prime}$. Take a nonzero
section $\displaystyle s_{0}$ in this subspace and define $\displaystyle\eta$
as above, then $\displaystyle\eta$ is injective as $\displaystyle C^{+}(x)$ is
open. For any $\displaystyle s\in H^{0}(X,{\mathcal{L}})$, $\displaystyle
z\in{\mathbb{C}}^{*}$ and $\displaystyle u\in C^{+}(x)$ we have:

$(z\cdot s)(u)=z\cdot s(z^{-1}\cdot u)=z\cdot(\eta(s)(z^{-1}\cdot
u)s_{0}(z^{-1}\cdot u))=\eta(s)(z^{-1}u)\,(z\cdot
s_{0})(u)=z^{w^{\prime}}(z\cdot\eta(s))(u)s_{0}(u),$

implying that $\displaystyle\eta(z\cdot s)=z^{w^{\prime}}(z\cdot\eta(s))$. For
any $\displaystyle s\in H^{0}(X,{\mathcal{L}})_{w_{k}}$, viewing
$\displaystyle\eta(s)$ as a regular function on $\displaystyle T_{x}X$ via
$\displaystyle C^{+}(x)\cong T_{x}X$, then we have:

$\displaystyle\displaystyle\eta(z\cdot
s)(u)=z^{w_{k}}\eta(s)(u),\,\,z\cdot\eta(s)(u)=\eta(s)(z^{-1}\cdot
u)=\eta(s)(z^{-1}u)$

for any $\displaystyle u\in T_{x}X$. Combined with $\displaystyle\eta(z\cdot
s)=z^{w^{\prime}}(z\cdot\eta(s))$ this shows that $\displaystyle\eta(s)$ is a
homogenous polynomial on $\displaystyle T_{x}X$ with $\displaystyle
deg(\eta(s))=w^{\prime}-w_{k}\,\geqslant 0$. This implies that $\displaystyle
w^{\prime}=w_{0}=0$ and $\displaystyle deg(\eta(s))=-w_{k}$. Thus
$\displaystyle\eta$ is $\displaystyle{\mathbb{C}}^{*}$-equivariant, and for
any nonzero section $\displaystyle s\in H^{0}(X,{\mathcal{L}})_{0}$,
$\displaystyle\eta(s)$ is a constant, which shows that $\displaystyle
C^{+}(x)=D_{+}(s)$, proving (1) and (2).

For a section $\displaystyle s\in H^{0}(X,{\mathcal{L}})$,write
$\displaystyle\eta(s)=\sum_{k\geqslant 0}\eta_{k}(s)$ as the sum of homogenous
functions. The $\displaystyle k$-th jet of $\displaystyle s$ at $\displaystyle
x$ equals the class of $\displaystyle\eta(s)|_{x}$ in
$\displaystyle{\mathcal{O}}_{x}/\mathfrak{m}_{x}^{k}$, whence under the
identification it is represented by
$\displaystyle\sum_{i=0}^{k-1}\eta_{i}(s)$. On the other hand we have proved
that $\displaystyle\eta$ maps $\displaystyle H^{0}(X,{\mathcal{L}})_{w_{k}}$
into the space of homogenous polynomials of degree $\displaystyle-w_{k}$. Thus
by the definition of fundamental forms we have
$\displaystyle\eta(H^{0}(X,{\mathcal{L}})_{w_{k}})={\mathbb{F}}_{x}^{-w_{k}}$
and Im$\displaystyle(\eta)={\mathbb{F}}_{x}$. Finally as
$\displaystyle{\mathcal{L}}$ is very ample, by [Ha77, Proposition 7.2]
Im$\displaystyle(\eta)$ generates $\displaystyle{\mathbb{C}}[C^{+}(x)]$ as a
$\displaystyle{\mathbb{C}}-$algebra. Thus Im$\displaystyle(\eta)$ contains the
space of linear functions $\displaystyle T_{x}^{*}X$, which has to be the
image of $\displaystyle H^{0}(X,{\mathcal{L}})_{w_{1}}$. It then follows that
$\displaystyle w_{1}=-1$ and
$\displaystyle\eta|_{H^{0}(X,{\mathcal{L}})_{w_{1}}}$ is an isomorphism. ∎

From Lemma 2.7, the fundamental form at $\displaystyle x$ is given by the
collection of linear injection
$\displaystyle\eta|_{H^{0}(X,{\mathcal{L}})_{w_{k}}}$ for each $\displaystyle
k$. Dually it is given by the collection of surjective linear map
$\displaystyle\Pi_{k}:\,Sym^{-w_{k}}(T_{x}X)\rightarrow
H^{0}(X,{\mathcal{L}})_{w_{k}}^{\vee}$ for each $\displaystyle k$, where
$\displaystyle\Pi_{0}:{\mathbb{C}}\rightarrow
H^{0}(X,{\mathcal{L}})_{0}^{\vee}$ maps 1 to the unique $\displaystyle
e_{0}\in H^{0}(X,{\mathcal{L}})_{0}^{\vee}$ such that $\displaystyle
e_{0}(s_{0})=1$ and $\displaystyle\Pi_{1}$ is an isomorphism by Lemma 2.7 (3).
Denote by $\displaystyle
f:X\rightarrow{\mathbb{P}}H^{0}(X,{\mathcal{L}})^{\vee}$ the projective
embedding. Then under the identification $\displaystyle C^{+}(x)\cong T_{x}X$,
we have $\displaystyle f(u)=[e_{0}+\sum_{k=1}^{r}\Pi_{k}(u,...,u)]$ for each
$\displaystyle u\in T_{x}X$ and $\displaystyle f(x)=f(0)=[e_{0}]$. At this
point we can deduce an easy corollary:

######  Corollary 2.8.

Let $\displaystyle X,{\mathcal{L}},x$ be as above, if $\displaystyle r=1$ then
$\displaystyle X$ is isomorphic to a projective space.

###### Proof.

Write $\displaystyle V=H^{0}(X,{\mathcal{L}})^{\vee}$ and $\displaystyle
W=H^{0}(X,{\mathcal{L}})^{\vee}_{w_{1}}$. Then we have $\displaystyle
V={\mathbb{C}}e_{0}\oplus W$ and $\displaystyle
C^{+}(x)=f(T_{x}X)=\\{[e_{0}+w]:w\in W\\}$ as $\displaystyle\Pi_{1}$ is
surjective. Thus the sink of the action equals $\displaystyle{\mathbb{P}}W$
and hence $\displaystyle f$ is an isomorphism onto
$\displaystyle{\mathbb{P}}V$. ∎

Now assume that $\displaystyle x$ is chosen as a general point. The
contraction homomorphism in Definition 2.5(2) defines a locally nilpotent
action of $\displaystyle T_{x}X$ on $\displaystyle\oplus_{k\geqslant
0}Sym^{k}T_{x}^{*}X$. Then by Theorem 2.6, $\displaystyle{\mathbb{F}}_{x}$ is
a finite dimensional $\displaystyle T_{x}X$-invariant subspace. Dually it
defines a nilpotent action of $\displaystyle T_{x}X$ on $\displaystyle
H^{0}(X,{\mathcal{L}})^{\vee}$ through each $\displaystyle\Pi_{k}$. More
precisely for each $\displaystyle 0\leqslant k\leqslant r$ and for any
$\displaystyle v\in T_{x}X$, denote
$\displaystyle\Gamma_{v}|_{H^{0}(X,{\mathcal{L}})_{w_{k}}^{\vee}}:H^{0}(X,{\mathcal{L}})_{w_{k}}^{\vee}\rightarrow
H^{0}(X,{\mathcal{L}})_{w_{k+1}}^{\vee}$ by:

(2.2)
$\Gamma_{v}\circ\Pi_{k}(v_{1},...,v_{k})=\Pi_{k+1}(v,v_{1},...,v_{k}),\,\,\,\,\text{$\displaystyle\forall
k\geqslant
1$}\text{\,\,\,and\,\,\,}\Gamma_{v}(e_{0})=\Pi_{1}(v),\Gamma_{v}(H^{0}(X,{\mathcal{L}})_{w_{r}}^{\vee})=0.$

Then $\displaystyle f|_{T_{x}X}$ can be also written as: $\displaystyle
f(w)=\sum_{k=0}^{r}\Gamma_{w}^{k}(e_{0})$ for any $\displaystyle w\in T_{x}X$.

In the following we denote by $\displaystyle
V_{k}=H^{0}(X,{\mathcal{L}})_{w_{k}}^{\vee}$ for each $\displaystyle k$. It
can be easily checked that the induced linear map
$\displaystyle\Gamma:T_{x}X\rightarrow\mathfrak{g}l(V)$ is a homomorphism
between Lie algebras. As $\displaystyle x$ is a general point, $\displaystyle
X\subset{\mathbb{P}}V$ is Euler-symmetric by Proposition 2.4, where the vector
group action on $\displaystyle X$ is given by a linear representation
$\displaystyle\rho_{x}:T_{x}X\rightarrow GL(V)$ defined as the following (see
[FH20, Proof of Theorem 3.7]):

(2.3)
$\rho_{x}(u)(v)=\sum_{l=0}^{r}\sum_{k=0}^{l}\tbinom{l}{k}\Gamma_{u}^{l-k}(v_{k}),$

for any $\displaystyle u\in T_{x}X$ and $\displaystyle
v=\sum_{k=0}^{r}v_{k}\in V$, where $\displaystyle v_{k}\in V_{k}$ for each
$\displaystyle k$.

######  Proposition 2.9.

Let $\displaystyle X,{\mathcal{L}},\rho_{x},f$ be as above, where
$\displaystyle x$ is chosen as a general point. Then:

(1) For each $\displaystyle 0\leqslant k\leqslant r$, $\displaystyle
w_{k}=-k$.

(2) For each $\displaystyle k\geqslant 1$ and for any $\displaystyle
u_{1},...,u_{k}\in T_{x}X$,
$\displaystyle\Pi_{k}(u_{1},...,u_{k})=\Gamma_{u_{1}}\circ...\circ\Gamma_{u_{k}}(e_{0})$.

(3) The induced action of $\displaystyle T_{x}X$ on
$\displaystyle{\mathbb{P}}V$ leaves $\displaystyle X$ invariant and lifts the
action of $\displaystyle T_{x}X$ on $\displaystyle C^{+}(x)$. Moreover denote
by $\displaystyle d\rho_{x}:T_{x}X\rightarrow\mathfrak{g}l(V)$ the
differential map of $\displaystyle\rho_{x}$, then $\displaystyle
d\rho_{x}|_{V_{k}}=(k+1)\,\Gamma|_{V_{k}}$ for each $\displaystyle k$. In
particular we have

$d\rho_{x}(u)\cdot V_{k}\subseteq V_{k+1},$

for any $\displaystyle u\in T_{x}X$, any $\displaystyle k\geqslant 0$ and
$\displaystyle d\rho_{x}(u)(e_{0})=\Pi_{1}(u)=\Gamma_{u}(e_{0})$.

###### Proof.

As $\displaystyle x$ is a general point, $\displaystyle{\mathbb{F}}_{x}$ is a
symbol system. Thus for any $\displaystyle l$ such that
$\displaystyle{\mathbb{F}}_{x}^{l+1}\not=0$ we have
$\displaystyle{\mathbb{F}}_{x}^{l}\not=0$ as well. Then from Lemma 2.7 (3), we
conclude that
$\displaystyle{\mathbb{F}}_{x}=\oplus_{k=0}^{r}{\mathbb{F}}_{x}^{r}$,
$\displaystyle w_{k}=-k$ and
$\displaystyle\eta(H^{0}(X,{\mathcal{L}})_{w_{k}})={\mathbb{F}}_{x}^{k}$ for
each $\displaystyle k$. (2) is a direct consequence of our definition in
(2.2). For (3) to show the first assertion, it suffices to check
$\displaystyle\rho_{x}(u^{\prime})\cdot f(u)=f(u+u^{\prime})$, for any
$\displaystyle u,u^{\prime}\in T_{x}X$. For any $\displaystyle u,u^{\prime}\in
T_{x}X$, we have:

$f(u+u^{\prime})=[\sum_{k=0}^{r}\Gamma_{u+u^{\prime}}^{k}(e_{0})]=[\sum_{l=0}^{r}\sum_{k=0}^{l}\tbinom{l}{k}\Gamma_{u}^{k}\circ\Gamma_{u^{\prime}}^{l-k}(e_{0})]=[\rho_{x}(u^{\prime})(\sum_{k=0}^{r}\Gamma_{u}^{k}(e_{0}))]=\rho_{x}(u^{\prime})\cdot
f(u).$

Now for any $\displaystyle u\in T_{x}X$ and $\displaystyle
v=\sum_{k=0}^{r}v_{k}\in V$:

$\displaystyle\displaystyle d\rho_{x}(u)(v)$
$\displaystyle\displaystyle=\frac{d}{d_{z}}_{|z=0}(\rho_{x}(zu)(v))=\frac{d}{d_{z}}_{|z=0}(\sum_{l=0}^{r}\sum_{k=0}^{l}\tbinom{l}{k}z^{l-k}\Gamma_{u}^{l-k}(v_{k}))=\sum_{l=1}^{r}l\cdot\Gamma_{u}(v_{l-1}).$

In other words we have $\displaystyle
d\rho_{x}|_{V_{k}}=(k+1)\,\Gamma|_{V_{k}}$, for any $\displaystyle k\geqslant
0$. In particular, $\displaystyle d\rho_{x}.V_{k}\subseteq V_{k+1}$ and
$\displaystyle d\rho_{x}(u)(e_{0})=\Gamma_{u}(e_{0})=\Pi_{1}(u)$ for any
$\displaystyle u\in T_{x}X$ ∎

Next we apply it to the case when both the sink and the source are isolated.
Assume that for a general pair of points $\displaystyle x,y$ on $\displaystyle
X$, there is a $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$
which is of Euler type at the source $\displaystyle x$ and its inverse action
is of Euler type at $\displaystyle y$. Take the linearization of the inverse
action on $\displaystyle{\mathcal{L}}$ such that the decomposition of the
associated weight subspaces equals $\displaystyle
H^{0}(X,{\mathcal{L}})=\oplus_{k=0}^{r}H^{0}(X,{\mathcal{L}})^{\prime}_{w^{\prime}_{k}}$,
where $\displaystyle 0=w_{0}^{\prime}>...>w_{r}^{\prime}$. Applying
Proposition 2.9, we have the following:

######  Corollary 2.10.

Let $\displaystyle X,{\mathcal{L}},x$ and $\displaystyle y$ be as above, then:

(1) We have $\displaystyle
H^{0}(X,{\mathcal{L}})_{w_{k}}=H^{0}(X,{\mathcal{L}})^{\prime}_{w^{\prime}_{r-k}}$
and $\displaystyle w_{k}=-r-w^{\prime}_{r-k}$.

(2) We have $\displaystyle
V_{r}=H^{0}(X,{\mathcal{L}})^{\prime}_{w_{0}^{\prime}}=\\{s\in
H^{0}(X,{\mathcal{L}}):D_{+}(s)=X^{-}(y)\\}$ and $\displaystyle
dim(H^{0}(X,{\mathcal{L}})^{\prime}_{w_{0}^{\prime}})=1$. Furthermore
$\displaystyle f(y)=[e_{r}]$ where $\displaystyle e_{r}$ is a nonzero element
in $\displaystyle V_{r}$.

(3) There is a vector group action of $\displaystyle T_{y}X$ on $\displaystyle
V$ namely $\displaystyle\rho_{y}:T_{y}X\rightarrow GL(V)$, such that the
induced action on $\displaystyle{\mathbb{P}}V$ leaves $\displaystyle X$
invariant and lifts the action of $\displaystyle T_{y}X$ on $\displaystyle
C^{-}(y)$. Moreover for the differential map $\displaystyle d\rho_{y}$ we have
$\displaystyle d\rho_{y}\cdot V_{0}=0$ and:

$d\rho_{y}(w)\cdot V_{k+1}\subseteq V_{k},$

for any $\displaystyle w\in T_{y}X$ and for any $\displaystyle k\geqslant 0$.

###### Proof.

(1) follows directly from the definition of $\displaystyle w_{k}^{\prime}$.
(2) follows by (1), Lemma 2.7 (1) and Proposition 2.9 (3). For (3) we dually
denote by $\displaystyle
V_{k}^{\prime}=(H^{0}(X,{\mathcal{L}})^{\prime}_{w_{k}^{\prime}})^{\vee}$ for
each $\displaystyle k$. Then by (1) we have $\displaystyle
V_{k}^{\prime}=V_{r-k}$. And from Proposition 2.9 (3) there is a vector group
action of $\displaystyle T_{y}X$ on $\displaystyle V$, such that the induced
action on $\displaystyle X$ extends the action of $\displaystyle T_{y}X$ on
$\displaystyle C^{-}(y)$. Moreover $\displaystyle d\rho_{y}\cdot
V_{k}^{\prime}\subseteq V_{k+1}^{\prime}$, equivalently we have $\displaystyle
d\rho_{y}\cdot V_{k+1}\subseteq V_{k}$ for any $\displaystyle k\geqslant 0$
and $\displaystyle d\rho_{y}\cdot V_{0}=0$. ∎

##  3\. Prolongations of projective subvarieties

In this section, we study prolongation of projective subvarieties in two steps
following [FH12][HM05]. Firstly we study the prolongation of the VMRT of
Euler-symmetric varieties at a general point. Then we calculate the dimension
of $\displaystyle\mathfrak{a}ut(\hat{S})^{(1)}$ for certain $\displaystyle
S\subset{\mathbb{P}}V$ with nonzero prolongation, which was explicitly
formulated in [FH12].

###  3.1. Prolongation of the VMRT

In this section we apply the results built in Section 2 to prove Theorem 1.4.
Throughout the section, $\displaystyle X$ is assumed to be a Fano manifold of
Picard number one. Let us recall the definition of VMRT.

######  Definition 3.1.

An irreducible component $\displaystyle{\mathcal{K}}$ of the space
Ratcurves${}^{n}(X)$ of rational curves on $\displaystyle X$ is called a
minimal rational component if the subvariety $\displaystyle{\mathcal{K}}_{x}$
of $\displaystyle{\mathcal{K}}$ parameterizing curves passing through a
general point $\displaystyle x\in X$ is non-empty and proper. Curves
parameterized by $\displaystyle{\mathcal{K}}$ will be called minimal rational
curves. Let $\displaystyle\rho:{\mathcal{U}}\rightarrow{\mathcal{K}}$ be the
universal family and $\displaystyle\mu:{\mathcal{U}}\rightarrow X$ the
evaluation map. The tangent map
$\displaystyle\tau:{\mathcal{U}}\dashrightarrow{\mathbb{P}}T(X)$ is defined by
$\displaystyle\tau(u)=T_{\mu(u)}(\mu(\rho^{-1}(\rho(u))))$. The closure
$\displaystyle{\mathcal{C}}\subset{\mathbb{P}}T(X)$ of its image is the total
space of variety of minimal rational tangents. The natural projection
$\displaystyle{\mathcal{C}}\rightarrow X$ is a proper surjective morphism and
a general fiber $\displaystyle{\mathcal{C}}_{x}\subset{\mathbb{P}}T_{x}X$ is
called the variety of minimal rational tangents (VMRT for short) at the point
$\displaystyle x\in X$.

The VMRT of an Euler-symmetric variety at a general point is related to its
fundamental forms as follows.

######  Definition 3.2.

Assume that for a general point $\displaystyle x$ on $\displaystyle X$, there
exists a $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ which is
of Euler type at $\displaystyle x$. Take a very ample line bundle
$\displaystyle{\mathcal{L}}$ on $\displaystyle X$ and denote by $\displaystyle
f:X\rightarrow{\mathbb{P}}H^{0}(X,{\mathcal{L}})^{\vee}$ the projective
embedding. Denote by $\displaystyle{\mathbb{F}}_{x}$ the fundamental forms of
$\displaystyle X$ at $\displaystyle x$ and take $\displaystyle\eta$ as in
Lemma 2.7. For each $\displaystyle k\geqslant 2$ denote:

$\displaystyle\displaystyle Bs({\mathbb{F}}_{x}^{k})$
$\displaystyle\displaystyle=\\{[w]\in{\mathbb{P}}T_{x}X:\phi(w,...,w)=0,\forall\phi\in{\mathbb{F}}_{x}^{k}\subset
Sym^{k}T_{x}^{*}X\\}$

Then $\displaystyle
Bs({\mathbb{F}}_{x}^{k})=\\{[w]\in{\mathbb{P}}T_{x}X:\eta(s)(w)=0,\forall s\in
H^{0}(X,{\mathcal{L}})_{w_{k}}\\}$ and we have the inclusions: $\displaystyle
Bs({\mathbb{F}}_{x}^{2})\subset Bs({\mathbb{F}}_{x}^{3})\subset...\subset
Bs({\mathbb{F}}_{x}^{r})$ as $\displaystyle{\mathbb{F}}_{x}$ is a symbol
system . Denote the base locus of fundamental forms at $\displaystyle x$ to be
$\displaystyle Bs({\mathbb{F}}_{x})=Bs({\mathbb{F}}_{x}^{l_{0}})$, where
$\displaystyle l_{0}$ is the smallest integer $\displaystyle l$ such that
$\displaystyle Bs({\mathbb{F}}_{x}^{l})$ is non-empty.

######  Proposition 3.3.

[FH20, Prop 4.4 and Prop 5.4(iii)]
Let $\displaystyle X,x,{\mathcal{L}}$ be as above. Let
$\displaystyle{\mathcal{K}}$ be the family of minimal rational curves on
$\displaystyle X$ and $\displaystyle{\mathcal{C}}\subset{\mathbb{P}}T(X)$ the
VMRT-structure on $\displaystyle X$. Then
$\displaystyle{\mathcal{C}}_{x}=Bs({\mathbb{F}}_{x})\subset{\mathbb{P}}T_{x}X$,
which is an irreducible, nonsingular and non-degenerate projective subvariety.

Next we aim to study the prolongation of infinitesimal linear automorphism of
$\displaystyle C_{x}\subseteq{\mathbb{P}}T_{x}X$, the most important example
of which is the case of IHSS.

######  Example 1.

Let $\displaystyle G$ be a simple algebraic group, $\displaystyle B\subset G$
a Borel subgroup, and $\displaystyle T\subset B$ a maximal torus. Denote by
$\displaystyle\Phi$ the root system, $\displaystyle\Delta\subset\Phi$ the
simple roots and $\displaystyle{\mathcal{D}}$ the Dynkin diagram. Let
$\displaystyle{\mathfrak{g}}=Lie(G)$ be the Lie algebra of $\displaystyle G$
and let $\displaystyle{\mathfrak{h}}\subset{\mathfrak{g}}$ be the Cartan
subalgebra. For a subset $\displaystyle I\subset\Delta$, denote by
$\displaystyle P_{I}$ the standard parabolic subgroup indexed by
$\displaystyle I$, and denote by $\displaystyle P_{I}^{-}\subset G$ the
opposite group of $\displaystyle P_{I}$. We write the quotient $\displaystyle
G/P_{I}$ as $\displaystyle{\mathcal{D}}(I)$, which is a smooth projective
rational variety of Picard number $\displaystyle|I|$. In the following we
always assume $\displaystyle I=\\{\alpha\\}$ for a simple root
$\displaystyle\alpha\in\Delta$. For any root $\displaystyle\beta\in\Phi$, the
multiplicity of the simple root $\displaystyle\alpha$ in $\displaystyle\beta$
is denoted by $\displaystyle m_{\alpha}(\beta)$. Denote by $\displaystyle
y=eP$ and $\displaystyle x=\dot{w_{0}}P$, where $\displaystyle w_{0}$ is the
longest element in the Weyl group of $\displaystyle G$.

Write
$\displaystyle{\mathfrak{g}}=\mathop{\oplus}\limits_{k\in{\mathbb{Z}}}{\mathfrak{g}}_{k}$,
where
$\displaystyle{\mathfrak{g}}_{k}=\mathop{\oplus}\limits_{m_{\alpha}(\beta)=k}{\mathfrak{g}}_{\beta}$
for each $\displaystyle k\in{\mathbb{Z}}$. Then $\displaystyle X=G/P_{\alpha}$
is called an IHSS if $\displaystyle{\mathfrak{g}}$ admits a short grading:
$\displaystyle{\mathfrak{g}}={\mathfrak{g}}_{-1}\oplus{\mathfrak{g}}_{0}\oplus{\mathfrak{g}}_{1}$,
or equivalently $\displaystyle R_{u}(P_{\alpha}^{-})$ and $\displaystyle
R_{u}(P_{\alpha})$ are vector groups (see for example Ar11). In this case,
$\displaystyle G/P_{\alpha}$ is an equivariant compactification of the vector
group $\displaystyle R_{u}(P_{\alpha}^{-})$, with origin $\displaystyle y$.
Define a $\displaystyle{\mathbb{C}}^{*}$-action on
$\displaystyle{\mathcal{D}}(I)$ by the cocharcter
$\displaystyle\sigma_{\alpha}:{\mathbb{G}}_{m}\rightarrow T$, such that
$\displaystyle\sigma_{\alpha}(\beta)=\delta_{\alpha,\beta},\forall\beta\in\Delta$.
The $\displaystyle{\mathbb{C}}^{*}$-action is equalized at the sink
$\displaystyle y$ and $\displaystyle C^{-}(y)=R_{u}(P^{-})\cdot y$. This shows
that $\displaystyle{\mathcal{D}}(I)$ is Euler-symmetric by Proposition 2.4.
Denote by $\displaystyle H\subset P_{\alpha}$ the Levi-part of $\displaystyle
P$, then $\displaystyle y$ is $\displaystyle H$-fixed. And
$\displaystyle{\mathfrak{g}}_{0}=Lie(H),{\mathfrak{g}}_{1}=Lie(R_{u}(P_{\alpha}))$,
$\displaystyle{\mathfrak{g}}_{-1}=Lie(R_{u}(P_{\alpha}^{-}))$. Finally the
prolongation of $\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{y}})$ can
be interpreted as adjoint actions inside the simple grading algebra
$\displaystyle{\mathfrak{g}}$ as follows.

(1) $\displaystyle
T_{y}X={\mathfrak{g}}/{\mathfrak{p}}\cong{\mathfrak{g}}_{-1}$ and
$\displaystyle{\mathcal{C}}_{y}\subset{\mathbb{P}}T_{y}X$ is the unique closed
orbit of the isotropy action of $\displaystyle H$ on
$\displaystyle{\mathbb{P}}T_{y}X\cong{\mathbb{P}}{\mathfrak{g}}_{-1}$.

(2)
$\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{y}})\cong{\mathfrak{g}}_{0}$
is given by the adjoint action of $\displaystyle{\mathfrak{g}}_{0}$ on
$\displaystyle{\mathfrak{g}}_{-1}$.

(3)
$\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})^{(1)}\cong{\mathfrak{g}}_{1}$
is given by the adjoint action:

(3.1) $\displaystyle\displaystyle{\mathfrak{g}}_{1}$
$\displaystyle\displaystyle\rightarrow
Hom({\mathfrak{g}}_{-1},{\mathfrak{g}}_{0})$
$\displaystyle\displaystyle\alpha$
$\displaystyle\displaystyle\rightarrow(\beta\rightarrow[\alpha,\beta]),$

where the image lies in $\displaystyle Sym^{2}T_{x}^{*}X\otimes T_{x}(X)$ as
$\displaystyle{\mathfrak{g}}_{-1}$ is abelian. We list IHSS and their VMRTs in
the following table.

Table 1. IHSS and their VMRTs IHSS $\displaystyle X=G/P$ | $\displaystyle{\mathbb{Q}}^{n}$ | $\displaystyle Gr(a,a+b)$ | $\displaystyle{\mathbb{S}}_{n}$ | $\displaystyle Lag(n,2n)$ | $\displaystyle E_{6}/P_{1}$ | $\displaystyle E_{7}/P_{7}$
---|---|---|---|---|---|---
VMRT $\displaystyle{\mathcal{C}}_{x}$ | $\displaystyle{\mathbb{Q}}^{n-2}$ | $\displaystyle{\mathbb{P}}^{a-1}\times{\mathbb{P}}^{b-1}$ | $\displaystyle Gr(2,n)$ | $\displaystyle{\mathbb{P}}^{n-1}$ | $\displaystyle{\mathbb{S}}_{5}$ | $\displaystyle E_{6}/P_{1}$
$\displaystyle{\mathcal{C}}_{x}\subset{\mathbb{P}}T_{x}(X)$ | Hyperquadric | Segre | Pl$\displaystyle\ddot{u}$cker | second Veronese | Spinor | Severi

Now back to the general situation, assume that for a general pair of points
$\displaystyle x,y$ on $\displaystyle X$, there is a
$\displaystyle{\mathbb{C}}^{*}$-action which is of Euler type at
$\displaystyle x$ and its inverse action is of Euler type at $\displaystyle
y$. Take a very ample line bundle $\displaystyle{\mathcal{L}}$ on
$\displaystyle X$ and denote $\displaystyle V=H^{0}(X,{\mathcal{L}})^{\vee}$.
Let $\displaystyle f:X\rightarrow{\mathbb{P}}V$ be the projective embedding.
Recall $\displaystyle\rho_{x},\rho_{y}$ the linear actions of $\displaystyle
T_{x}X$ and $\displaystyle T_{y}X$ on $\displaystyle V$ respectively. We
denote by
$\displaystyle{\mathfrak{g}}_{1}=\text{I}m(d\rho_{x})\subset{\mathfrak{g}}l(V)$
and
$\displaystyle{\mathfrak{g}}_{-1}=\text{I}m(d\rho_{y})\subset{\mathfrak{g}}l(V)$.
By definition
$\displaystyle{\mathfrak{g}}_{1},{\mathfrak{g}}_{-1}\subset{\mathfrak{a}}ut(\hat{X})$,
thus
$\displaystyle\forall\alpha\in{\mathfrak{g}}_{1},\forall\beta\in{\mathfrak{g}}_{-1}$,
$\displaystyle\gamma:=[\alpha,\beta]$ lies in
$\displaystyle{\mathfrak{a}}ut(\hat{X})$ as well. Denote by $\displaystyle
G_{\gamma}:=\\{exp(z\gamma)\,|\,z\in{\mathbb{C}}\\}\subset GL(V)$ the one-
parameter subgroup. Then the action of $\displaystyle G_{\gamma}$ on
$\displaystyle{\mathbb{P}}V$ leaves $\displaystyle X$ invariant. Moreover we
have:

######  Lemma 3.4.

$\displaystyle x$ and $\displaystyle y$ are fixed by $\displaystyle
G_{\gamma}$. Denote by $\displaystyle\Phi_{\gamma}:G_{\gamma}\rightarrow
GL(T_{x}X)$ the induced isotropy action of $\displaystyle G_{\gamma}$ on
$\displaystyle T_{x}X$, then for any $\displaystyle w\in T_{x}X$ we have:

(3.2) $\Pi_{1}((d\Phi_{\gamma})(\gamma)(w))=[\gamma,d\phi_{x}(w)]\cdot
e_{0}\in V_{1}.$

###### Proof.

By Proposition 2.9(iii) and Corollary 2.10(3), $\displaystyle\alpha\cdot
V_{k}\subset V_{k+1},$ and $\displaystyle\beta\cdot V_{k+1}\subset V_{k}$ for
each $\displaystyle 0\leqslant k\leqslant r-1$. Thus $\displaystyle\gamma\cdot
V_{k}\subset V_{k}$ for any $\displaystyle 0\leqslant k\leqslant r$, whence
$\displaystyle G_{\gamma}\cdot V_{k}\subset V_{k}$ for each $\displaystyle k$.
This particularly shows that $\displaystyle x$ and $\displaystyle y$ are fixed
by $\displaystyle G_{\gamma}$ as $\displaystyle x=[e_{0}],y=[e_{r}]$ and
$\displaystyle V_{0},V_{r}$ are both of dimension one. It also implies that
the action of $\displaystyle G_{\gamma}$ commutes with the
$\displaystyle{\mathbb{C}}^{*}$-action. Now for any nonzero tangent vector
$\displaystyle w\in T_{x}X$, under the identification $\displaystyle
C^{+}(x)\cong T_{x}X$, consider the holomorphic arc $\displaystyle\theta_{w}$
on X through $\displaystyle x$:

$\displaystyle\displaystyle\theta_{w}:{\mathbb{C}}$
$\displaystyle\displaystyle\rightarrow X\subset{\mathbb{P}}V$
$\displaystyle\displaystyle z$ $\displaystyle\displaystyle\rightarrow
f(zw)=[\sum_{k=0}^{r}z^{k}\Gamma_{w}^{k}(e_{0})]$

Denote the closure of its image to be $\displaystyle C_{w}$. Then
$\displaystyle C_{w}$ is a non-trival $\displaystyle{\mathbb{C}}^{*}$-orbit
closure with source $\displaystyle x$. As the action of $\displaystyle
G_{\gamma}$ commutes with the $\displaystyle{\mathbb{C}}^{*}$-action,
$\displaystyle g\cdot C_{w}$ is also a non-trival
$\displaystyle{\mathbb{C}}^{*}$-orbit closure through $\displaystyle g\cdot
x=x$ for any $\displaystyle g\in G_{\gamma}$. This implies that $\displaystyle
g\cdot f(w)\in C^{+}(x)$ for any $\displaystyle w$, in other words we have
$\displaystyle g\cdot C^{+}(x)\subset C^{+}(x)$.

Now we calculate the action of $\displaystyle G_{\gamma}$ on $\displaystyle
T_{x}X$ via $\displaystyle T_{x}X\cong C^{+}(x)$. Write
$\displaystyle\gamma\cdot e_{0}=ce_{0}$ for some $\displaystyle
c\in{\mathbb{C}}$, then for any $\displaystyle g_{t}=exp(t\gamma)$ we have:

$\displaystyle\displaystyle f(g_{t}\cdot w)$
$\displaystyle\displaystyle=[\sum_{k=0}^{r}\Gamma_{g_{t}\cdot
w}^{k}(e_{0})]=[e_{0}+\Gamma_{g_{t}.w}^{1}(e_{0})+\sum_{k=2}^{r}\Gamma_{g_{t}.w}^{k}(e_{0})],$
$\displaystyle\displaystyle g_{t}\cdot f(w)$
$\displaystyle\displaystyle=g_{t}\cdot[\sum_{k=0}^{r}\Gamma_{w}^{k}(e_{0})]=[e_{0}+e^{-tc}g_{t}\cdot\Gamma_{w}^{1}(e_{0})+\sum_{k=2}^{r}e^{-tc}g_{t}\cdot\Gamma_{w}^{k}(e_{0})].$

Thus from $\displaystyle f(g_{t}.w)=g_{t}\cdot f(w)$ we conclude that:

(3.3) $\Pi_{1}(g_{t}\cdot w)=\Gamma_{g_{t}\cdot
w}^{1}(e_{0})=exp(t\,\gamma|_{V_{1}}-tc\,\,\text{Id}|_{V_{1}})\cdot\Pi_{1}(w),$

from which we see that the action of $\displaystyle G_{\gamma}$ on
$\displaystyle T_{x}X$ via $\displaystyle T_{x}X\cong C^{+}(x)$ is linear.
Then as the differential map of the isomorphism $\displaystyle T_{x}X\cong
C^{+}(x)$ at $\displaystyle 0\in T_{x}X$ equals I$\displaystyle d|_{T_{x}X}$,
we conclude that $\displaystyle\Phi_{\gamma}(g)(w)=g\cdot w$. Thus:

$\displaystyle\displaystyle\Pi_{1}(d\Phi_{\gamma)}(\gamma)(w))$
$\displaystyle\displaystyle=\frac{d}{dt}_{|t=0}(\Pi_{1}(g_{t}\cdot
w))=\frac{d}{dt}_{|t=0}(e^{-tc}e^{t\gamma}\cdot\Pi_{1}(w))$
$\displaystyle\displaystyle=-c\,\Pi_{1}(w)+\gamma.\Pi_{1}(w)=[\gamma,d\phi_{x}(w)].e_{0}$

from $\displaystyle\Pi_{1}(w)=\Gamma_{w}(e_{0})=d\phi_{x}(w)\cdot e_{0}$. ∎

As $\displaystyle G_{\gamma}$ fixes $\displaystyle x$, it acts on the family
of minimal rational curves through $\displaystyle x$. Thus the image of
$\displaystyle\Phi_{\gamma}$ is contained in $\displaystyle
Aut^{0}(\hat{{\mathcal{C}}_{x}})$ and $\displaystyle
d\Phi_{\gamma}(\gamma)\in{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})$. Then
under the identification $\displaystyle\Pi_{1}$, we can rewrite Lemma 3.4 as
follows, which is a generalization of the map (3.1).

######  Corollary 3.5.

Consider the linear map $\displaystyle\lambda:{\mathfrak{g}}_{-1}\rightarrow
Sym^{2}(V_{1}^{*})\otimes V_{1}$ given by:

(3.4)
$\displaystyle\displaystyle\lambda(d\phi_{y}(\beta)):\,\,\,\,\,\,\,\,\,V_{1}\times
V_{1}$ $\displaystyle\displaystyle\longrightarrow V_{1}$
$\displaystyle\displaystyle(\Pi_{1}(\alpha),\Pi_{1}(\xi))$
$\displaystyle\displaystyle\longrightarrow[[d\phi_{y}(\beta),d\phi_{x}(\alpha)],d\phi_{x}(\xi)].e_{0},$

for any $\displaystyle\beta\in T_{y}X$ and for any $\displaystyle\alpha,\xi\in
T_{x}X$. Then
Im$\displaystyle(\lambda)\subset{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})^{(1)}$,
under the identiication $\displaystyle\Pi_{1}$.

###### Proof.

It suffices to check $\displaystyle\lambda(d\phi_{y}(\alpha)$ is symmetric,
which follows from the fact that $\displaystyle{\mathfrak{g}}_{1}$ is abelian.
∎

Now Theorem 1.4 is a corollary of the following proposition.

######  Proposition 3.6.

$\displaystyle\lambda$ induces an isomorhpism from
$\displaystyle{\mathfrak{g}}_{-1}$ onto
$\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})^{(1)}$.

###### Proof.

By [HM05, Theorem 1.1.3] there is a natural inclusion:
$\displaystyle{\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})^{1}\hookrightarrow(T_{x}X)^{\vee}$,
whence it suffices to show that $\displaystyle\lambda$ is injective.

Assume otherwise that $\displaystyle\lambda(d\phi_{y}(\beta))=0$ for some
nonzero $\displaystyle\beta\in T_{y}X$. For any $\displaystyle\alpha\in
T_{x}X$, denote by
$\displaystyle\gamma_{\alpha}=[d\phi_{y}(\beta),d\phi_{x}(\alpha)]$. We now
show that $\displaystyle\gamma_{\alpha}=l(\alpha)\,\text{Id}_{V}$ for some
$\displaystyle l\in(T_{x}X)^{\vee}$. To do this it suffices to show that
$\displaystyle G_{\gamma_{\alpha}}$ acts on $\displaystyle V$ by scalar
multiplications. By Lemma 3.4 we have $\displaystyle
d\Phi_{\gamma_{\alpha}}(\gamma_{\alpha})=0$. From (3.3) it implies that
$\displaystyle G_{\gamma_{\alpha}}$ acts trivally on $\displaystyle C^{+}(x)$,
and consequently on $\displaystyle X$ as $\displaystyle C^{+}(x)$ is open
dense in $\displaystyle X$. Thus for any nonzero vector $\displaystyle
v\in\hat{X}$, $\displaystyle v$ is an eigenvector of the linear action of
$\displaystyle G_{\gamma_{\alpha}}$ on $\displaystyle V$. For any
$\displaystyle g\in G_{\gamma_{\alpha}}$ denote by
$\displaystyle\\{V_{g(c)}:c\in J_{g}\\}$ the characteristic subspaces of the
action of $\displaystyle g$ on $\displaystyle V$, then $\displaystyle
X\subset\cup_{c\in J_{g}}{\mathbb{P}}V_{g(c)}$. As $\displaystyle X$ is non-
degenrate and irreducible, there exists some $\displaystyle c\in J_{g}$ such
that $\displaystyle V_{g(c)}=V$, i.e., $\displaystyle g$ acts on
$\displaystyle V$ by scalar multiplications.

Now denote by $\displaystyle
U_{\beta}=\\{exp(d\phi_{y}(t\beta)):t\in{\mathbb{C}}\\}\subset GL(V)$ the
1-dimensional vector subgroup of I$\displaystyle m(\phi_{y})$ and
$\displaystyle V_{l}=\\{exp(d\phi_{x}(\alpha)):l(\alpha)=0\\}$ the vector
subgroup of Im$\displaystyle(\phi_{x})$. Then by Corollary 2.10 $\displaystyle
x$ is fixed by $\displaystyle U_{\beta}$ as $\displaystyle x=[e_{0}]$ and
$\displaystyle d\phi_{y}(\beta)\cdot e_{0}=0$. Thus $\displaystyle V_{l}.x$ is
also fixed by $\displaystyle U_{\beta}$ as the action of $\displaystyle
U_{\beta}$ and $\displaystyle V_{l}$ on $\displaystyle X$ commutes by our
definition of $\displaystyle l$. On the other hand as
$\displaystyle\beta\not=0$, $\displaystyle U_{\beta}$ acts freely on
$\displaystyle C^{-}(y)\cong T_{y}X$. This implies that $\displaystyle
V_{l}\cdot x\subset C^{+}(x)\backslash C^{-}(y)$. We prove that this induces a
contradiction:

If $\displaystyle l=0$ then $\displaystyle V_{l}=Im(\rho_{x})$ and
$\displaystyle V_{l}\cdot x=C^{+}(x)$. But $\displaystyle C^{+}(x)\cap
C^{-}(y)$ is non-empty as they are both open dense in $\displaystyle X$.

If $\displaystyle l\not=0$ then $\displaystyle V_{l}\cdot x$ is a hyperplane
in $\displaystyle C^{+}(x)\cong T_{x}X$. By the proof of Lemma 2.7,
$\displaystyle D_{y}:=X\backslash C^{-}(y)$ is an irreducible divisor of
$\displaystyle X$. So as an open subset of $\displaystyle D_{y}$,
$\displaystyle C^{+}(x)\backslash C^{-}(y)$ is an irreducible divisor of
$\displaystyle C^{+}(x)$. This implies that $\displaystyle C^{+}(x)\backslash
C^{-}(y)$ equals the hyperplane $\displaystyle V_{l}\cdot x$. On the other
hand by Corollary 2.10 we can write $\displaystyle C^{+}(x)\backslash
C^{-}(y)=\\{w\in T_{x}X:\eta(s_{r})(w)=0\\}$ where $\displaystyle s_{r}$ is a
nonzero section in $\displaystyle H^{0}(X,{\mathcal{L}})_{w_{r}}$. Thus
$\displaystyle
Bs({\mathbb{F}}_{x}^{r})=\\{[w]\in{\mathbb{P}}T_{x}X:\eta(s_{r})(w)=0\\}$ is a
hyperplane in $\displaystyle{\mathbb{P}}T_{x}X$. But then
$\displaystyle{\mathcal{C}}_{x}=Bs({\mathbb{F}}_{x})\subset
Bs({\mathbb{F}}_{x}^{r})$ is linear degenerate in
$\displaystyle{\mathbb{P}}T_{x}X$, contradicting Proposition 3.3. ∎

###  3.2. Projective subvarieties with nonzero prolongations

Let us recall the classification result of projective subvarieties with
nonzero prolongations by Fu and Hwang as follows.

######  Theorem 3.7.

[FH12, Main Theorem] and [FH18, Theorem 7.13] Let $\displaystyle
S\subset{\mathbb{P}}V$ be an irreducible nonsingular nondegenerate variety
such that $\displaystyle{\mathfrak{a}}ut(\hat{S})^{1}\not=0$. Then
$\displaystyle S\subset{\mathbb{P}}V$ is projectively equivalent to one of the
followings:

(1) The VMRT of an IHSS.

(2) The VMRT of a symplectic Grassmannian.

(3) A nonsingular linear section of $\displaystyle
Gr(2,5)\subset{\mathbb{P}}^{9}$ of codimension $\displaystyle\leqslant 2$.

(4) A nonsingular $\displaystyle{\mathbb{P}}^{4}$-general linear section of
$\displaystyle{\mathbb{S}}_{5}\subset{\mathbb{P}}^{15}$ of codimension
$\displaystyle\leqslant 3$.

(5) Biregular projections of (1) and (2) with nonzero prolongations, which are
completely described in Section 4 of [FH12] .

######  Remark 3.8.

As noted in [FH18, Proposition 2.11], all nonsingular sections of
$\displaystyle Gr(2,5)\subset{\mathbb{P}}^{9}$ with codimension $\displaystyle
s\leqslant 3$ are projectively equivalent.

The main result of this subsection is the following result based on Theorem
3.7.

######  Proposition 3.9.

Let $\displaystyle S\subset{\mathbb{P}}V$ be one of the projective
subvarieties in Theorem 3.7 (2)(3)(4)(5), then:

(3.5) $dim({\mathfrak{a}}ut(\hat{S})^{(1)})<dim(V).$

We will prove this proposition case by case based on Theorem 3.7.

####  3.2.1. Case (2) and (5)

In these cases, the prolongation of $\displaystyle{\mathfrak{a}}ut(\hat{S})$
was explicitly formulated in [FH12]. First we consider Case (2) and the case
of biregular projections of (2):

######  Lemma 3.10.

Let $\displaystyle W$ and $\displaystyle Q$ be vector spaces of dimensions
$\displaystyle k\geqslant 2$ and $\displaystyle m$ respectively. Set
$\displaystyle L=Sym^{2}(Q)\subset V=Sym^{2}(W\oplus Q)$ and $\displaystyle
U=V/L$. For $\displaystyle\phi\in Sym^{2}(W\oplus Q)$ denote by
$\displaystyle\phi^{\\#}\in Hom(W^{\vee}\oplus Q^{\vee},W\oplus Q)$ the
corresponding homomorphism via the natrual inclusion $\displaystyle
Sym^{2}(W\oplus Q)\subset Hom((W^{\vee}\oplus Q^{\vee},W\oplus Q)$. For
$\displaystyle L_{2}\subset U$, let $\displaystyle\text{Im}(L_{2})$ be the
linear space of $\displaystyle\\{\text{Im}(\phi^{\\#}):\overline{\phi}\in
L_{2}\\}$. Define $\displaystyle\text{Im}_{W}(L_{2})=P_{Q}(Im(L_{2}))\subset
W$, where $\displaystyle P_{Q}:W\oplus Q\rightarrow W$ is the projection to
the first factor, then:

(i) Denote by $\displaystyle
p_{L}:{\mathbb{P}}V\dashrightarrow{\mathbb{P}}(V/L)$ the projection from
$\displaystyle{\mathbb{P}}L$. Let $\displaystyle v_{2}:{\mathbb{P}}(W\oplus
Q)\rightarrow{\mathbb{P}}(Sym^{2}(W\oplus Q))$ be the second Veronese
embedding, $\displaystyle Z$ the proper image of $\displaystyle Im(v_{2})$.
Then $\displaystyle Z\subset{\mathbb{P}}V/L={\mathbb{P}}U$ is isomorphic to
the VMRT of the sympletic Grassmannian $\displaystyle Gr_{w}(k,\Sigma)$ at a
general point and $\displaystyle{\mathfrak{a}}ut(\hat{Z})^{(1)}\cong
Sym^{2}(W^{\vee})$.

(ii) If $\displaystyle Z\cap{\mathbb{P}}L_{2}=\emptyset$, then
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L_{2}}(Z)})^{(1)}\cong
Sym^{2}(W/Im_{W}(L_{2}))^{\vee}$.

(iii) $\displaystyle dim({\mathfrak{a}}ut(\hat{Z})^{(1)})<dim(V/L)$. Let
$\displaystyle L_{2}\subset U$ be as in (ii), if
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L_{2}}(Z)})^{(1)}\not=0$, then:

$dim({\mathfrak{a}}ut(\widehat{p_{L_{2}}(Z)})^{(1)})<dim(U/L_{2}).$

###### Proof.

(i) and (ii) are from [FH12, Proposition 4.18].

Under the identification: $\displaystyle V=Sym^{2}(W\oplus Q)\subset
Hom(W^{\vee}\oplus Q^{\vee},W\oplus Q)$, we write $\displaystyle
U=V/L=Sym^{2}(W)\oplus Hom(Q^{\vee},W)$, where we identify $\displaystyle
Sym^{2}(W)$ inside $\displaystyle Hom(W^{\vee},W)$. Thus $\displaystyle
dim(V/L)>dim({\mathfrak{a}}ut(\widehat{Z})^{(1)})$ as $\displaystyle
Hom(Q^{\vee},W)\not=0$. Take a basis of $\displaystyle Im_{W}(L_{2})$ to be
$\displaystyle e_{1},...,e_{t}$ and extend it to a basis of $\displaystyle W$:
$\displaystyle e_{1},...,e_{t},e_{t+1},...,e_{k}$. Then by the definition of
$\displaystyle Im_{W}(L_{2})$, we have:

$\displaystyle\displaystyle L_{2}\subset$
$\displaystyle\displaystyle\\{(\phi,\eta)\in Sym^{2}(W)\oplus
Hom(Q^{\vee},W)\,|\,Im(\phi^{\\#})\subset
Im_{W}(L_{2})\,\,and\,\,Im(\eta)\subset Im_{W}(L_{2})\\}$
$\displaystyle\displaystyle\cong Sym^{2}(Im_{W}(L_{2}))\oplus
Hom(Q^{\vee},Im_{W}(L_{2})),$

whence $\displaystyle dim(L_{2})\leqslant\frac{t(t+1)}{2}+mt$. Then:

$\displaystyle\displaystyle
dim(U/L_{2})-dim({\mathfrak{a}}ut(\widehat{p_{L_{2}}(Z)})^{(1)})$
$\displaystyle\displaystyle\geqslant\frac{k(k+1)}{2}+km-\frac{t(t+1)}{2}-tm-\frac{(k-t)(k-t+1)}{2}$
$\displaystyle\displaystyle=(m+t)(k-t)>0,$

where $\displaystyle k-t>0$ as
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L_{2}}(Z)})^{(1)}\not=0$. ∎

By [FH12, Main Theorem (C)], the other cases of (5) are biregular projections
of the VMRT of $\displaystyle Gr(a,a+b),{\mathbb{S}}_{n}$ and $\displaystyle
Lag(n,2n)$ respectively. We prove these cases by the following three lemmas.

######  Lemma 3.11.

Let $\displaystyle A$ and $\displaystyle B$ be vector spaces with
$\displaystyle a=dim(A)\geqslant b=dim(B)\geqslant 3$. Let $\displaystyle
V=Hom(A,B)$. For a subspace $\displaystyle L\subset V$, set
$\displaystyle\text{Im}(L)=\\{\text{Im}(\phi)\subset B:\phi\in L\\}$.
$\displaystyle Ker(L)=\bigcap\limits_{\phi\in L}Ker(\phi)$. Then:

(i) $\displaystyle S=\\{[\phi]\in{\mathbb{P}}V:rank(\phi)\leqslant
1\\}\subset{\mathbb{P}}V$ is projectively isomorphic to the VMRT of
$\displaystyle Gr(a,a+b)$.

(iii) Let $\displaystyle L\subset V$ such that $\displaystyle L\cap
Sec(S)=\emptyset$, then
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\cong
Hom(B/Im(L),Ker(L))$

(iii) Let $\displaystyle L\subset V$ be as in (ii), if
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\not=0$ then
$\displaystyle dim({\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)})<dim(V/L)$.

###### Proof.

(i) and (ii) are from [FH12, Proposition 4.10]. For (iii), denote
$\displaystyle s=dim(Ker(L))$ and $\displaystyle t=dim(Im(L))$, by the
definition we have

$L\subset\\{\phi\in Hom(A,B):\phi|_{Ker(L)}=0,Im(\phi)\subset
Im(L)\\}=Hom(A/Ker(L),Im(L)),$

implying $\displaystyle dim(L)\leqslant(a-s)t$. Thus

$dim(V/L)-dim({\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)})=ab-
dim(L)-(b-t)s\geqslant ab-(a-s)t-(b-t)s=(a-t)(b-t)+st>0,$

where $\displaystyle s<a$ and $\displaystyle t<b$ as
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\not=0$. ∎

######  Lemma 3.12.

Let $\displaystyle W$ be a vector space of dimension $\displaystyle n\geqslant
6$. $\displaystyle V=\wedge^{2}W$. For each $\displaystyle\phi\in\wedge^{2}W$,
denote by $\displaystyle\phi^{\\#}\in Hom(W^{\vee},W)$ via the inclusion
$\displaystyle\wedge^{2}W\subset W\otimes W=Hom(W^{\vee},W)$. For a subspace
$\displaystyle L\subset V$, define $\displaystyle Im(L)\subset W$ as the
linear span of $\displaystyle\\{Im(\phi^{\\#})\subset W,\phi\in L\\}$. Then:

(i) $\displaystyle S=\\{[\phi]\in V:rk(\phi)\leqslant
2\\}\subset{\mathbb{P}}V$ is isomorphic to the VMRT of
$\displaystyle{\mathbb{S}}_{n}$.

(ii) If $\displaystyle L\subset V$ such that $\displaystyle{\mathbb{P}}L\cap
Sec(S)=\emptyset$, then
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\cong\wedge^{2}(W/Im(L))^{\vee}$.

(iii) Let $\displaystyle L\subset V$ be as in (ii), if
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\not=0$, then
$\displaystyle dim(\widehat{p_{L}(S)})^{(1)})<dim(V/L)$.

###### Proof.

(i) and (ii) are from [FH12, Proposition 4.11]. For (iii) take a basis of
$\displaystyle Im(L)$ to be $\displaystyle e_{1},...,e_{t}$ and extend it to a
basis of $\displaystyle W$: $\displaystyle e_{1},...,e_{t},e_{t+1},...,e_{n}$.
Denote the dual basis of $\displaystyle W^{\vee}$ to be $\displaystyle
f_{1},...,f_{n}$ such that $\displaystyle f_{i}(e_{j})=\delta_{i,j}$ for any
$\displaystyle 1\leqslant i,j\leqslant n$. Identify $\displaystyle
Hom(W^{\vee},W)$ with $\displaystyle M_{n\times n}({\mathbb{C}})$ through:

(3.6) $\displaystyle\displaystyle Hom(W^{\vee},W)$
$\displaystyle\displaystyle\longrightarrow M_{n\times n}({\mathbb{C}})$
$\displaystyle\displaystyle{\mathcal{A}}$
$\displaystyle\displaystyle\longrightarrow A=(a_{ij}:1\leqslant i,j\leqslant
n)$

such that $\displaystyle{\mathcal{A}}(f_{i})=\sum_{j=1}^{n}a_{ij}e_{j}$. Then
$\displaystyle V$ corresponds to all skew-symmetric matrices. Now

$\displaystyle\displaystyle L\subset\\{{\mathcal{A}}\in
V:Im({\mathcal{A}})\subset Im(L)\\}=\\{{\mathcal{A}}\in
V:a_{ij}=0\,\,\,\text{if \,\,\,}i\geqslant r+1\text{\,\,\,or\,\,\,}j\geqslant
r+1\\},$

thus $\displaystyle dim(L)\leqslant\frac{t(t-1)}{2}$. Then:

$\displaystyle\displaystyle
dim(V/L)-dim({\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)})$
$\displaystyle\displaystyle=\frac{n(n-1)}{2}-dim(L)-\frac{(n-t)(n-t-1)}{2}$
$\displaystyle\displaystyle\geqslant\frac{n(n-1)}{2}-\frac{t(t-1)}{2}-\frac{(n-t)(n-t-1)}{2}=t(n-t)>0,$

where $\displaystyle t>0$ as $\displaystyle L\not=0$ and $\displaystyle t<n$
as $\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\not=0$. ∎

######  Lemma 3.13.

Let $\displaystyle W$ be a vector space of dimension $\displaystyle n\geqslant
3$. $\displaystyle V=Sym^{2}W$. For each $\displaystyle\phi\in Sym^{2}W$,
denot by $\displaystyle\phi^{\\#}\in Hom(W^{\vee},W)$ via the inclusion
$\displaystyle Sym^{2}W\subset W\otimes W=Hom(W^{\vee},W)$. For a subspace
$\displaystyle L\subset V$, define $\displaystyle Im(L)\subset W$ as the
linear span of $\displaystyle\\{Im(\phi^{\\#})\subset W,\phi\in L\\}$. Then:

(i) $\displaystyle S=\\{[\phi]\in V:rk(\phi)\leqslant
1\\}\subset{\mathbb{P}}V$ is isomorphic to the VMRT of $\displaystyle
Lag(n,2n)$.

(ii) If $\displaystyle L\subset V$ such that $\displaystyle{\mathbb{P}}L\cap
Sec(S)=\emptyset$, then
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\cong
Sym^{2}(W/Im(L))^{\vee}$.

(iii) Let $\displaystyle L\subset V$ be as in (ii), if
$\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\not=0$, then
$\displaystyle dim(\widehat{p_{L}(S)})^{(1)})<dim(V/L)$.

###### Proof.

(i) and (ii) are from [FH12, Proposition 4.12] . For (iii), as in Lemma 3.11
we take a basis of $\displaystyle Im(L)$ to be $\displaystyle e_{1},...,e_{r}$
and extend it to a basis of $\displaystyle W$ to be $\displaystyle
e_{1},...,e_{r},e_{r+1},...,e_{n}$. Denote the dual basis to be $\displaystyle
f_{1},...,f_{n}$. Keep the identification (3.5) then $\displaystyle V$
corresponds to all symmetric matrices and we have

$\displaystyle\displaystyle L\subset\\{{\mathcal{A}}\in
V:Im({\mathcal{A}})\subset Im(L)\\}=\\{{\mathcal{A}}\in
V:a_{ij}=0\,\,\,\text{if \,\,\,}i\geqslant r+1\text{\,\,\,or\,\,\,}j\geqslant
r+1\\},$

implying $\displaystyle dim(L)\leqslant\frac{r(r+1)}{2}$ and thus

$\displaystyle\displaystyle
dim(V/L)-dim({\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)})$
$\displaystyle\displaystyle=\frac{n(n+1)}{2}-dim(L)-\frac{(n-r)(n-r+1)}{2}$
$\displaystyle\displaystyle\geqslant\frac{n(n+1)}{2}-\frac{r(r+1)}{2}-\frac{(n-r)(n-r+1)}{2}=r(n-r)>0,$

where $\displaystyle r>0$ as $\displaystyle L\not=0$ and $\displaystyle r<n$
as $\displaystyle{\mathfrak{a}}ut(\widehat{p_{L}(S)})^{(1)}\not=0$. ∎

####  3.2.2. Case (3)

Let $\displaystyle X=Gr(2,5)\subset{\mathbb{P}}^{9}$. For each $\displaystyle
k=1,2$, denote by $\displaystyle X_{k}\subset{\mathbb{P}}^{9-k}$ the
nonsingular linear section of codimension $\displaystyle k$. Then Case
$\displaystyle X_{1}$ follows from [FH12, Section 3.4] and Case $\displaystyle
X_{2}$ follows from [BFM20, Lemma 4.6].

####  3.2.3. Case (4)

Let $\displaystyle S={\mathbb{S}}_{5}\subset{\mathbb{P}}^{15}$. For each
$\displaystyle k=1,2,3$, denote by $\displaystyle
S_{k}\subset{\mathbb{P}}^{15-k}$ the nonsingular
$\displaystyle{\mathbb{P}}^{4}$-general linear section of codimension
$\displaystyle k$ as described in [FH18, Proposition 2.12].

(i) Case $\displaystyle S_{1}$ follows from [FH12, Section 3.3].

(ii) By [FH18, Proposition 7.6] $\displaystyle S_{3}$ is quadratically
symmetric. The VMRT of $\displaystyle S_{3}$ at a general point is a
nonsingular linear section of $\displaystyle Gr(2,5)\subset{\mathbb{P}}^{9}$
of codimension 3, which has zero prolongations by Theorem 3.7. Then by the
proof of [FH12, Theorem 6.15] we conclude that
$\displaystyle{\mathfrak{a}}ut(\hat{S_{3}})\cong{\mathbb{C}}$.

(iii) To prove Case $\displaystyle S_{2}$ we recall the following
characterization of $\displaystyle S_{2}$ proved by Kuznetsov.

######  Theorem 3.14.

[Ku18, Proposition 6.1 and Lemma 6.7] Let $\displaystyle S_{K}\subset S$ be a
nonsingular linear section of $\displaystyle S$ of codimension 2, then the
followings are equivalent:

(a) $\displaystyle S_{K}$ is projectively equivalent to $\displaystyle S_{2}$;

(b) The Hilbert space $\displaystyle F_{4}(S_{K})$ of linear 4-spaces on
$\displaystyle S_{K}$ is non-empty;

(c) There exists a line $\displaystyle L$ in $\displaystyle S_{K}$ such that

(3.7)
${\mathcal{N}}_{L/S_{K}}\cong{\mathcal{O}}_{{\mathcal{L}}}(-2)\oplus{\mathcal{O}}_{L}(1)^{\oplus
6}$

Moreover, such line is unique and is equal to the intersection of all 4-spaces
on $\displaystyle S_{K}$.

Now assume otherwise that $\displaystyle
dim({\mathfrak{a}}ut(\hat{S_{2}})^{(1)})=dim(V)$. Take $\displaystyle L$ as
the line defined in Theorem 3.14. For any point $\displaystyle x=[\hat{x}]\in
L$, take a linear function $\displaystyle l\in V^{\vee}$ such that
$\displaystyle l(\hat{x})\not=0$. Then by [HM05, Proposition 2.3.1] there
exists a unique
$\displaystyle{\mathcal{A}}\in{\mathfrak{a}}ut(\hat{S_{2}})^{(1)}$ such that
$\displaystyle{\mathcal{A}}_{\alpha,\alpha}=l(\alpha)\alpha$ for any
$\displaystyle\alpha\in\hat{S_{2}}$. Moreover denote by $\displaystyle
P_{\alpha}=T_{\alpha}(\hat{S_{2}})$ the tangent space of
$\displaystyle\hat{S_{2}}$ at $\displaystyle\alpha$, then we have:

(3.8) $2{\mathcal{A}}_{\alpha,\beta}=l(\alpha)\beta+l(\beta)\alpha,$

for any $\displaystyle\alpha\in\hat{S_{2}}$ and any $\displaystyle\beta\in
P_{\alpha}$. Denote by $\displaystyle s$ the semisimple part of
$\displaystyle{\mathcal{A}}_{\hat{x}}$. By the proof of [HM05, Theorem 1.1.3],
the one parameter subgroup $\displaystyle\\{exp(2ts):t\in{\mathbb{C}}\\}$
induces a $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle S_{2}$
which is of Euler type at $\displaystyle x$. We shall deduce the contradiction
from the $\displaystyle{\mathbb{C}}^{*}$-action.

First we claim that there are exactly three different weight subspaces of the
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle V$. In fact by
$\displaystyle{\mathcal{A}}_{\hat{x}}(V)\subset P_{\hat{x}}$ and by (3.8), the
linear action of $\displaystyle{\mathbb{C}}^{*}$ on $\displaystyle V$ has at
most three different weight subspaces. On the other hand denote by
$\displaystyle{\mathcal{L}}={\mathcal{O}}_{{\mathbb{P}}V}(1)|_{S_{2}}$. As
$\displaystyle S_{2}$ is linear normal in $\displaystyle{\mathbb{P}}V$,
$\displaystyle(S_{2},{\mathcal{L}},x)$ satisfies the conditions in Lemma 2.7.
Thus by Corollary 2.8 it has exactly three weighted subspaces. Under the
setting of Section 2 we have $\displaystyle r=2$. Denote by $\displaystyle
W=H^{0}(S_{2},{\mathcal{L}})^{\vee}_{-1}$, $\displaystyle
U=H^{0}(S_{2},{\mathcal{L}})^{\vee}_{w_{2}}$ and $\displaystyle
f:S_{2}\rightarrow{\mathbb{P}}V$ the projective embedding. Then we have:
$\displaystyle dim(V)=14$, $\displaystyle dim(W)=dim(T_{x}S_{2})=8$ and
$\displaystyle dim(U)=dim(V)-1-dim(W)=5$. We now check that the
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle S_{2}$ satisfies the
following two properties.

(a) There are exactly three irreducible components of $\displaystyle
X^{{\mathbb{C}}^{*}}$: the isolated source $\displaystyle\\{x\\}$, the unique
component $\displaystyle Y_{1}$ contained in $\displaystyle{\mathbb{P}}W$ and
the unique component $\displaystyle Y_{2}$ contained in
$\displaystyle{\mathbb{P}}U$. First note that for any $\displaystyle
Y\in{\mathcal{Y}}$, $\displaystyle Y\subset{\mathbb{P}}V_{k}$ for some k. If
$\displaystyle Y\subset{\mathbb{P}}W$, then a
$\displaystyle{\mathbb{C}}^{*}$-orbit whose sink lies in $\displaystyle Y$ has
its source equal to $\displaystyle x$. Thus by Proposition 2.3 such
$\displaystyle Y$ is unique, and $\displaystyle C^{-}(Y)$ is a line bundle
over $\displaystyle Y$. If $\displaystyle Y\subset{\mathbb{P}}U$ then we see
that $\displaystyle v^{+}(Y)=0$ whence $\displaystyle Y$ is the unique sink of
the $\displaystyle{\mathbb{C}}^{*}$-action.

(b) We have $\displaystyle dim(Y_{1})>0$. Otherwise assume that $\displaystyle
Y_{1}=\\{y\\}$ is a single point. If $\displaystyle dim(Y_{2})=0$ then by
Proposition 2.3 and (a), we have $\displaystyle v^{+}(Y_{1})=v^{-}(Y_{1})=1$
and thus $\displaystyle
dim(S_{2})=dim(T_{y}S_{2})=v^{+}(Y_{1})+v^{-}(Y_{1})=2<8$, which is a
contradiction. If $\displaystyle dim(Y_{2})>0$ then we have $\displaystyle
D_{x}=S_{2}\backslash C^{+}(x)=C^{+}(y)\cup Y_{2}$. This implies that
$\displaystyle Y_{2}$, as a divisor of $\displaystyle D_{x}$, is of dimension
6, contradicting the fact that $\displaystyle
Y_{2}\subset{\mathbb{P}}U\cong{\mathbb{P}}^{4}$.

Now as $\displaystyle L$ is the intersection of all 4-spaces in $\displaystyle
X_{2}$, it is $\displaystyle{\mathbb{C}}^{*}$-invariant. Moreover as
$\displaystyle x\in L$ and the action is of Euler type at $\displaystyle x$,
we conclude that $\displaystyle L$ is a non-trival
$\displaystyle{\mathbb{C}}^{*}$-orbit closure with source $\displaystyle
x=[e_{0}]$. Denote the orbit to be $\displaystyle{\mathbb{C}}^{*}\cdot
f(w)=\\{[e_{0}+z\Pi_{1}(w)+z^{-w_{2}}\Pi_{2}(w,w)]:z\in{\mathbb{C}}^{*}\\}$
for some nonzero $\displaystyle w\in T_{x}S_{2}$ and denote by $\displaystyle
y$ the sink of the orbit. Then we must have $\displaystyle\Pi_{2}(w,w)=0$ and
$\displaystyle y\in Y_{1}$. Otherwise the sink of the orbit would be
$\displaystyle[\Pi_{2}(w,w)]\in{\mathbb{P}}U$. Then the line $\displaystyle L$
will be contained in $\displaystyle{\mathbb{P}}({\mathbb{C}}e_{0}\oplus U)$,
contradicting the fact that $\displaystyle\Pi_{1}(w)\not=0$ as
$\displaystyle\Pi_{1}$ is injective by Lemma 2.7 (3). Now take any point
$\displaystyle y^{\prime}\in Y_{1}$, dentoe by $\displaystyle L_{y^{\prime}}$
the unique non-trival $\displaystyle{\mathbb{C}}^{*}$-orbit closure with sink
$\displaystyle y^{\prime}$ and source $\displaystyle x$. Then $\displaystyle
L_{y^{\prime}}$ is exactly the line connecting $\displaystyle x$ and
$\displaystyle y^{\prime}$. By [ORSW22, Lemma 2.16] and by the proof of
[ORSW22, Proposition 2.17], the splitting type of $\displaystyle
T_{S_{2}}|_{L_{y^{\prime}}}$ is determined by the weights of the isotropy
action of $\displaystyle{\mathbb{C}}^{*}$ on $\displaystyle
T_{y^{\prime}}S_{2}$. As $\displaystyle Y_{1}$ is irreducible, the weights of
$\displaystyle{\mathbb{C}}^{*}$ on $\displaystyle T_{y^{\prime}}S_{2}$ remain
invariant as $\displaystyle y^{\prime}$ varies in $\displaystyle Y_{1}$. This
implies that the splitting type of $\displaystyle T_{S_{2}}|_{L_{y^{\prime}}}$
also remains invariant, contradicting the uniqueness of $\displaystyle L$ as
$\displaystyle dim(Y_{1})>0$. Thus we conclude that $\displaystyle
dim({\mathfrak{a}}ut(\hat{S_{2}})^{(1)})<dim(V)$. This completes the proof of
Proposition 3.9.

##  4\. Proof of main result

In this section we will prove Theorem 1.2. Let’s first recall the Cartan-
Fubini type extension theorem proved by Hwang and Mok [HM01]. We will use the
following version taken from [FH12, Theorem 6.8]

######  Theorem 4.1.

Let $\displaystyle X_{1}$ and $\displaystyle X_{2}$ be two Fano manifolds of
Picard number 1, different from projective spaces. Let
$\displaystyle{\mathcal{K}}_{1}$ and $\displaystyle{\mathcal{K}}_{2}$ be
families of minimal rational curves on $\displaystyle X_{1}$ and
$\displaystyle X_{2}$ respectively. Assume that for a general point
$\displaystyle x\in X_{1}$, the VMRT
$\displaystyle{\mathcal{C}}_{x}\subset{\mathbb{P}}T_{x}(X_{1})$ is irreducible
and nonsingular. Let $\displaystyle U_{1}\subset X_{1}$ and $\displaystyle
U_{2}\subset X_{2}$ be connected analytical open subsets. Suppose that there
exists a biholomorphic map $\displaystyle\phi:U_{1}\rightarrow U_{2}$ such
that for a general point $\displaystyle x\in U_{1}$, the differential
$\displaystyle
d\phi_{x}:{\mathbb{P}}T_{x}(U_{1})\rightarrow{\mathbb{P}}T_{\phi(x)}(U_{2})$
sends $\displaystyle{\mathcal{C}}_{x}$ isomorphically to
$\displaystyle{\mathcal{C}}_{\phi(x)}$. Then there exists a biregular morphism
$\displaystyle\Phi:X_{1}\rightarrow X_{2}$ such that
$\displaystyle\phi=\Phi|U_{1}$.

The extension theorem enables us to characterize an Euler-symmetric variety by
its VMRT. We present a $\displaystyle{\mathbb{C}}^{*}$-equivariant version for
our convenience.

######  Corollary 4.2.

Let $\displaystyle X,X^{\prime}$ be two Fano manifolds of Picard number 1, and
$\displaystyle{\mathcal{K}},{\mathcal{K}}^{\prime}$ are families of minimal
rational curves on $\displaystyle X,X^{\prime}$ respectively.
$\displaystyle{\mathcal{C}}\subset{\mathbb{P}}T(X)$ and
$\displaystyle{\mathcal{C}}^{\prime}\subset{\mathbb{P}}T(X^{\prime})$ the
associated VMRT sturctures on $\displaystyle X$ and $\displaystyle
X^{\prime}$. Assume that for a general point $\displaystyle x$ on
$\displaystyle X$ and for a general point $\displaystyle x^{\prime}$ on
$\displaystyle X^{\prime}$ there are $\displaystyle{\mathbb{C}}^{*}$-actions
on $\displaystyle X$ and $\displaystyle X^{\prime}$ such that the actions are
of Euler type at $\displaystyle x$ and $\displaystyle x^{\prime}$
respectively. If $\displaystyle{\mathcal{C}}_{x}$ is projectively isomorphic
to $\displaystyle{\mathcal{C}}^{\prime}_{x}$, then there is a
$\displaystyle{\mathbb{C}}^{*}$-equivariant isomorphism
$\displaystyle\Phi:X\rightarrow X^{\prime}$ maps $\displaystyle x$ to
$\displaystyle x^{\prime}$.

###### Proof.

Assume the projective isomorphism between
$\displaystyle{\mathcal{C}}_{x}\subset{\mathbb{P}}T_{x}X$ and
$\displaystyle{\mathcal{C}}^{\prime}_{x^{\prime}}\subset{\mathbb{P}}T_{x^{\prime}}X^{\prime}$
is given by the linear isomorphism $\displaystyle\phi:T_{x}X\rightarrow
T_{x^{\prime}}X^{\prime}$. Then the identifications $\displaystyle
C^{+}(x)\cong T_{x}X$ and $\displaystyle C^{+}(x^{\prime})\cong
T_{x^{\prime}}X^{\prime}$ induce an isomorphism
$\displaystyle\overline{\phi}:C^{+}(x)\rightarrow C^{+}(x^{\prime})$. By
Proposition 3.3, their VMRTs at general points are both irreducible and
nonsingular. To extend $\displaystyle\overline{\phi}$, it suffices to check
the differential map of $\displaystyle\overline{\phi}$ preserves the VMRT at a
general point. As $\displaystyle x$ and $\displaystyle x^{\prime}$ are both
taken as general points, by Proposition 2.10 (3) the action of $\displaystyle
T_{x}X$ on $\displaystyle C^{+}(x)$ and the action of $\displaystyle
T_{x^{\prime}}X^{\prime}$ on $\displaystyle C^{+}(x^{\prime})$ can be extended
to an action on $\displaystyle X$ and $\displaystyle X^{\prime}$ respectively.
This shows the VMRT structures over $\displaystyle C^{+}(x)$ and
$\displaystyle C^{+}(x^{\prime})$ are both locally flat. Thus by our
definition of $\displaystyle\overline{\phi}$, for any point $\displaystyle
y\in C^{+}(x)$, $\displaystyle d\overline{\phi}_{y}$ must map
$\displaystyle{\mathcal{C}}_{y}$ isomorphically onto
$\displaystyle{\mathcal{C}}^{\prime}_{\overline{\phi}(y)}$. This yields the
existence of $\displaystyle\Phi$. Finally by our assumption
$\displaystyle\overline{\phi}$ is $\displaystyle{\mathbb{C}}^{*}$-equivariant,
thus $\displaystyle\Phi$ is also $\displaystyle{\mathbb{C}}^{*}$-equivariant
as $\displaystyle C^{+}(x)$ is open dense. ∎

Now let $\displaystyle X=G/P={\mathcal{D}}(I)$ be a rational homogenous space
of Picard number one defined by a simple algebraic group $\displaystyle G$ and
a parabolic subgroup $\displaystyle P$ given by $\displaystyle I=\\{\alpha\\}$
for a simple positive root $\displaystyle\alpha\in\Delta$. We review some
facts about $\displaystyle{\mathbb{C}}^{*}$-actions on rational homogenous
spaces, most of which are taken from [ORSW21].

Up to composing with a character, a $\displaystyle{\mathbb{C}}^{*}$-action on
$\displaystyle X$ is given by a cocharacter
$\displaystyle\sigma:{\mathbb{G}}_{m}\rightarrow T$ and the left
multiplication of the maximal torus $\displaystyle T$ on $\displaystyle G$.
For a simple root $\displaystyle\beta$, we denote the cocharacter
$\displaystyle\sigma_{\beta}$ by assigning
$\displaystyle\sigma_{\beta}(\eta)=\delta_{\eta,\beta}$ for any
$\displaystyle\eta\in\Delta$. For a cocharacter $\displaystyle\sigma$, denote
$\displaystyle{\mathfrak{g}}_{k}=\bigoplus\limits_{\theta\in\Phi:\sigma(\theta)=k}{\mathfrak{g}}_{\theta}$.

######  Proposition 4.3.

Let $\displaystyle X=G/P={\mathcal{D}}(I)$ be a rational homogenous space of
Picard number one, then:

(i) A $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ given by a
cocharacter $\displaystyle\sigma\in X_{*}(T)$ is equalized if and only if the
grading of $\displaystyle{\mathfrak{g}}$ is short, i.e.,
$\displaystyle{\mathfrak{g}}={\mathfrak{g}}_{1}\oplus{\mathfrak{g}}_{0}\oplus{\mathfrak{g}}_{-1}$.
In this case, up to a conjuagation we can assume
$\displaystyle\sigma=\sigma_{\beta}$ for some $\displaystyle\beta\in\Delta$.

(ii) A $\displaystyle{\mathbb{C}}^{*}$-action on
$\displaystyle{\mathcal{D}}(I)$ given by the cocharacter
$\displaystyle\sigma_{\beta}$ is equalized with an isolated sink if and only
if $\displaystyle\alpha=\beta$ and $\displaystyle\sigma_{\alpha}$ defines a
short grading of $\displaystyle{\mathfrak{g}}$, equivalently $\displaystyle X$
is an IHSS and $\displaystyle\sigma=\sigma_{\alpha}$. In this case, the
$\displaystyle{\mathbb{C}}^{*}$-action is equalized of weight -1 at the sink
$\displaystyle x^{\prime}=eP$, and it has an isolated source if and only if
$\displaystyle X={\mathcal{D}}(I)$ is an IHSS of tube type, where the isolated
source is $\displaystyle y^{\prime}=\dot{w_{o}}x^{\prime}$.

(iii) Assme that $\displaystyle X={\mathcal{D}}(I)$ is an IHSS, with the
$\displaystyle{\mathbb{C}}^{*}$-action given by
$\displaystyle\sigma_{\alpha}$. The maximal dimensional Bruhat cell is
$\displaystyle C^{-}(x^{\prime})=R_{u}(P^{-})\cdot x^{\prime}$ and
$\displaystyle X$ is an equivariant compactification of $\displaystyle
R_{u}(P^{-})$. If $\displaystyle X$ is of tube type, then $\displaystyle
C^{+}(y^{\prime})=R_{u}(P)\cdot y^{\prime}$ and $\displaystyle X$ an
equivariant compactification of $\displaystyle R_{u}(P)$.

Now assume $\displaystyle X={\mathcal{D}}(I)$ to be an IHSS where
$\displaystyle I=\\{\alpha\\}$ for some simple root $\displaystyle\alpha$ and
consider the $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ given
by the cocharacter $\displaystyle-\sigma_{\alpha}$. Then the action is of
Euler type at $\displaystyle x=eP$ and its inverse action is of Euler type at
$\displaystyle y=\dot{w_{0}}\cdot x$. By (iii) the
$\displaystyle{\mathbb{C}}^{*}$\- action satisfies the condition of
Proposition 2.4 (ii).

We are ready to prove our main result.

###### Proof of Theorem 1.2.

If $\displaystyle X={\mathcal{D}}(I)$ is an IHSS of tube type, denote the
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ by
$\displaystyle-\sigma_{\alpha}$. Then $\displaystyle X$ is the equivariant
compactification of $\displaystyle R_{u}(P^{-})$ and $\displaystyle R_{u}(P)$
respectively, where $\displaystyle x=eP$ is fixed by $\displaystyle R_{u}(P)$
and $\displaystyle y=\dot{w_{0}}x$ is fixed by $\displaystyle R_{u}(P^{-})$.
Consider the morphism $\displaystyle R_{u}(P)\times R_{u}(P^{-})\rightarrow
X\times X:(u,v)\rightarrow(uv\cdot x,u\cdot y)$, its image is a dense and
constructible, whence contains a dense open subset. Moreover for any element
$\displaystyle(u\cdot y,uv\cdot x)$ in the image we can define a
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X$ by:

$\displaystyle\displaystyle{\mathbb{C}}^{*}\times X$
$\displaystyle\displaystyle\longrightarrow X$
$\displaystyle\displaystyle(\,\,t\,\,\,,\,\,x\,\,)$
$\displaystyle\displaystyle\longrightarrow uv\,t\,(uv)^{-1}\cdot x,$

such that the $\displaystyle{\mathbb{C}}^{*}$-action is of Euler type at the
source $\displaystyle\tilde{y}=uv\cdot y=u\cdot y$ and its inverse action is
of Euler type at $\displaystyle\tilde{x}=uv\cdot x$.

Conversely assume that for a general pair of points $\displaystyle x,y$ on
$\displaystyle X$, there is a $\displaystyle{\mathbb{C}}^{*}$-action which is
of Euler type at $\displaystyle x$ and the inverse action is of Euler type at
$\displaystyle y$. Take $\displaystyle{\mathcal{K}}$ to be a family of minimal
ratioinal curves on $\displaystyle X$, and let
$\displaystyle{\mathcal{C}}\subset{\mathbb{P}}T(X)$ be the VMRT structure. By
Theorem 1.4, for the projective embedding
$\displaystyle{\mathcal{C}}_{x}\subset{\mathbb{P}}T_{x}X$ we have
$\displaystyle
dim({\mathfrak{a}}ut(\hat{{\mathcal{C}}_{x}})^{(1)})=dim(T_{x}X)$. Thus from
Theorem 3.7 and Proposition 3.9, $\displaystyle{\mathcal{C}}_{x}$ is
projectively isomorphic to the VMRT of an IHSS. Denote the IHSS by
$\displaystyle X^{\prime}={\mathcal{D}}(I)$ and consider the
$\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle X^{\prime}$ given by
$\displaystyle-\sigma_{\alpha}$ with the isolated source $\displaystyle
x^{\prime}$. Then $\displaystyle(X,x,X^{\prime},x^{\prime})$ satisfies the
condition of Corollary 4.2 and hence $\displaystyle X$ is
$\displaystyle{\mathbb{C}}^{*}$-equivariant isomorphic to $\displaystyle
X^{\prime}$. As the $\displaystyle{\mathbb{C}}^{*}$-action on $\displaystyle
X$ has isolated sink and source, $\displaystyle X^{\prime}$ must be an IHSS of
tube type by Proposition 4.3 (ii). ∎

## Acknowledgements

I am very grateful to my advisor Baohua Fu for sharing valuable ideas on
Euler-symmetric varieties with me. I am also indebeted to him for many helpful
suggestions during the preparation of the paper. I would like to thank Cong
Ding for reading the first draft of this paper and for useful discussions and
suggestions. I am grateful to Zhijun Luo for helpful discussions.

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