1. Introduction
We work over the field of complex numbers. Open subsets refer to those in the Euclidean topology. The projectivization
${\mathbb P} V$
of a vector space V is the set of one-dimensional subspaces of V.
Recall the following from [Reference Fu and Hwang2, Definition 3.2].
Definition 1.1. Let W be a vector space and let
$ \mathrm{Sym} W^* = \oplus _{k \in {\mathbb Z}} \mathrm{Sym} ^k W^*$
be the graded ring of symmetric forms on W. Here, we use the convention
$ \mathrm{Sym} ^k W^* =0$
if
$k<0$
.
-
(i) For $w \in W$
and
$\zeta \in \mathrm{Sym} ^k W^*$
, let
$\iota _w \zeta \in \mathrm{Sym} ^{k-1} W^*$
be the contraction defined by $$ \begin{align*}\iota_w \zeta (w_1, \ldots, w_{k-1}) := \zeta (w, w_1, \ldots, w_{k-1}).\end{align*} $$
-
(ii) A graded vector subspace
$$ \begin{align*}S = \oplus_{k \in {\mathbb Z}} S^k \subset \mathrm{Sym} W^*, \ S^k \subset \mathrm{Sym}^k W^*,\end{align*} $$is called a symbol system of rank r on W if for each k,
(1.1) $$ \begin{align} \iota_w s \in S^{k-1} \text{ for any } w \in W \text{ and } s \in S^k, \end{align} $$and
$$ \begin{align*}S^0 = {\mathbb C}, \ S^1 = W^*, \ S^r \neq 0, \ S^{j} =0 \text{ for all } j>r.\end{align*} $$
We have also the following natural equivalence relation.
Definition 1.2. Let
$S \subset \mathrm{Sym} W^*$
(resp.
$\widetilde {S} \subset \mathrm{Sym} \widetilde {W}^*$
) be a symbol system on a vector space W (resp.
$\widetilde {W}$
). We say that the two symbol systems are equivalent if there is a linear isomorphism
$f: W \to \widetilde {W}$
such that the induced isomorphism
$f^*: \mathrm{Sym} \widetilde {W}^* \to \mathrm{Sym} W^*$
satisfies
$f^*(\widetilde {S}) = S.$
The most important examples of symbol systems are systems of fundamental forms (see Definition 3.5) at a general point of a projective variety, from the following result of E. Cartan (p. 68 of [Reference Landsberg and Manivel6]).
Theorem 1.3. Let
$Z \subset {\mathbb P} V$
be a projective subvariety. Then for a general point
$x \in Z$
, the system of fundamental forms
$S_{x}$
at x, as a graded subspace of
$ \mathrm{Sym} T^*_x Z,$
is a symbol system on the tangent space
$T_x Z$
. If furthermore Z is linearly nondegenerate in
${\mathbb P} V$
, then
$\dim S_{x} = \dim V$
for a general point
$x \in Z$
.
Conversely, [Reference Fu and Hwang2, Theorem 3.7] says that a symbol system
$S \subset \mathrm{Sym} W^*$
determines a projective variety
$Z^S \subset {\mathbb P} V^S$
(up to a projective equivalence), called the Euler-symmetric variety (see Definition 4.1), such that the system of fundamental forms at any general point
$x \in Z^S$
is equivalent to S.
Let
$\widehat {Z} \subset V$
be the affine cone of a projective variety
$Z \subset {\mathbb P} V$
. Let
${\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}) \subset {\mathfrak g}{\mathfrak l}(V) $
be the Lie algebra of the linear automorphism group
Our main result is the following relation between
${\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z})$
and the system of fundamental forms of Z at a general point.
Theorem 1.4. Let
$Z \subset {\mathbb P} V$
be a linearly nondegenerate projective variety and let
$x \in Z$
be a general point such that (by Theorem 1.3) the system of fundamental forms at x is a symbol system
$S_x \subset \mathrm{Sym} T^*_x Z$
of dimension equal to
$\dim V$
. Then
and the equality holds if and only if
$Z\subset {\mathbb P} V$
is projectively equivalent to the Euler-symmetric variety
$Z^{S_x} \subset {\mathbb P} V^{S_x}$
by a linear isomorphism
$V \cong V^{S_x}$
.
Euler-symmetric varieties are equivariant compactifications of vector groups (see [Reference Fu and Hwang2, Theorem 3.7]), hence have large automorphism groups. According to Theorem 1.4, they have maximal linear automorphism groups among projective varieties admitting the same symbol algebras.
The proof of Theorem 1.4 consists of two steps. The first step involves a suitable Lie algebra of symmetries of the system of fundamental forms. What should be the Lie algebra of symmetries of a symbol system
$S \subset \mathrm{Sym} W^*$
? From Definition 1.2, a naive guess would be the subalgebra consisting of elements of
${\mathfrak g}{\mathfrak l}(W)$
such that the induced endomorphisms of
$ \mathrm{Sym} ^* W$
preserve both the gradation and the subspace
$S \subset \mathrm{Sym} W^*$
. It turns out that this is not the correct Lie algebra of symmetries: one has to consider hidden symmetries which do not preserve the gradation. We give a correct definition in Definition 2.7, associating in a canonical way a finite-dimensional graded Lie algebra
${\mathfrak h}^S = \oplus _{k \in {\mathbb Z}} {\mathfrak h}^S_k$
to each symbol system
$S.$
Then the first step in the proof of Theorem 1.4 is the following.
Theorem 1.5. Let
$Z \subset {\mathbb P} V$
and
$x \in Z$
be as in Theorem 1.4 such that the system of fundamental forms at x is a symbol system
$S_x \subset \mathrm{Sym} T^*_x M$
of dimension equal to
$\dim V$
. Then the Lie algebra
${\mathfrak h} := {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z})$
admits a filtration
such that
$\cap _{k \in {\mathbb Z}} {\mathfrak h}^k_x =0$
and the graded object
${\mathfrak h}^k_x/{\mathfrak h}^{k+1}_x$
admits a natural inclusion into
${\mathfrak h}^{S_x}_k$
for each integer k. In particular, the dimension of
${\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z})$
is at most
$\dim {\mathfrak h}^{S_x}$
.
The second step in the proof of Theorem 1.4 is the following.
Theorem 1.6. In Theorem 1.5, we have
$\dim {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}) = \dim {\mathfrak h}^{S_x}$
if and only if
$Z \subset {\mathbb P} V$
is projectively equivalent to the Euler-symmetric variety
$Z^{S_x} \subset {\mathbb P} V^{S_x}$
. In this case, the linear isomorphisms
${\mathfrak h}^k_x/{\mathfrak h}^{k+1}_x \cong {\mathfrak h}^{S_x}_k, k \in {\mathbb Z},$
come from a Lie algebra isomorphism
${\mathfrak h} \cong {\mathfrak h}^{S_x}$
.
Clearly, Theorems 1.5 and 1.6 imply Theorem 1.4. We have the following application of Theorem 1.4.
Corollary 1.7. Fix a symbol system
$S \subset \mathrm{Sym} W^*$
. Let
$\Delta = \{ t \in {\mathbb C} \mid |t| < 1\}$
be the unit disc. Let
${\mathcal Z} \subset \Delta \times {\mathbb P} V$
be an irreducible closed analytic subset such that the fiber
is projectively equivalent to the Euler-symmetric variety
$Z^S \subset {\mathbb P} V^S$
for all
$t \in \Delta , t \neq 0$
. Assume that for a general point z in an irreducible component
$Z^{\prime }_0$
of
$Z_0$
, the system of fundamental forms
$S_z \in \mathrm{Sym} T_z^* Z^{\prime }_0$
is equivalent to
$S \subset \mathrm{ Sym} W^*$
as a symbol system. Then
$Z^{\prime }_0 \subset {\mathbb P} V$
is projectively equivalent to
$Z^S \subset {\mathbb P} V^S$
and
$Z_0 = Z^{\prime }_0$
.
Proof. The assumption on the system of fundamental forms at
$z \in Z^{\prime }_0$
implies that
$\dim S_z = \dim V$
, hence
$Z^{\prime }_0 \subset {\mathbb P} V$
is linearly nondegenerate. Thus,
${\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}_0) \subset {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}^{\prime }_0)$
. Then by the upper-semi-continuity, we have
for
$t \neq 0$
. Thus, Theorem 1.4 implies that
$Z^{\prime }_0 \subset {\mathbb P} V$
is projectively equivalent to
$Z^S \subset {\mathbb P} V^S$
. It follows that
$Z^{\prime }_0 = Z_0$
.
Some special cases of Corollary 1.7 have been proved previously: [Reference Hong and Kim3, Proposition 5.10], [Reference Hwang and Li4, Theorem 6.7], [Reference Hwang and Li5, Corollary 4.7], and [Reference Mok7, Proposition 2.3]. These special cases play an important role in the problem of recognizing rational homogeneous spaces and horospherical varieties of Picard number 1 from their varieties of minimal rational tangents, because the latter are Euler-symmetric varieties. The proofs in these special cases used specific geometric properties of
$Z^S$
in each case. Our proof of Corollary 1.7 gives a uniform argument to cover all these cases, and potentially further examples of varieties of minimal rational tangents. This was our initial motivation for Theorem 1.6. But Corollary 1.7 could be useful in a broader context because Euler-symmetric varieties form a large class of projective varieties. Among others, any subspace
$S^2 \subset \mathrm{Sym} ^2 W^*$
gives a symbol system
$S:= S^0\, +\, S^1\, +\, S^2$
, hence an Euler-symmetric variety
$Z^S$
. For other interesting examples and a survey of results on Euler-symmetric varieties, see [Reference Arzhantsev and Zaitseva1, Section 5.2].
When
$S_x\ {\subset}\ \mathrm{Sym} T^*_x M$
is isomorphic to the system of fundamental forms of an equivariantly embedded Hermitian symmetric space, Theorem 1.5 can be deduced from the result of Se-ashi in [Reference Se-ashi8, Theorem 5.1.2]. Se-ashi’s argument requires special properties of the Lie algebra of the automorphism group of Hermitian symmetric spaces and seems very difficult to be extended to the general setting of Theorem 1.5. It was, however, while we were examining his argument that we realized the expression of the Lie algebra
${\mathfrak h}^S$
and the filtration of
${\mathfrak h}$
in Theorem 1.5. We give a precise definition of the Lie algebra
${\mathfrak h}^S$
in Section 2 and prove Theorems 1.5 and 1.6 in Sections 3 and 4, respectively.
2. Infinitesimal symmetries of symbol systems
Definition 2.1. Let W be a vector space.
-
(i) Let
$$ \begin{align*}{\mathfrak v}(W) := \oplus_{k \in {\mathbb Z}} {\mathfrak b}_k (W), \ {\mathfrak v}_k(W):= (\mathrm{Sym}^{k+1} W^*) \otimes W\end{align*} $$be the graded Lie algebra of polynomial vector fields on W.
-
(ii) We have a natural representation of ${\mathfrak v}(W)$
on
$ \mathrm{Sym} W^*$
: an element
$\sigma \otimes w \in {\mathfrak v}_i(W)$
with
$ \sigma \in \mathrm{Sym} ^{i+1} W^*$
and
$w \in W$
sends
$\zeta \in \mathrm{Sym} ^{k} W^*$
to $$ \begin{align*}(\sigma \otimes w) \cdot \zeta := k \ \sigma \odot \iota_w \zeta,\end{align*} $$where $\iota _w \zeta $
is the contraction from Definition 1.1 and
$\odot $
represents the symmetric product. This is equivalent to the representation of the Lie algebra of polynomial vector fields on the space of polynomial functions by derivative.
-
(iii) We write ${\mathfrak a}(W) = \mathrm{Sym} W^*$
when we regard
$ \mathrm{Sym} W^*$
as an abelian Lie algebra. Define $$ \begin{align*}{\mathfrak g}(W):= \oplus_{k \in {\mathbb Z}} {\mathfrak g}_k(W), \ {\mathfrak g}_k(W) := {\mathfrak a}_k(W) \oplus {\mathfrak v}_k(W)\end{align*} $$as the semidirect product of the abelian Lie algebra ${\mathfrak a}(W)$
with the Lie algebra
${\mathfrak v}(W)$
via the representation in
$\mathrm{(ii)}$
.
-
(iv) We have a natural representation of ${\mathfrak g}(W)$
on the space
$ \mathrm{Sym} W^*$
, where the subalgebra
${\mathfrak a}(W) = \mathrm{ Sym} W^*$
acts on
$ \mathrm{Sym} W^*$
by the symmetric multiplication and the subalgebra
${\mathfrak v}(W) \subset {\mathfrak g}(W)$
acts by
$\mathrm{(ii)}$
. This is equivalent to viewing
${\mathfrak g}(W)$
as the Lie algebra of differential operators of order less than or equal to
$1$
on W with polynomial coefficients and considering its action on the space of polynomial functions on W as differential operators.
Lemma 2.2. The representation in Definition 2.1
$\mathrm{(iv)}$
gives an inclusion
for each
$k \in {\mathbb Z}$
.
Proof. In terms of a basis
$x_1, \ldots , x_n, n= \dim W,$
of
$W^*$
, we can write an element D of
${\mathfrak g}_k(W)$
as
where
$h(x)$
is a homogeneous polynomial of degree k and
$g^i(x)$
is a homogeneous polynomial of degree
$k+1$
for each
$1 \leq i \leq n$
. Suppose
$D\cdot ( \mathrm{Sym} ^0 W^* + \mathrm{Sym} ^1 W^*) =0$
. Then
implies that
$g^j(x) = -h(x) x_j$
for each
$1\leq j \leq n$
and
$D = h(x) (1 - \sum ^n_{i=1} x_i \frac {\partial }{\partial x_i})$
. But then
$0 = D \cdot 1 = h(x)$
implies
$D =0$
.
Notation 2.3. Let X be a complex manifold and let
$x \in X$
be a point.
-
(i) For the ring ${\mathcal O}_{X,x}$
of germs of holomorphic functions at x and an integer k, we denote by
$\mathbf {m}^k_{X,x}$
the ideal of functions vanishing at x to order at least k. Here, we use the convention
$\mathbf {m}^{k}_{X,x} = {\mathcal O}_{X,x}$
for
$k \leq 0$
. -
(ii) For the Lie algebra ${\mathfrak t}_{X,x}$
of germs of holomorphic vector fields at x and an integer k, let
${\mathfrak t}_{X,x}^k$
be the Lie subalgebra of vector fields vanishing at x to order at least
$k+1$
. Here, we use the convention
${\mathfrak t}^{k}_{X,x} = {\mathfrak t}_{X,x}$
for
$k \leq -1$
.
We skip the proof of the following elementary fact.
Lemma 2.4. For a complex manifold X and a point
$x \in X$
, the following holds for all integers
$k, \ell $
:
-
(i) We have canonical identifications $\mathbf {m}^{k}_{X,x}/\mathbf {m}^{k+1}_{X,x} = \mathrm{Sym} ^{k} T^*_x X $
and $$ \begin{align*}{\mathfrak t}_{X,x}^{k}/{\mathfrak t}_{X,x}^{k+1} = (\mathbf{m}_{X,x}^{k+1}/\mathbf{m}_{X,x}^{k+2})\otimes T_x X = (\mathrm{Sym}^{k+1} T^*_x X) \otimes T_x X.\end{align*} $$
-
(ii) The Lie bracket of vector fields satisfies $[{\mathfrak t}^k_{X,x}, {\mathfrak t}^{\ell }_{X,x}] \subset {\mathfrak t}_{X,x}^{k + \ell }.$
In other words, the Lie algebra
${\mathfrak t}_{X,x}$
is a filtered Lie algebra under $$ \begin{align*}{\mathfrak t}_{X,x}= {\mathfrak t}_{X,x}^{-1} \supset {\mathfrak t}_{X,x}^0 \supset {\mathfrak t}_{X,x}^1 \supset \cdots.\end{align*} $$
-
(iii) The derivatives of functions with respect to vector fields satisfy ${\mathfrak t}^k_{X,x} \cdot \mathbf {m}^{\ell }_{X,x} \subset \mathbf {m}^{\ell + k}_{X,x}$
.
Definition 2.5. Let
$\pi : L \to M$
be a line bundle on a complex manifold M and let
$0_M \subset L$
be its zero section.
-
(i) For an open subset $U \subset M$
, let
${\mathcal F}(U)$
be the space of
$\pi $
-linear functions on
$L|_U = \pi ^{-1}(U)$
, namely, holomorphic functions on
$\pi ^{-1}(U)$
which induce linear functionals on the one-dimensional vector space
$L_x =\pi ^{-1}(x)$
for every
$x \in U$
. The germ
${\mathcal F}_x$
can be identified with
${\mathcal O}_{M,x} \otimes L^*_x$
. For each integer k, define the subspace
${\mathcal F}_x^k \subset {\mathcal F}_x$
by $$ \begin{align*}{\mathcal F}_x^k := \mathbf{m}^k_{M,x} \otimes L^*_x \ \subset \ {\mathcal O}_{M,x} \otimes L^*_x = {\mathcal F}_x.\end{align*} $$
-
(ii) For an open subset $U \subset M$
, let
${\mathfrak l}(U)$
be the Lie algebra of
$\pi $
-linear vector fields on
$L|_U= \pi ^{-1}(U)$
, namely, holomorphic vector fields on
$\pi ^{-1}(U)$
which generate biholomorphic transformations of
$\pi ^{-1}(U)$
preserving the line bundle structure
$\pi |_{\pi ^{-1}(U)}: \pi ^{-1}(U) \to U$
. -
(iii) Regard a point $x \in M$
as a point of
$0_M \subset L$
. We have a natural Lie algebra homomorphism
$\rho : {\mathfrak l}_x \to {\mathfrak t}_{L,x}$
from the germ of
${\mathfrak l}$
at
$x \in M$
to the germ of vector fields at
$x \in L$
. For each integer k, define $$ \begin{align*}{\mathfrak l}_x^k := {\mathfrak l}_x \cap \rho^{-1} (\mathbf{t}^k_{L,x}).\end{align*} $$
Lemma 2.6. In Definition 2.5, the following holds for each
$x \in M$
and all integers
$k, \ell $
:
-
(i) There is a canonical identification
$$ \begin{align*}{\mathcal F}_x^k/{\mathcal F}_x^{k+1} = (\mathbf{m}_{M,x}^{k}/\mathbf{m}_{M,x}^{k+1}) \otimes L^*_x = \mathrm{Sym}^{k} T^*_x M \otimes L^*_x.\end{align*} $$
-
(ii) Lie brackets of vector fields satisfy $[{\mathfrak l}_x^{k}, {\mathfrak l}_x^{\ell }] \subset {\mathfrak l}_x^{k + \ell }$
for all
$k, \ell $
. -
(iii) Taking derivative of functions with respect to vector fields determines a representation of ${\mathfrak l}_x$
on
${\mathcal F}_x$
that satisfies $$ \begin{align*}{\mathfrak l}_x^k \cdot {\mathcal F}_x^{\ell} \subset {\mathcal F}_x^{k + \ell}.\end{align*} $$
-
(iv) Let ${\mathfrak o}_x \subset {\mathfrak l}_x$
be the subalgebra consisting of vector fields in
${\mathfrak l}_x$
tangent to fibers of
$\pi $
. Then
${\mathfrak o}_x$
is an abelian ideal of the Lie algebra
${\mathfrak l}_x$
and there is a canonical isomorphism of vector spaces
${\mathcal O}_{M,x} \cong {\mathfrak o}_x$
. -
(v) The differential $\mathrm{d} \pi : T L \to T M$
of the map
$\pi $
induces a Lie algebra homomorphism
${\mathfrak l}_x \to {\mathfrak t}_{M,x}$
, inducing a canonical exact sequence $$ \begin{align*}0 \to {\mathfrak o}_x \subset {\mathfrak l}_x \stackrel{\mathrm{d} \pi}{\longrightarrow} {\mathfrak t}_{M,x} \to 0,\end{align*} $$which yields the exact sequence
$$ \begin{align*}0 \to {\mathfrak o}^k_x \to {\mathfrak l}^k_x \to {\mathfrak t}^k_{M,x} \to 0,\end{align*} $$for each integer k, where ${\mathfrak o}_x^k \subset {\mathfrak o}_x$
is the subspace corresponding to
$\mathbf {m}_{X,x}^k$
under the isomorphism
${\mathfrak o}_x\cong {\mathcal O}_{M,x}$
in (iv).
-
(vi) Taking graded objects of the filtration in $\mathrm{(v)}$
gives the exact sequence $$ \begin{align*}\begin{array}{ccccccccc} 0 & \to & {\mathfrak o}_x^k/{\mathfrak o}_x^{k+1} & \to & {\mathfrak l}_x^k/{\mathfrak l}_x^{k+1} & \to & {\mathfrak t}_{M,x}^k/{\mathfrak t}_{M,x}^{k+1} & \to & 0 \\ & & \| & & \| & & \| & & \\ 0 & \to & \mathrm{Sym}^k T^*_x M & \to & {\mathfrak l}_x^k /{\mathfrak l}_x^{k+1} & \to & (\mathrm{Sym}^{k+1} T^*_x M) \otimes T_x M & \to & 0.\end{array}\end{align*} $$
-
(vii) Fix a vector space W of dimension equal to $\dim M$
. Using the notation of Definition 2.1, we can choose isomorphisms
$T_x X \cong W$
and
$ L^*_x \cong {\mathbb C}$
such that by the induced isomorphisms $$ \begin{align*}{\mathcal F}^k_x/{\mathcal F}^{k+1}_x \cong \mathrm{Sym}^k W^*, \ {\mathfrak t}^k_{M,x}/{\mathfrak t}^{k+1}_{M,x} \cong {\mathfrak v}_k(W), \ {\mathfrak l}^k_x/{\mathfrak l}^{k+1}_x \cong {\mathfrak g}_k(W),\end{align*} $$the representation of the graded Lie algebra $\mathrm {gr}({\mathfrak l}_x) = \oplus _{k\in {\mathbb Z}} {\mathfrak l}^k_x/{\mathfrak l}^{k+1}_x $
on
$\mathrm {gr}({\mathcal F}_x) = \oplus _{k \in {\mathbb Z}} {\mathcal F}^k_x/{\mathcal F}^{k+1}_x$
induced by the representation in
$\mathrm{(iii)}$
can be identified with the representation of
${\mathfrak g} (W)$
on
$ \mathrm{Sym} W^*$
in Definition 2.1
$\mathrm{(vi)}$
.
Proof. Fix a trivialization
$L|_U \cong U \times {\mathbb C}$
of the line bundle on an open neighborhood
$U \subset M$
of x equipped with a holomorphic coordinate system
$z_1, \ldots , z_n, n= \dim M,$
and a linear coordinate y on
${\mathbb C}$
. Then an element of
${\mathcal F}(U)$
is of the form
$ f(z_1, \ldots , z_n) y$
for a holomorphic function f on U and an element of
${\mathfrak l}(U)$
is of the form
for some holomorphic functions
$g^i$
and h on U. It follows that the derivative of the function
$f(z)y$
with respect to this vector field is equal to the result of applying the differential operator
to the function
$f(z)$
. From Lemma 2.4 and the above coordinates description, all the statements (i)–(vii) are easy to see. For example, the isomorphism
${\mathcal O}_{M,x} \cong {\mathfrak o}_x$
in (iv) identifies
$h(z) \in {\mathcal O}_{M,x}$
with
$h(z) y\frac {\partial }{\partial y} \in {\mathfrak o}_x$
.
Definition 2.7. Let us use the notation of Definitions 1.1 and 2.1. For a symbol system
$S = \oplus _{k \in {\mathbb Z}} S^k \subset \mathrm{ Sym} W^*$
of rank
$r \geq 1$
, using the representation of
${\mathfrak g}(W)$
on
$ \mathrm{Sym} W^*$
in Definition 2.1, define a Lie subalgebra
${\mathfrak h}^S \subset {\mathfrak g}(W)$
by
We call
${\mathfrak h}^S$
the Lie algebra of infinitesimal symmetries of the symbol system S.
Lemma 2.8. Set
${\mathfrak h}^S_k := {\mathfrak h}^S \cap {\mathfrak g}_k(W)$
in Definition 2.7. Then
-
(i) ${\mathfrak h}^S_{-1}= {\mathfrak g}_{-1}(W) = W$
; -
(ii) ${\mathfrak h}^S_k = 0 \text { if } k \leq -2 \text { or } k> r$
; -
(iii) ${\mathfrak h}^S= \oplus _{k \in {\mathbb Z}} {\mathfrak h}^S_k $
is a graded Lie subalgebra of
${\mathfrak g}$
.
Proof. The equality
${\mathfrak h}^S_{-1}= {\mathfrak g}_{-1}(W) = {\mathfrak v}_{-1}(W) = W$
is immediate from (1.1), proving (i). In (ii), it is obvious that
${\mathfrak h}^S_k = {\mathfrak g}_k(W) =0$
for
$k \leq -2$
. By Lemma 2.2, a nonzero element of
${\mathfrak g}_{r+i}(W), i>0$
cannot send
It follows that
${\mathfrak h}^S_{r+i} =0$
for
$i>0$
, proving (ii). Then (i) and (ii) imply
and it is easy to see that
${\mathfrak h}^S\cap \oplus _{k\geq 0}{\mathfrak g}_k(W)=\oplus _{k\geq 0}{\mathfrak h}^S_k$
, proving (iii).
3. Fundamental forms and infinitesimal symmetries
Definition 3.1. Let
$Z \subset {\mathbb P} V$
be a projective variety in the projectivization of a vector space V. Let
$M \subset Z$
be the smooth locus of Z and let
$\widehat {M} \subset \widehat {Z} \subset V$
be their affine cones. Let
$\pi : L \to M$
be the tautological line bundle on M with the fiber
$L_x$
equal to the one-dimensional subspace
$\widehat {x} \subset V$
corresponding to
$x \in M \subset {\mathbb P} V$
. Write
$T_v \widehat {M} \subset V$
for the affine tangent space of
$\widehat {M}$
at a point
$0 \neq v \in \widehat {M}$
.
-
(i) Define
$$ \begin{align*}{\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat{Z}) = {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat{M}) :=\{ A \in {\mathfrak g}{\mathfrak l}(V) \mid A \cdot v \in T_v \widehat{M} \text{ for all } 0 \neq v \in \widehat{M}\},\end{align*} $$called the Lie algebra of infinitesimal linear automorphisms of Z. It is the Lie algebra of the Lie group
$$ \begin{align*}\{ g \in \mathrm{GL}(V) \mid g\cdot \widehat{M} = \widehat{M} \}.\end{align*} $$
-
(ii) We have a natural morphism $\beta : L \to \widehat {M} \subset V$
contracting the zero section
$0_M$
to
$0 \in V$
and sending
$L_x$
to
$\widehat {x}$
isomorphically. For
$A \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M})$
, denote by
$\vec {A}$
the unique vector field on L such that for
$w \in L \setminus 0_M$
and
$v = \beta (w), $
$$ \begin{align*}\mathrm{d}_w \beta (\vec{A}_w) = A \cdot v \ \in T_v \widehat{M}.\end{align*} $$
Proposition 3.2. In Definition 3.1, for each
$A \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M})$
, the vector field
$\vec {A}$
on L belongs to
${\mathfrak l}(M)$
, namely, it is a
$\pi $
-linear vector field on L in the sense of Definition 2.5
$\mathrm{(ii)}$
. This determines a Lie algebra homomorphism
for each
$x \in M$
. If furthermore
$M \subset {\mathbb P} V$
is linearly nondegenerate, then
$\Psi _x$
is injective.
Proof. The 1-parameter subgroup
$\exp ({\mathbb C} A) \subset \mathrm{GL}(V)$
preserves
$\widehat {M}$
and the line bundle structure
$L \to M$
. Thus,
$\vec {A}$
belongs to
${\mathfrak l}(M)$
, defining the homomorphism
$\Psi _x$
for any
$x \in M$
. Assuming that M is linearly nondegenerate, suppose
$\Psi _x(A) =0$
for some
$A \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M}).$
Then
$A \cdot \beta (w) =0$
for any
$w \in L $
over a neighborhood of
$x \in M$
. This implies that
$A =0$
by the linear nondegeneracy, proving the injectivity of
$\Psi _x$
.
Definition 3.3. In Proposition 3.2, write
${\mathfrak h}= {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M})$
and pick a point
$x \in M$
.
-
(i) Define for each integer k,
$$ \begin{align*}{\mathfrak h}_x^k := \Psi_x^{-1}( {\mathfrak l}_x^k )\end{align*} $$such that ${\mathfrak h} = {\mathfrak h}_x^{-1} \supset {\mathfrak h}_x^0 \supset {\mathfrak h}_x^1 \supset \cdots $
gives a structure of filtered Lie algebra on
${\mathfrak h}$
. We denote by
${\mathfrak h}_x$
this filtered algebra.
-
(ii) Let
$$ \begin{align*}\mathrm {gr}({\mathfrak h}_x) := \oplus_{k \in {\mathbb Z}} \mathrm {gr}({\mathfrak h}_x)_k, \ \mathrm {gr}({\mathfrak h}_x)_{k}:= {\mathfrak h}_x^k/{\mathfrak h}_x^{k+1}\end{align*} $$be the associated graded Lie algebra.
-
(iii) Via Lemma 2.6(iii), we have a representation of ${\mathfrak h}_x$
on
${\mathcal F}_x$
induced by
$\Psi _x$
, which is compatible with the filtrations. Write $$ \begin{align*}\Phi_x: \mathrm {gr}({\mathfrak h}_x) \to \mathrm{End}(\mathrm {gr}({\mathcal F}_x))\end{align*} $$for the associated representation, which is compatible with the gradations.
Lemma 3.4. Assume that M is linearly nondegenerate in
${\mathbb P} V$
. Then
-
(i) the homomorphism $\Phi _x$
in Definition 3.3 is injective; -
(ii) ${\mathfrak h}_x^m =0$
for some positive integer m; -
(iii) $\dim {\mathfrak h} = \dim \mathrm {gr}({\mathfrak h}_x).$
Proof. The injectivity of
$\Psi _x$
in Proposition 3.2 shows that the induced homomorphism
$\mathrm {gr}({\mathfrak h}_x) \to \mathrm {gr}({\mathfrak l}_x)$
is injective. Since
$\mathrm {gr}({\mathfrak l}_x)$
acts on
$\mathrm {gr}({\mathcal F}_x)$
faithfully, we obtain (i). For each
$x \in M,$
there is some integer m such that the vanishing order of a nonzero vector field
$\vec {A} \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M})$
is bounded by m. Thus, (ii) also follows from the injectivity of
$\Psi _x$
. Finally, (iii) is immediate from (ii).
Definition 3.5. Let us use the terminology of Definitions 2.5 and 3.1.
-
(i) Via the morphism $\beta : L \to V$
, each element
$\lambda \in V^*$
determines a
$\pi $
-linear function
$\lambda _L \in {\mathcal F}(M)$
. For
$x \in M$
, we write
$\lambda _x \in {\mathcal F}_x$
for the germ of
$\lambda _L$
at
$x \in M$
. -
(ii) For each integer k, define
$$ \begin{align*}V_x^{*k} := \{ \lambda \in V^* \mid \lambda_x \in {\mathcal F}_x^k \}\end{align*} $$and let r be the smallest integer satisfying $V_x^{*r+1} = V_x^{*r+i}$
for all
$i \geq 1$
, inducing a filtration $$ \begin{align*}V^* = V_x^{*0} \supset V_x^{*1} \supset V_x^{*2} \supset \cdots \supset V_x^{*r} \supset V_x^{*r+1}\end{align*} $$satisfying $\cap _{k \in {\mathbb Z}} V_x^{*k} = V_x^{* r+1}$
.
-
(iii) Using Lemma 2.6, we have the induced homomorphism
$$ \begin{align*}\mathrm{FF}^k_x: V_x^{*k}/V_x^{* k+1} \to {\mathcal F}_x^{k}/{\mathcal F}_x^{k+1} = \mathrm{ Sym}^k T^*_x M \otimes L_x^*,\end{align*} $$called the kth fundamental form of M at $x \in M$
.
-
(iv) For each k, let $S^k_{x} \subset \mathrm{Sym} ^k T^*_x M$
be the image of the kth fundamental form. The graded subspace $$ \begin{align*}S_{x} = \oplus_{k \in {\mathbb Z}} S^k_{x} \subset \mathrm{Sym} T^*_x M\end{align*} $$is called the system of fundamental forms of M (or its closure Z) at $x \in M$
.
Lemma 3.6. For
$A \in {\mathfrak g}{\mathfrak l}(V)$
, we have the dual representation, namely, for
$\lambda \in V^*$
, the evaluation of
$A \cdot \lambda \in V^*$
at
$v \in V$
is
Let
$A \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M})$
and let
$\vec {A} \in {\mathfrak l}(M)$
be the
$\pi $
-linear vector field on L from Definition 3.1. For
$\lambda \in V^*$
and
$\lambda _L \in {\mathcal F}(M)$
in Definition 3.5, we have
where the left-hand side denotes the derivative of the
$\pi $
-linear function
$\lambda _L$
on L with respect to the vector field
$\vec {A}$
and the right-hand side denotes the
$\pi $
-linear function on L corresponding to the element
$-A \cdot \lambda \in V^*$
.
Proof. For any point
$w \in L \setminus 0_M$
and
$v = \beta (w) \in \widehat {M} \setminus 0$
, choose an arc
$\gamma : \Delta := \{ t \in {\mathbb C} \mid |t| < \epsilon \} \to \widehat {M}$
with the tangent vector
$A\cdot v \in T_v \widehat {M}$
of the form
Then
This proves the lemma.
Lemma 3.7. For
$A \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {M}) = {\mathfrak h}$
and
$x \in M$
, if
$A \in {\mathfrak h}^k_x$
for some
$k \in {\mathbb Z}$
in the notation of Definition 3.3, then
$A \cdot V_x^{*\ell } \subset V_x^{* \ell +k}$
for any
$\ell \in {\mathbb Z}$
in the notation of Definition 3.5.
Proof. For
$\lambda \in V_x^{* \ell }$
, the germ
$\lambda _x \in {\mathcal F}_x$
of
$\lambda _L$
at x belongs to
${\mathcal F}^{\ell }_x$
. Then Lemma 2.6(iii) implies
$\vec {A}_x \cdot \lambda _x \in {\mathcal F}^{\ell + k}_x$
. Thus,
$A \cdot \lambda \in V_x^{*\ell +k}$
by Lemma 3.6.
We are ready to prove Theorem 1.5.
Proof of Theorem 1.5.
Let M be the smooth locus of
$Z \subset {\mathbb P} V$
and let
$x \in M$
be as in Theorem 1.5. We use the filtrations in Definitions 3.3 and 3.5. For
$A \in {\mathfrak h}_x^k$
satisfying
$ A \notin {\mathfrak h}_x^{k+1}$
, let
$ 0 \neq \mathrm {gr}(A) \in \mathrm {gr}({\mathfrak h}_x)_k$
be the corresponding element. Then for each integer
$\ell ,$
Lemmas 3.6 and 3.7 give a commutative diagram
where
$\Phi _x$
is the representation in Definition 3.3. Thus,
$\Phi _x(\mathrm {gr}(A))$
sends
$S_x^{\ell }$
to
$S_x^{\ell +k}$
for all
$\ell $
.
Fix a vector space W of dimension equal to
$\dim M$
. Choosing an isomorphism
$T_x M \cong W,$
we can identify
$\mathrm {gr}({\mathfrak l}_x)$
with
${\mathfrak g}(W)$
and
$S_x = \oplus _{k \in {\mathbb Z}} S_x^k \subset \mathrm{Sym} T^*_x M$
with a symbol system
$S = \oplus _{k \in {\mathbb Z}} S^k \subset \mathrm{Sym} W^*$
. Then the image of
$\Phi _x(\mathrm {gr}(A))$
lies in
${\mathfrak g}_k(W)$
and since it sends
$S^{\ell }$
to
$S^{\ell +k}$
for all
$\ell $
, it belongs to
${\mathfrak h}^{S}_k \subset {\mathfrak g}_k(W)$
. Consequently, we have a homomorphism
${\mathfrak h}_x^k/{\mathfrak h}_x^{k+1} \to {\mathfrak h}^{S}_k$
for each
$k \in {\mathbb Z}$
.
By the assumption that
$Z \subset {\mathbb P} V$
is linearly nondegenerate, the homomorphism
$\Phi _x$
is injective from Lemma 3.4(i) and so is the above homomorphism
${\mathfrak h}_x^k/{\mathfrak h}_x^{k+1} \to {\mathfrak h}^{S}_k$
for each
$k \in {\mathbb Z}$
. Then (ii) and (iii) of Lemma 3.4 give
$\cap _{k\in {\mathbb Z}} {\mathfrak h}^k_x = 0$
and
$\dim {\mathfrak h} = \dim \mathrm {gr}({\mathfrak h}_x) \leq \dim {\mathfrak h}^S$
, completing the proof of the theorem.
4. Infinitesimal symmetries of Euler-symmetric varieties
We recall the following from [Reference Fu and Hwang2, Definition 3.6]. In [Reference Fu and Hwang2, Definition 2.1], Euler-symmetric varieties are defined in terms of
${\mathbb C}^*$
-actions. But the two definitions are equivalent by [Reference Fu and Hwang2, Theorem 3.7].
Definition 4.1. Let
$S = \oplus _{k\in {\mathbb Z}} S^k \subset \mathrm{Sym} W^*$
be a symbol system of rank r.
-
(i) Define $S_\bot := \oplus _{k \in {\mathbb Z}} S^k_\bot \subset \mathrm{Sym} W$
with $$ \begin{align*}S^k_\bot:=\{\xi\in\mathrm{Sym}^k W \mid f(\xi)=0 \text{ for all } f\in S_k\}.\end{align*} $$
-
(ii) Define $V^S:= \oplus _{k \in {\mathbb Z}} V^S_k$
with
$V^S_k:= \mathrm{Sym} ^k W/S^k_\bot $
. Note that
$V^S_0= {\mathbb C}, V^S_1 = W$
and
$V^S_k =0$
for
$k<0$
or
$k>r$
. We can identify
$V^S_k$
with
$(S^k)^*$
and
$V^S$
with
$S^*$
. -
(iii) For $\xi \in \mathrm{Sym} ^k W$
, denote by
$[\xi ] \in V^S_k$
its coset class modulo
$S^k_\bot .$
Define a morphism
$\phi ^S: {\mathbb C} \oplus W \to \oplus _{k \in {\mathbb Z}} V^S_k = V^S$
by sending
$(t, v) \in {\mathbb C} \oplus W$
to $$ \begin{align*}\phi^S(t, v) := \sum_{k \in {\mathbb Z}} [t^{r-k} v^k] \ \in \oplus_{k \in {\mathbb Z}} V^S_k = V^S.\end{align*} $$Here, note that when $k> r,$
we have
$S^k_\bot = \mathrm{Sym} ^k W$
and thus
$[t^{r-k} v^k]\in V^S_k=0$
for all
$t\in {\mathbb C}$
and
$v\in W$
. In particular, the summand is nonzero only for
$0 \leq k \leq r$
and we need not worry about the meaning of
$t^i, v^i$
for
$i<0$
.
-
(iv) The closure of the image of the morphism $\phi ^S$
is denoted by
$\widehat {Z}^S \subset V^S$
. The projective subvariety
$Z^S \subset {\mathbb P} V^S$
whose affine cone is
$\widehat {Z}^S$
is called the Euler-symmetric variety determined by the symbol system S.
Recall the following properties of Euler-symmetric varieties from [Reference Fu and Hwang2, Definition 2.1 and Theorem 3.7].
Proposition 4.2. A projective variety
$Z \subset {\mathbb P} V$
is projectively equivalent to
$Z^S \subset {\mathbb P} V^S$
in Definition 4.1 if and only if the following three conditions are satisfied:
-
(i) $\dim S = \dim V^S = \dim V$
. -
(ii) The system of fundamental forms $S_z \subset \mathrm{Sym} T^*_z Z$
at a general point
$z \in Z$
is isomorphic to
$S \subset \mathrm{Sym} W^*$
. -
(iii) For a general point $z \in Z$
, there is a multiplicative subgroup
${\mathbb C}^* \subset \mathrm{GL}(V)$
that preserves
$\widehat {Z}$
and
$\widehat {z} \subset \widehat {Z}$
such that the induced
${\mathbb C}^*$
-action on the tangent space
$T_z Z$
is by scalar multiplication.
Lemma 4.3. In Definition 4.1, for
$u, v \in W$
, define
Then for any choice of
$v \in W$
and
the affine tangent space
$T_\xi (\widehat {Z}^S) \subset V^S$
is spanned by
$\xi \in V^S$
itself and the vectors
Proof. This is immediate from
Lemma 4.4. The representation of
${\mathfrak g}(W)$
on
$ \mathrm{Sym} W^*$
in Definition 2.1 has its dual representation on
$ \mathrm{ Sym} W$
. The following holds for the dual representation:
-
(i) If $A \in {\mathfrak g}_k(W)$
, then
$A \cdot \mathrm{Sym} ^{\ell } W \subset \mathrm{Sym} ^{\ell -k} W$
. -
(ii) If $A = y^k \in {\mathfrak a}_k(W)$
for some
$y \in W^*$
, then
$A \cdot v^k = - y(v)^k$
for any
$v \in W$
. -
(iii) If $A \in {\mathfrak a}_k(W)$
and
$j \geq k$
, then
$A \cdot v^j = (A \cdot v^k) v^{j-k}$
for any
$v \in W$
. -
(iv) If $A = y^{k+1} \otimes u \in {\mathfrak v}_k(W)$
for some
$y \in W^*$
and
$u \in W$
, then
$A \cdot v^{k+1} = - y(v)^{k+1} u$
for any
$v \in W$
. -
(v) If $A \in {\mathfrak v}_k(W)$
and
$j \geq k$
, then
$A \cdot v^j = (j-k) (A \cdot v^{k+1}) \odot v^{j-k-1}$
for any
$v \in W$
. Here, the right-hand side is zero when
$j=k$
.
Proof. From Definition 2.1, we know that the representation of
${\mathfrak g}(W)$
on
$ \mathrm{Sym} W^*$
satisfies
for all
$k, \ell \in {\mathbb Z}$
. This implies (i) by duality.
(ii) follows from
To check (iii), we may assume by linearity
$A = y^k$
for some
$y \in W^*$
. Setting
$s= j-k$
, we need to show
$\langle \xi , A \cdot v^j\rangle = \langle \xi , (A \cdot v^k) v^s \rangle $
for any
$\xi \in \mathrm{Sym} ^s W^*$
. By linearity, we may check it for
$\xi = x^s$
with
$x \in W^*$
. Then
where the last equality uses (ii). This proves (iii).
(iv) is a consequence of the following equality for any
$z \in W^*$
:
To check (v), as in the proof of (iii), we may check
for any
$\xi = x^s$
with
$s = j-k$
and
$x \in W^*$
. Moreover, we may check it for
$A = y^{k+1} \otimes u$
with
$y \in W^*$
and
$u \in W$
. But
where the last equality uses (iv). Furthermore, when
$s = j-k=0$
, the right-hand side becomes zero. This proves (v).
Lemma 4.5. Fix a symbol system
$S \subset \mathrm{Sym} W^*$
and let
${\mathfrak h}^S \subset {\mathfrak g}(W)$
be the Lie algebra of infinitesimal symmetries of S.
-
(i) Under the natural representation of ${\mathfrak g}(W)$
on
$ \mathrm{Sym} W^*,$
the subalgebra
${\mathfrak h}^S \subset {\mathfrak g}(W)$
preserves the subspace
$S \subset \mathrm{Sym} W^*$
, inducing a faithful representation of
${\mathfrak h}^S$
on S. -
(ii) The dual representation of ${\mathfrak g}(W)$
on
$ \mathrm{Sym} W$
gives a faithful representation of
${\mathfrak h}^S$
on
$ \mathrm{Sym} W$
, which preserves
$S_\bot \subset \mathrm{Sym} W$
. It induces a representation of
${\mathfrak h}^S$
on
$V^S$
, which is dual to the representation on S in
$\mathrm{(i)}$
.
Proof. We only verify the faithfulness of the representation of
${\mathfrak h}^S$
on S, and other statements are straightforward. By Lemma 2.2, the Lie algebra
${\mathfrak g}(W)$
, hence its subalgebra
${\mathfrak h}^S$
, is a vector subspace of
$ \mathrm{ Hom} (\oplus _{k=0}^1 \mathrm{Sym} ^k W^*, \mathrm{Sym} W^*)$
. Since
$S^k= \mathrm{Sym} ^k W^*$
for
$k=0, 1$
, the natural homomorphism
is injective. It follows that the representation of
${\mathfrak h}^S$
on S is faithful.
Proposition 4.6. Let
$S \subset \mathrm{Sym} W^*$
and
$\widehat {Z}^S \subset V^S$
be as in Definition 4.1. We can regard
${\mathfrak h}^S$
as a subalgebra of
${\mathfrak g}{\mathfrak l}(V^S)$
by Lemma 4.5. Then for any
$A \in {\mathfrak h}^S_k, k \in {\mathbb Z,} $
and a general
$\xi \in \widehat {Z}^S$
, we have
$A \cdot \xi \in T_\xi \widehat {Z}^S$
. It follows that
$A \in {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}^S) \subset {\mathfrak g}{\mathfrak l}(V^S).$
Proof. By the invariance of
$\widehat {Z}^S \subset V^S$
under the scalar multiplication on V, it suffices to check
$A \cdot \xi \in T_{\xi } \widehat {Z}^S$
for any
$v \in W$
and
$\xi := \phi ^S(1, v) \in \widehat {Z}^S.$
Recall that
By Lemma 4.4, we have
$A \cdot V^S_j \subset V^S_{j-k}$
. Thus,
Applying the direct sum decomposition
we can write
with
$A'\in \mathrm{Sym} ^k W^*$
and
$A"\in \mathrm{Sym} ^{k+1}W^*\otimes W$
. Set
Lemma 4.4(v) shows
$A" \cdot v^k = 0$
, hence
For
$j\geq k+1$
, we have by Lemma 4.4
In particular, one can see that when
$j>\max \{r, k\}$
the vector
$c v^{j-k}+(j-k) u\odot v^{j-k-1}\in S^{j-k}_\bot $
and thus the corresponding coset class in
$V_{j-k}= \mathrm{Sym} ^{j-k} W/S^{j-k}_\bot $
is zero. Combining the above equality with (4.2) and (4.3), we have
where we use (4.1) in the last line. By Lemma 4.3, this implies
$A\cdot \xi \in T_{\xi } \widehat {Z}^S$
, proving the proposition.
Proposition 4.7. For the Euler-symmetric variety
$Z^S \subset {\mathbb P} V^S$
associated with a symbol system
$S \subset \mathrm{Sym} W^*$
, the Lie algebra homomorphism
${\mathfrak h}^S \to {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}^S)$
obtained from Proposition 4.6 is an isomorphism.
Proof. We know that the homomorphism
${\mathfrak h}^S \to {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}^S)$
is injective because
${\mathfrak h}^S \subset {\mathfrak g}{\mathfrak l}(V^S)$
from Lemma 4.5. By Proposition 4.2, the system of fundamental forms of
$Z^S$
at a general point is isomorphic to S as a symbol system. Consequently, Theorem 1.5 implies
$\dim {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}^S) \leq \dim {\mathfrak h}^S$
. Hence, the homomorphism must be surjective.
Proof of Theorem 1.6.
Proposition 4.7 shows that
${\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}^S) \cong \dim {\mathfrak h}^S$
for the Euler-symmetric variety
$\widehat {Z}^S \subset {\mathbb P} V^S$
. Conversely, suppose that
$Z \subset {\mathbb P} V$
is linearly nondegenerate and
$\dim {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}) = \dim {\mathfrak h}^{S_x}$
for the system of fundamental forms
$S_x \subset \mathrm{Sym} T_x^* M$
for a point
$x \in Z$
chosen in Theorem 1.5. To prove that
$Z \subset {\mathbb P} V$
is projectively equivalent to
$Z^{S_x} \subset {\mathbb P} V^{S_x}$
, it suffices to check the three conditions in Proposition 4.2. The condition (i) is obvious by the linear nondegeneracy of
$Z \subset {\mathbb P} V$
.
By
$\dim {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}) = \dim {\mathfrak h}^{S_x}$
and Lemma 3.4, the inclusion
$ {\mathfrak h}_x^k/{\mathfrak h}_x^{k+1} \subset {\mathfrak h}^{S_x}_k$
in Theorem 1.5 must be an isomorphism for all
$k \in {\mathbb Z}$
. From
${\mathfrak h}_x^{-1}/{\mathfrak h}_x^0 \cong {\mathfrak h}^{S_x}_{-1} = T_x Z$
, we see that the orbit of
$x \in Z$
under the group
$ \exp {\mathfrak a}{\mathfrak u}{\mathfrak t}(\widehat {Z}) \subset \mathrm{GL}(V)$
contains an open neighborhood
$O \subset Z$
of x. It follows that the condition (ii) in Proposition 4.2 is satisfied by any
$z \in O$
.
It is easy to see that
lies in
${\mathfrak h}^{S}_0 $
in Definition 2.7. From
$ {\mathfrak h}_x^0/{\mathfrak h}_x^{1} \cong {\mathfrak h}^{S_x}_0,$
there must be an element
$E_x \in {\mathfrak h}_x^0$
representing
$\mathrm{Id}_{T_x M}.$
Then the algebraic closure of
$\exp ({\mathbb C} E_x)$
must contain a multiplicative subgroup
${\mathbb C}^* \subset \mathrm{Aut}(\widehat {Z}) \subset \mathrm{GL}(V)$
such that the induced
${\mathbb C}^*$
-action on the tangent space
$T_x Z$
is by scalar multiplication. Thus, the condition (iii) in Proposition 4.2 is satisfied by any
$z \in O.$
This proves the theorem.
Funding statement
J.-M.H. was supported by the Institute for Basic Science (IBS-R032-D1). Q.L. was supported by the National Natural Science Foundation of China (Grant No. 12571043) and by the Natural Science Foundation of Shandong Province (Grant No. ZR2025QA08).
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