On Deformations of Nodal Hypersurfaces

TL;DR

This paper extends the infinitesimal Torelli theorem, originally for smooth hypersurfaces, to include nodal hypersurfaces, broadening the understanding of their deformation theory.

Contribution

It introduces a generalization of the infinitesimal Torelli theorem to nodal hypersurfaces, which were not covered by previous results.

Findings

Infinitesimal Torelli theorem holds for certain classes of nodal hypersurfaces.

Extension of deformation theory to singular hypersurfaces.

New techniques for analyzing nodal hypersurface deformations.

Abstract

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

Equations

$$J(f)=(\frac{\partial f}{\partial x_0},\cdots ,\frac{\partial f}{\partial x_n})$$

$$\phi :(S/J(f){)}_d\to \text{Hom}((S/J(f){)}_{d-n-1},(S/J(f){)}_{2d-n-1}),[P]\mapsto ([Q]\mapsto [PQ]).$$

$$\mathcal{P}:\mathcal{U}\ni g\mapsto (F^{n-1}{H}_0^{n-1}(X_g),F^{n-2}{H}_0^{n-1}(X_g))\in \mathcal{F}$$

$$K^{\bullet }(f):0\to {\Omega }^0\to {\Omega }^1\to \cdots \to {\Omega }^{n+1}\to 0$$

$$\sum _{j=0}^n{a}_j\frac{\partial f}{\partial x_j}=0$$

$$\frac{\partial f}{\partial x_{i}}\frac{\partial f}{\partial x_{j}}+\biggl{(}-\frac{\partial f}{\partial x_{i}}\biggr{)}\biggl{(}\frac{\partial f}{\partial x_{j}}\biggr{)}=0,\qquad i<j.$$

$$m\le \frac{nd-1}{2}.$$

$$ct(X_f)=\max \{q:\dim (S/J(f){)}_k=\dim (S/J(f_s){)}_k\text{ for all }k\le q\}$$

$$mdr(X_f)=\min \{q:H^n(K^{\bullet }(f){)}_{q+n}\ne 0\}.$$

$$ct(X_f)=mdr(X_f)+d-2,$$

$$\dim (S/J(f){)}_k=\dim (S/J(f_s){)}_k,k=d-n-1,2d-n-1.$$

$$ct(X_{f})=mdr(X_{f})+d-2\geq\biggl{(}\frac{nd-1}{2}-n\biggr{)}+d-2>2d-n-1,$$

$$G{x}_i=\sum _{k=0}^n{H}_{ik}\frac{\partial f}{\partial x_k},i=0,\cdots ,n,$$

$$0=x_i(x_{j}G)-x_j(x_{i}G)=\sum _{k=0}^n(x_i{H}_{jk}-x_j{H}_{ik})\frac{\partial f}{\partial x_k}.$$

$$t+3-d+n\le (2d-n-2)+3+n-d=d+1\le \frac{nd-1}{2},$$

$$x_i{H}_{jk}-x_j{H}_{ik}=0,i\ne j$$

$$H^i(X_f)=H^i({\mathbb{P}}^n),i<n-1,$$

$$H^{n-1}({\mathbb{P}}^n)\to H^{n-1}(X_f)$$

$$H_0^{n-1}(X_f)=\text{Coker}(H^{n-1}({\mathbb{P}}^n)\to H^{n-1}(X_f)),$$

$$H_0^{n-1}(X_f^{\ast })=\text{Coker}(H^{n-1}({\mathbb{P}}^n)\to H^{n-1}(X_f^{\ast })).$$

$${\overline{R}}_f:H^n(U_f)\stackrel{\sim }{\to }{H}_0^{n-1}(X_f^{\ast })$$

$$i^{\ast }:H^{n-1}(X_f)\to H^{n-1}(X_f^{\ast })$$

$$i_0^{\ast }:H_0^{n-1}(X_f)\to H_0^{n-1}(X_f^{\ast }).$$

$$\to H^{n-1}(X_f,X_f^{\ast })\to H^{n-1}(X_f)\stackrel{i^{\ast }}{\to }{H}^{n-1}(X_f^{\ast })\to H^n(X_f,X_f^{\ast })$$

$$H^{n-1}(X_f,X_f^{\ast })=\underset{i=1}{\overset{r}{⨁}}{H}^{n-1}(X_f,X_f\setminus \{x_i\})=0,$$

$$R_f=(i_0^{\ast }{)}^{-1}\circ {\overline{R}}_f:H^n(U_f)\to H_0^{n-1}(X_f).$$

$$R_f:F^p{H}^n(U_f)\stackrel{\sim }{\to }{F}^{p-1}{H}_0^{n-1}(X_f)$$

$$R_f:G{r}_F^{p+1}{H}^n(U_f)\stackrel{\sim }{\to }G{r}_F^p{H}_0^{n-1}(X_f),p=n-1,n-2.$$

$$H_0^{n-1}(X_f)=H^{n-1}(X_f),H_0^{n-1}(X_f^{\ast })=H^{n-1}(X_f^{\ast })$$

$$\cdots \to H^{n-1}(X_f,X_f^{\ast })\to H^{n-1}(X_f)\stackrel{i^{\ast }}{\to }{H}^{n-1}(X_f^{\ast })\to H^n(X_f,X_f^{\ast })\to \cdots .$$

Full text

1. Introduction

Deformations of smooth hypersurfaces provide examples of great interest and importance in the theory of variation of Hodge structures, especially because of the generic Torelli theorem, see [13], Chapter 6. In a recent thesis [15], Y. Zhao considers deformations of nodal surfaces in the $3$-dimensional complex projective space $\mathbb{P}^{3}$ and shows that the infinitesimal Torelli theorem still holds.

Let $S=\mathbb{C}[x_{0},\cdots,x_{n}]=\bigoplus_{d=0}^{\infty}S_{d}$ be the graded ring of polynomials and let $f\in S_{d}$ be a homogeneous polynomial of degree $d$. Denote by $X_{f}:f=0$ the hypersurface in $\mathbb{P}^{n}$ defined by $f$. Moreover, let

[TABLE]

be the graded ideal generated by the first derivatives of $f$, also called the Jacobian ideal of $f$. We consider the following map

[TABLE]

As a matter of fact, Y. Zhao [15] proves the infinitesimal Torelli theorem by showing that the map $\varphi$ is injective when $n=3$ and $X_{f}$ is a nodal surface. This result can be extended to higher dimensional cases.

Theorem 1.1**.**

Assume $n\geq 3$ is an integer and $d\geq n+1$. Let $f\in S$ be a homogeneous polynomial of degree $d$ such that $X_{f}:f=0$ is a nodal hypersurface in $\mathbb{P}^{n}$. Then the map $\varphi$ is injective.

As it is proved in [15], Chapter 3, Example 3.1.3, $(S/J(f))_{d}$ parameterizes the equivalence classes of deformations of the pair $(\mathbb{P}^{n},X_{f})$. Alternatively, let $GL=GL(n+1,\mathbb{C})$ be the general linear group of rank $n+1$. Then $GL$ acts on $S_{d}$ by coordinate transformations and for any $f\in S_{d}$, the tangent space at $f$ of the orbit $GL\cdot f$ is given by $J(f)_{d}$, see [1], Chapter 4, Formula (4.16). It follows that $(S/J(f))_{d}$ can be seen as the set of directions in $S_{d}$ that are transversal to the orbit $GL\cdot f$ at $f$. In addition, any smooth analytic subset $\mathcal{U}\subseteq S_{d}$ can be seen as a family of hypersurfaces in $\mathbb{P}^{n}$. If $f\in\mathcal{U}$ and $T_{f}\mathcal{U}\cap J(f)_{d}=\{0\}$, then we call $\mathcal{U}$ an effective deformation of $f$. From this point of view, $(S/J(f))_{d}$ is the maximal set of effective deformations of $f$.

Now let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ and let $n(f)$ be the number of nodes in $X_{f}$. Then we have a moduli space, denoted by $\mathfrak{B}_{f}\subseteq S_{d}$, parameterizing all nodal hypersurfaces in $\mathbb{P}^{n}$ having exactly $n(f)$ nodes. By the discussion following Corollary 3.8 in [2], Chapter 1, we have that $\mathfrak{B}_{f}$ is a constructible subvariety of $S_{d}$ and the topological type of $(\mathbb{P}^{n},X_{g})$ is locally trivial for $g\in\mathfrak{B}_{f}$. Moreover, for any $g$ lying in the connected component of $\mathfrak{B}_{f}$ containing $f$, $(\mathbb{P}^{n},X_{g})$ is topologically equivalent to $(\mathbb{P}^{n},X_{f})$.

Now assume $\mathcal{U}\subseteq\mathfrak{B}_{f}$ is a connected smooth subvariety and $f\in\mathcal{U}$. For any $g\in\mathcal{U}$, $X_{g}$ is homeomorphic to $X_{f}$ by the local topological triviality of the pair $(\mathbb{P}^{n},X_{g})$. So there is a natural identification $H^{n-1}_{0}(X_{g})\cong H^{n-1}_{0}(X_{f})$, where $H^{n-1}_{0}(X_{g})$ is the primitive cohomology of $X_{g}$ defined by $H^{n-1}_{0}(X_{g})=\text{Coker}(H^{n-1}(\mathbb{P}^{n})\to H^{n-1}(X_{g}))$. In particular, $\dim H^{n-1}_{0}(X_{g})$ is constant for $g\in\mathcal{U}$.

Moreover, $H^{n-1}_{0}(X_{g})$ has a natural mixed Hodge structure, since $X_{g}$ is a singular algebraic variety, see [11], Part II, Chapter 5. It turns out that $\dim F^{n-1}H^{n-1}_{0}(X_{g})$ and $\dim F^{n-2}H^{n-1}_{0}(X_{g})$ are also constant for $g\in\mathcal{U}$ (in most cases), see Corollary 3.6 below. Thus, we have the following well-defined map

[TABLE]

where $\mathcal{F}$ is the corresponding flag manifold of subspaces of $H^{n-1}_{0}(X_{f})$.

By relating the primitive cohomology with the graded pieces of the algebra $S/J(f)$ and applying Theorem 1.1, we prove the following, as a generalization of [15], Chapter 3.

Theorem 1.2**.**

Assume $n\geq 3$ is odd or $n\geq 6$ is even. Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$ and let $\mathcal{U}\subseteq\mathfrak{B}_{f}$ be a smooth subvariety of $\mathfrak{B}_{f}$ which gives an effective deformation of $X_{f}$. Then the map $\mathcal{P}$ above is well-defined and the differential $d\mathcal{P}$ is injective at $f$.

Thus, loosely speaking, the infinitesimal Torelli theorem also holds for nodal hypersurfaces.

Note that for smooth hypersurfaces, the generic Torelli theorem holds, see [13], Part II, Chapter 6, Section 6.3.2, and it remains an interesting question whether this is also the case for nodal hypersurfaces. Recall that in the proof of the generic Torelli theorem for smooth hypersurfaces, the essential part is to show that a generic homogeneous polynomial can be reconstructed from its Jacobian ideal, which also holds for nodal hypersurfaces by Theorem 1.1 in [14], because a generic $f$ of degree $d>3$ with the associated hypersurface $X_{f}$ having a fixed number of nodes is not of Sabastiani-Thom type, which is the only exception for $f$ not to be reconstructed from $J(f)$; another key ingredient in the smooth case is the symmetriser lemma, which is still open for nodal hypersurfaces.

The author would like to thank an anonymous referee, whose remarks make the exposition of this paper improved.

2. Syzygies of the Jacobian ideal

Let $K^{\bullet}(f)$ be the Koszul complex of $\frac{\partial f}{\partial x_{0}},\cdots,\frac{\partial f}{\partial x_{n}}$ with the natural grading $\deg(x_{j})=1$ and $\deg(dx_{j})=1$:

[TABLE]

where $\Omega^{1}=\sum_{i=0}^{n}Sdx_{i}$ and $\Omega^{p}=\bigwedge^{p}\Omega^{1}$, and the differentials are given by the wedge product with $df=\sum_{i=0}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}$.

The homogeneous component of the cohomology group $H^{n}(K^{\bullet}(f))_{n+r}$ describes the syzygies

[TABLE]

with $a_{j}\in S_{r}$ modulo the trivial syzygies generated by

[TABLE]

We may restate the main result in [5] or Theorem 9 in [6] in the following form.

Lemma 2.1**.**

Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d>2$ and $n\geq 3$, then $H^{n}(K^{\bullet}(f))_{m}=0$ for any

[TABLE]

Let $f_{s}\in S_{d}$ be such that $X_{f_{s}}:f_{s}=0$ is a smooth hypersurface. It is well-known that $\dim(S/J(f_{s}))_{k}$ depends only on $n,d$ and $k$, see [1], Chapter 7, Proposition 7.22. In the introduction part of [4], the following two notions are given:

[TABLE]

and

[TABLE]

They have the following relation

[TABLE]

see loc. cit.. We have the following.

Lemma 2.2**.**

Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$ and $n\geq 3$, then

[TABLE]

In particular, $\dim(S/J(f))_{k}$ does not depend on the concrete equation of the polynomial $f$ for $k=d-n-1,2d-n-1$.

Proof.

We only need to check that $2d-n-1\leq ct(X_{f})$. Indeed, by Lemma 2.1, we immediately have

[TABLE]

where the last inequality follows from $n\geq 3$ and $d\geq n+1$. ∎

2.3. Proof of Theorem 1.1

To prove Theorem 1.1, we first prove the following.

Lemma 2.4**.**

Assume $X_{f}:f=0$ is a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$ and $n\geq 3$. Let $G\in S_{t}$ such that $t<2d-n-1$ and $Gx_{j}\in J(f)$ for all $j=0,\cdots,n$, then $G\in J(f)$.

Proof.

Assume

[TABLE]

with $H_{ik}\in S_{t+2-d},i,k=0,\cdots,n$, then

[TABLE]

Note that

[TABLE]

so by Lemma 2.1, we get $x_{i}H_{jk}-x_{j}H_{ik}\in J(f)$ for all $i,j,k=0,\cdots,n$ while all these polynomials have degree $t+3-d<(2d-n-1)+3-d=d-n+2\leq d-1$, so they must all vanish identically; in particular,

[TABLE]

thus, $x_{i}|H_{ik}$. It follows that $G\in J(f)$ as desired. ∎

*Proof of Theorem 1.1: * We first remark that Theorem 1.1 holds when $d=n+1$. In fact, in this case, $J(f)_{d-n-1}=J(f)_{0}=0$ and $(S/J(f))_{d-n-1}=S_{0}=\mathbb{C}$ consists of constants. Since $1\in(S/J(f))_{d-n-1}$ and $\varphi([P])(1)=[P]$, one sees easily that $\varphi$ is injective.

Thus, in the sequel of the proof, we will focus on the case $d>n+1$.

Aiming at a contradiction, we assume that there exists $P\in S_{d}\setminus J(f)_{d}$ such that $\varphi([P])=0$.

Then there exists a $Q\in S_{l},0\leq l<d-n-1$ such that $PQ\notin J(f)$ and $l$ is chosen to be maximal. By the maximality of $l$, we have $(PQ)x_{j}\in J(f)$ for all $j=0,\cdots,n$. Note that $PQ\in S_{l+d}$ and $l+d<2d-n-1$, hence by Lemma 2.4, $PQ\in J(f)$, contradiction.

3. Hodge theory for nodal hypersurfaces

Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$ and $n\geq 3$. The cohomology groups under consideration below all have $\mathbb{C}$ as coefficients unless otherwise explicitly pointed out.

By the Lefschetz hyperplane theorem for singular varieties (see [9]), we have

[TABLE]

and

[TABLE]

is injective. Let

[TABLE]

be the primitive cohomology of $X_{f}$. Then $H^{n-1}_{0}(X_{f})$ admits a mixed Hodge structure. Moreover, let $U_{f}=\mathbb{P}^{n}\setminus X_{f}$ be the complement of $X=X_{f}$, then $H^{n}(U_{f})$ also admits a mixed Hodge structure and $H^{n}(U_{f})$ and $H^{n-1}_{0}(X_{f})$ are closely related.

3.1. Relation between $H^{*}(U_{f})$ and $H^{*}(X_{f})$

Let $X_{f}^{*}$ be the smooth locus of $X_{f}$ and let

[TABLE]

Then $H^{n-1}_{0}(X_{f}^{*})$ has a natural mixed Hodge structure. Moreover, as is shown in [2], Chapter 6, Corollary 3.11, there is a natural residue isomorphism

[TABLE]

which is also an isomorphism of mixed Hodge structures of type $(-1,-1)$.

Let $i:X_{f}^{*}\to X_{f}$ be the inclusion. We have the naturally induced homomorphisms in cohomology

[TABLE]

and

[TABLE]

Moreover, $i^{*},i^{*}_{0}$ are also morphisms of mixed Hodge structures. Our discussion will be divided into two cases, regarding whether $n$ is odd or even.

3.1.1. Case 1: $n$ is odd

When $n$ is odd, the variety $X_{f}$ is a $\mathbb{Q}$-homology manifold, i.e., for any point $x\in X_{f}$, $H^{i}(X_{f},X_{f}\setminus\{x\},\mathbb{Q})=\mathbb{Q}$ if $i=2n$ and [math] otherwise. Moreover, we have the following claim.

Claim 3.2**.**

$i^{*}$* and $i_{0}^{*}$ are both isomorphisms.*

Proof.

Indeed, we have a long exact sequence of mixed Hodge structures with respect to the pair $(X_{f},X_{f}^{*})$:

[TABLE]

Let $x_{i},i=1,\cdots,r$ be all the nodes in $X_{f}$, then $X_{f}^{*}=X_{f}\setminus\{x_{1},\cdots,x_{r}\}$, and furthermore, by the excision theorem

[TABLE]

since $X_{f}$ is a $\mathbb{Q}$-homology manifold and $n-1\neq 0,2n$ for $n\geq 3$. Similarly, $H^{n}(X_{f},X_{f}^{*})=0$. Thus, it follows from (5) that $i^{*}$ and $i^{*}_{0}$ are both isomorphisms. ∎

Note that the weights of $H^{n-1}(X_{f})$ are $\leq n-1$ since $X_{f}$ is compact while the weights of $H^{n-1}(X_{f}^{*})$ are $\geq n-1$ since $X_{f}^{*}$ is smooth (see [11], p. 131, Table 5.1), hence both $H^{n-1}(X_{f}^{*})$ and $H^{n-1}(X_{f})$ have pure Hodge structures of weight $n-1$ and it follows from the isomorphism (4) that $H^{n}(U_{f})$ has a pure Hodge structure of weight $n+1$.

Let

[TABLE]

Then $R_{f}$ is an isomorphism of mixed Hodge structures of type $(-1,-1)$. It follows that we have isomorphisms

[TABLE]

for all $p$. In particular, there are isomorphisms

[TABLE]

3.2.1. Case 2: $n$ is even

When $n$ is even, $X_{f}$ is no longer a $\mathbb{Q}$-homology manifold. However, there is still an explicit description of the relations between $H^{n}(U_{f})$ and $H^{n-1}_{0}(X_{f})$. Note that in this case $H^{n-1}(\mathbb{P}^{n})=0$ and thus

[TABLE]

and $i^{*}=i^{*}_{0}$. Moreover, there exists an exact sequence of mixed Hodge structures

[TABLE]

To make use of this exact sequence, we first give the following claim.

Claim 3.3**.**

For $k=n-1,n$, $H^{k}(X_{f},X_{f}^{*})$ has a pure Hodge structure of type $(\rho_{k},\rho_{k})$ for some $\rho_{k}\in\mathbb{N}$.

Proof.

Let $a_{1},\cdots,a_{m}$ be the nodes in $X_{f}$ and $B_{i}\ni a_{i},i=1,\cdots,m$ be a small ball in $\mathbb{P}^{n}$ around $a_{i}$ such that $B_{i}\cap B_{j}=\emptyset$ for $i\neq j$.

By the excision theorem and conic structure theorem (see [2], Chapter 1, Theorem 5.1),

[TABLE]

where $K_{i}$ is the link of $X_{f}$ around $a_{i}$ ($i=1,\cdots,m$).

For each $i$, $K_{i}$ has the homotopy type of the unit sphere bundle of tangent bundle of $S^{n-1}$. Indeed, locally around $a_{i}$, $X_{f}$ is defined as $z_{1}^{2}+\cdots+z_{n}^{2}=0$, where $(z_{1},\cdots,z_{n})$ is the local coordinate system of $\mathbb{P}^{n}$ centered at $a_{i}$. Then $K_{i}$ can be described as

[TABLE]

where $\epsilon>0$ is small. Let

[TABLE]

and

[TABLE]

then

[TABLE]

which is the unit sphere bundle of tangent bundle of $S^{n-1}$.

It follows that

[TABLE]

Note also that $H^{k-1}(K_{i})=H^{k}(B_{i}\cap X_{f},B_{i}\cap X_{f}\setminus\{a_{i}\})=H^{k}(X_{f},X_{f}\setminus\{a_{i}\})$ admits a natural mixed Hodge structure. In particular,

[TABLE]

where $Gr^{W}_{w}H^{k-1}(K_{i})$ is a pure Hodge structure of weight $w$ and

[TABLE]

is the Hodge decomposition. By the Hodge symmetry, we have

[TABLE]

It follows that there exists $\rho_{k,i}\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

and

[TABLE]

In particular, $H^{k-1}(K_{i})$ is pure of type $(\rho_{k,i},\rho_{k,i})$.

Note that the mixed Hodge structure on $H^{k-1}(K_{i})$ depends only on the local structure of $X_{f}$ around $a_{i}$ (see [8], Theorem 3.4). Since all the $a_{i}$’s are nodes, $H^{k-1}(K_{i})$ is naturally isomorphic to $H^{k-1}(K_{j})$ as mixed Hodge structures for any $i,j$, hence there exists $\rho_{k}\in\mathbb{N}$ such that

[TABLE]

and thus $H^{k}(X_{f},X_{f}^{*})$ is pure of type $(\rho_{k},\rho_{k})$ for $k=n-1,n$. ∎

By Proposition (C28) in [2], Appendix C (see also [8], Proposition 3.8) , it follows that $2\rho_{n-1}\leq n-2$. Thus,

[TABLE]

Moreover, by the discussions above Example 3.18 in [2], Chapter 6, $H^{n-1}(K_{i})$ has weight $n$, namely, $2\rho_{n}=n$, and thus for $n\geq 6$

[TABLE]

Therefore, it follows from (7) that we have an isomorphism

[TABLE]

for $n\geq 4$. Furthermore, we have isomorphisms

[TABLE]

and

[TABLE]

for $n\geq 6$; but for $n=4$, we only have injections

[TABLE]

and

[TABLE]

Using the residue isomorphism (4), we denote

[TABLE]

for $n\geq 4$ (and $n$ is even). Then clearly, $F^{n-1}(U_{f},X_{f})=F^{n-1}H^{n}(U_{f})$ for $n\geq 6$.

We still denote by $\overline{R}_{f}$ its restriction to $F^{n-1}(U_{f},X_{f})$. Then

[TABLE]

is an isomorphism and we have an isomorphism

[TABLE]

3.3.1. Conclusion

In conclusion, no matter whether $n$ is even or odd, we always have isomorphisms

[TABLE]

and

[TABLE]

where F^{n-1}(U_{f},X_{f})=\overline{R}_{f}^{-1}\biggl{(}i_{0}^{*}(F^{n-2}H^{n-1}_{0}(X_{f}))\biggr{)} is a subspace of $F^{n-1}H^{n}(U_{f})$ containing $F^{n}H^{n}(U_{f})$; and $R_{f}=(i_{0}^{*})^{-1}\circ\overline{R}_{f}$.

3.4. Cohomology of $X_{f}$

Denote by

[TABLE]

where $\widehat{(\ )}$ means that the term is omitted. As is shown in [2], Chapter 6, any cohomology class in $F^{p}H^{n}(U_{f})$ can be represented by a form

[TABLE]

with $h\in S_{(n-p+1)d-n-1}$. Hence, by (8), we see that any element in $Gr_{F}^{n-1}H^{n-1}_{0}(X_{f})$ can be represented by

[TABLE]

with $h_{1}\in S_{d-n-1}$ and similarly, by (9), any element in $Gr_{F}^{n-2}H^{n-1}_{0}(X_{f})$ can be represented by

[TABLE]

with $h_{2}\in S_{2d-n-1}$.

Such results agree with [7], Theorem 2.2, where the following formulae are given

[TABLE]

for $n>3$ and for $n=3$,

[TABLE]

where $I(f)$ is the saturation of $J(f)$, which is also equal to the radical of $J(f)$ for a nodal hypersurface (see [4], Remark 2.2).

Putting all the discussions above in this section together, we obtain the following.

Proposition 3.5**.**

Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$. Then

1. (i)

when $n\geq 3$, there is an isomorphism

[TABLE] 2. (ii)

when $n>4$, there is an isomorphism

[TABLE] 3. (iii)

when $n=3$, there is an isomorphism

[TABLE] 4. (iv)

when $n=4$, there is an isomorphism

[TABLE]

where $S^{\prime}\subseteq S_{2d-n-1}$ is a vector subspace containing $J(f)_{2d-n-1}$ obtained via

[TABLE]

where $\omega$ is the isomorphism

[TABLE]

established in **[7]**, Theorem 2.2, and $F^{n-1}(U_{f},X_{f})$ is obtained in (9).

In all the formulae above, $R_{f}$ denotes the residue map.

As a corollary, we have the following.

Corollary 3.6**.**

Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$. Then

1. (i)

if $n\geq 3$, the dimension

[TABLE]

depends only on $n,d$. 2. (ii)

if $n\geq 3$ is odd or $n\geq 6$ is even, the dimension

[TABLE]

depends only on $n,d$ and possibly the number of nodes in $X_{f}$.

Proof.

Note that

[TABLE]

and

[TABLE]

If $n>4$, the results follow from Proposition 3.5 and Lemma 2.2, and the dimensions depend only on $n,d$. When $n=3$, $X_{f}$ is a $\mathbb{Q}$-homology manifold and the Hodge numbers of $X_{f}$ depend only on $n,d$ and the number of nodes in $X_{f}$, see also [3]. ∎

4. Variations of mixed Hodge structures

Let $X_{f}:f=0$ be a nodal hypersurface in $\mathbb{P}^{n}$ of degree $d\geq n+1$. When $n$ is odd, assume $n\geq 3$ while when $n$ is even, assume $n\geq 6$.

4.1. Topological triviality

Recall that $\mathfrak{B}_{f}\subseteq S_{d}$ parameterizes all nodal hypersurfaces with the same number of nodes as $X_{f}$. Let $\mathcal{U}\subseteq\mathfrak{B}_{f}$ be a contractible smooth subvariety containing $f$ such that it gives an effective deformation for $X_{f}$. Set

[TABLE]

which can be seen as the union of all nodal hypersurfaces parameterized by $\mathcal{U}$.

Then by the First Thom Isotopy Lemma (see [2], Chapter 1, Section 3), there is a homeomorphism $\Phi$ satisfying the following commutative diagram

[TABLE]

where $p_{1},p_{2}$ are natural projections. In fact, $\Phi$ can be obtained by integrating some well-controlled stratified vector field; for a proof, see [10]. From now on, we fix such a homeomorphism.

In particular, for any $g\in\mathcal{U}$, there is a canonical homeomorphism $\Phi_{g}:\mathbb{P}^{n}\to\mathbb{P}^{n}$, which induces homeomorphisms $\Phi_{g,X}:X_{f}\to X_{g}$ and $\Phi_{g,U}:U_{f}\to U_{g}$ with $\Phi_{f}=\text{\rm Id}$.

Moreover, we have an induced isomorphism of groups

[TABLE]

Hence $\dim H^{n-1}_{0}(X_{g})$ is constant for $g\in\mathcal{U}$.

In addition, by Corollary 3.6, under our assumption on $n$, the dimensions

[TABLE]

are constant with respect to $g\in\mathcal{U}$. Via the identification $\Phi_{g,X}^{*}:H^{n-1}_{0}(X_{g})\xrightarrow{\sim}H^{n-1}_{0}(X_{f})$, it follows that $(F^{n-1}H^{n-1}_{0}(X_{g}),F^{n-2}H^{n-1}_{0}(X_{g}))$ can be identified with $(\Phi_{g,X}^{*}F^{n-1}H^{n-1}_{0}(X_{g}),\Phi_{g,X}^{*}F^{n-2}H^{n-1}_{0}(X_{g}))$, which are two subspaces of $H^{n-1}_{0}(X_{f})$ of fixed dimension. Therefore, we have the well-defined map as in (2)

[TABLE]

where $\mathcal{F}$ is the following flag manifold

[TABLE]

When $n$ is odd, all the Hodge numbers of $X_{g}$ are constant for $g\in\mathcal{U}$, and $\mathcal{P}$ is just two components of the period map in the theory of variation of Hodge structures, see [12], Part III, Chapter 10.

4.2. Infinitesimal deformation

Now we consider the differential of $\mathcal{P}$. Note that a component of $d\mathcal{P}_{f}$ is the map

[TABLE]

for the properties of tangent spaces of flag manifolds, we refer to [12], Part III, Chapter 10 and for analogous treatments for smooth hypersurfaces, see [13], Part II, Chapter 6. Recall that Proposition 3.5 implies that any element in $F^{n-1}H^{n-1}_{0}(X_{f})$ is of the form

[TABLE]

The following holds.

Lemma 4.3**.**

For $h\in T_{f}\mathcal{U}\subseteq S_{d}$, we have

[TABLE]

Its proof is a little lengthy and we postpone it to the end of this section; instead, we first derive Theorem 1.2 from Lemma 4.3.

4.4. Proof of Theorem 1.2

From Lemma 4.3 and Proposition 3.5, the image of $d\mathcal{P}_{f}$ is contained in

[TABLE]

Moreover, we get the following commutative diagram

[TABLE]

where $\varphi$ is given in (1). $i_{1}$ is the composite $T_{f}\mathcal{U}\subseteq S_{d}\to S_{d}/J(f)_{d}$, which is injective since $\mathcal{U}$ is an effective deformation. $i_{2}$ is defined as follows: for $\eta\in\text{\rm Hom}(Gr_{F}^{n-1}H^{n-1}_{0}(X_{f}),Gr_{F}^{n-2}H^{n-1}_{0}(X_{f}))$ and $h_{1}\in(S/J(f))_{d-n-1}$,

[TABLE]

where $\Lambda_{f}$ is the isomorphism given in Proposition 3.5.

By Theorem 1.1, $\varphi$ is injective, hence $\varphi\circ i_{1}$ is injective. Thus it follows from (10) that $d\mathcal{P}_{f}$ is injective, hence Theorem 1.2 follows.

Remark 4.5**.**

The result is probably also true for $n=4$. We exclude this case because we do not know whether the dimension $\dim F^{n-2}H^{n-1}_{0}(X_{g})$ or equivalently $\dim Gr_{F}^{n-2}H^{n-1}_{0}(X_{g})$ is constant for $g\in\mathcal{U}$ in this case.

4.6. Proof of Lemma 4.3

The proof is almost the same as that in [13], Part II, Chapter 6 where variations of smooth hypersurfaces are considered. However, to avoid any possible confusion, we give the details here.

From the topological triviality of the family $X_{g},g\in\mathcal{U}$, it follows that there exists a small contractible neighbourhood $\mathcal{N}\ni f$ in $\mathcal{U}$, such that for any $g\in\mathcal{N}$, $X_{g}$ is a deformation retract of

[TABLE]

Set

[TABLE]

Then $U_{\mathcal{N}}$ is a deformation retract of $U_{g}$ for every $g\in\mathcal{N}$. Let for $g\in\mathcal{N}$

[TABLE]

be the natural inclusion, then the induced homomorphism in cohomology

[TABLE]

is an isomorphism.

The differential $d\mathcal{P}_{f}$ can be computed as follows: for any $h\in T_{f}\mathcal{U}\subseteq S_{d}$, choose a curve $g(t):(-\epsilon,\epsilon)\to\mathcal{N}\subseteq\mathcal{U}$ such that $g(0)=f$ and $\frac{dg}{dt}(0)=h$. For any element in $F^{n-1}H^{n-1}_{0}(X_{f})$ of the form

[TABLE]

let

[TABLE]

give an element of $F^{n-1}H^{n-1}_{0}(X_{g(t)})$. Then

[TABLE]

We have

[TABLE]

where $\Phi_{g,U}^{*}$ is the homomorphism induced by the map $\Phi_{g,U}:U_{f}\to U_{g}$. Note that $\Phi_{g(t),U}^{*}:H^{n}(U_{g(t)})\to H^{n}(U_{f})$ is equal to the composition

[TABLE]

Hence,

[TABLE]

Note that $\tau_{g(t)}^{*}$ acting on forms is a restriction map, it follows that

[TABLE]

Therefore,

[TABLE]

Now the proof of Lemma 4.3 is complete.

Remark 4.7**.**

To prove Theorem 1.2, it is essential for us to obtain a diagram like (10). In fact, when Y. Zhao [15] proves the infinitesimal Torelli theorem for nodal surfaces, he uses such a diagram implicitly; however, he does not give any proofs. We believe that a detailed proof is indeed needed and this is a special reason why our discussions above always include the case $n=3$.