Generalized adjoint forms on algebraic varieties

Abstract

We prove a full generalization of the Castelnuovo’s free pencil trick. We show its analogies with Rizzi and Zucconi (Differential forms and quadrics of the canonical image. arXiv:1409.1826, Theorem 2.1.7); see also Pirola and Zucconi (J Algebraic Geom 12(3):535–572, Theorem 1.5.1). Moreover we find a new formulation of the Griffiths’s infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic 1-forms.

1 Introduction

Let X be an m-dimensional smooth projective variety and $\mathcal {F}$ be a rank n locally free sheaf over it. A way to study $\mathcal {F}$ is to study its extensions $0\rightarrow \mathcal {L}\rightarrow \mathcal {E}\rightarrow \mathcal {F}\rightarrow 0$ which, up to isomorphism, are parametrized by $\text {Ext}^1(\mathcal {F},\mathcal {L})$. In [2, 3, 5, 6, 10–13] and [1] the adjoint forms associated to $\xi \in \mathrm{{Ext}}^{1}(\mathcal {O}_X, \mathcal {F})$ are deeply studied and many applications are given. Let us recall the notion of adjoint form in the case $\mathcal {L}=\mathcal {O}_X$.

Given $\xi \in \mathrm{{Ext}}^{1}(\mathcal {O}_X, \mathcal {F})$, take an $(n+1)$-dimensional subspace W of the kernel of the cup-product homomorphism $\partial _{\xi }:H^0(X,\mathcal {F})\rightarrow H^1(X,\mathcal {O}_X)$. Denote by $\lambda ^{i}W$ the image of $\bigwedge ^iW$ through the natural homomorphism $\lambda ^i:\bigwedge ^i H^0(X,\mathcal {F})\rightarrow H^0(X,\bigwedge ^{i}\mathcal {F})$. If $\mathcal {B}:=\langle \eta _{1},\ldots ,\eta _{n+1}\rangle$ is a basis of W and $s_1,\ldots , s_{n+1}\in H^0(X,\mathcal {E})$ are liftings of $\eta _{1},\ldots ,\eta _{n+1}$, respectively, then the map $\Lambda ^{n+1}:\bigwedge ^{n+1} H^0(X,\mathcal {E})\rightarrow H^0(X,\bigwedge ^{n+1}\mathcal {E})$ gives the top form $\Omega :=\Lambda ^{n+1}(s_1\wedge s_2 \wedge \ldots \wedge s_{n+1})\in H^0(X,\det \mathcal {E})$. The section $\Omega$ corresponds to a top form $\omega _{\xi ,W,{\widehat{\mathcal {B}}}}\in H^0(X,\det \mathcal {F})$ via the isomorphism $\det \mathcal {F}\simeq \det \mathcal {E}$, where ${\widehat{\mathcal {B}}}=\langle s_1,\ldots , s_{n+1}\rangle$; the form $\omega _{\xi ,W,{\widehat{\mathcal {B}}}}$ is called an adjoint form of W and $\xi$. To the basis $\mathcal {B}$ there are also naturally associated $n+1$ elements $\omega _i:=\lambda ^{n}(\eta _1\wedge \ldots \wedge \eta _{i-1}\wedge {\widehat{\eta _i}}\wedge \eta _{i+1}\wedge \ldots \wedge \eta _{n+1})$, $i=1,\ldots ,n+1$, obtained by the basis $\langle \eta _1\wedge \ldots \wedge \eta _{i-1}\wedge {\widehat{\eta _i}}\wedge \eta _{i+1}\wedge \ldots \wedge \eta _{n+1}\rangle _{i=1}^{n+1}$ of $\bigwedge ^n W$. Note that if we change the liftings $s_1,\ldots , s_{n+1}\in H^0(X,\mathcal {E})$ with other liftings ${\widetilde{s}_1},\ldots , {\widetilde{s}_{n+1}}$, then $\omega _{\xi ,W,\widehat{\mathcal {B}}}$ is a linear combination of $\omega _{\xi ,W,\widetilde{\mathcal {B}}}$ and $\omega _1$,..., $\omega _{n+1}$. The natural problem of this theory is to characterize the condition $\omega _{\xi ,W,\widehat{\mathcal {B}}}\in \lambda ^nW$ in terms of the fixed divisor $D_W$ of $|\lambda ^nW|\subset \mathbb {P} H^0(X,\det \mathcal {F})$ and of the base locus $Z_W$ of the moving part $M_W\in \mathbb {P} H^0(X,\det \mathcal {F}\otimes _{\mathcal {O}_X}\mathcal {O}_{X}(-D_W))$, where $|\lambda ^nW|=D_W+|M_W|$.

In this paper we consider the general case where $\mathcal {L}$ is an invertible sheaf not necessarily equal to $\mathcal {O}_X$. In this case $\det \mathcal {E}=\mathcal {L}\otimes \det \mathcal {F}$ and liftings $s_1,\ldots , s_{n+1}\in H^0(X,\mathcal {E})$ of $\eta _{1},\ldots ,\eta _{n+1}\in H^0(X,\mathcal {F})$ determine $\Omega :=\Lambda ^{n+1}(s_1\wedge s_2 \wedge \ldots \wedge s_{n+1})\in H^0(X,\det \mathcal {E})$ which is now called a generalized adjoint form. We define as before $\omega _i:=\lambda ^{n}(\eta _1\wedge \ldots \wedge \eta _{i-1}\wedge {\widehat{\eta _i}}\wedge \eta _{i+1}\wedge \ldots \wedge \eta _{n+1})$, $i=1,\ldots , n+1$ and we characterize the case where $\Omega$ belongs to the image of $H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E})$ by the natural tensor product map. The game is more complicated than in the above-mentioned papers because the linear system $|\lambda ^nW|$ is inside $\mathbb {P} H^0(X,\det \mathcal {F})$ and we have to relate the fixed divisor $D_W$ of $|\lambda ^nW|$ and the base locus $Z_W$ of the moving part $M_W\in \mathbb {P} H^0(X,\det \mathcal {F}\otimes _{\mathcal {O}_X}\mathcal {O}_{X}(-D_W))$ to forms which are not anymore inside $H^0(X,\det \mathcal {F})$. Nevertheless the result is analogue to the one of [11, Theorem 1.5.1] and [13, Theorem 2.1.7]:

Theorem [A] Let X be an m-dimensional complex compact smooth variety. Let $\mathcal {F}$ be a rank n locally free sheaf on X and $\mathcal {L}$ an invertible sheaf. Consider an extension $0\rightarrow \mathcal {L}\rightarrow \mathcal {E}\rightarrow \mathcal {F}\rightarrow 0$ corresponding to $\xi \in \mathrm{{Ext}}^1(\mathcal {F},\mathcal {L})$. Let $W=\langle \eta _1,\ldots ,\eta _{n+1}\rangle$ be an $n+1$-dimensional sublinear system of $\ker ({\partial _\xi })\subset H^0(X,\mathcal {F})$. Let $\Omega \in H^0(X,\det \mathcal {E})$ be a generalized adjoint form associated to W as above. It holds that if $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$ then $\xi \in \ker (H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})\rightarrow H^1(X,\mathcal {F}^\vee \otimes \mathcal {L}(D_W)))$.

Theorem [A], called Adjoint Theorem, can be thought as a general version of the well-known Castelnuovo’s free pencil trick; c.f. see Theorem 2.8.

We have also a viceversa of the Adjoint Theorem; see: Theorem 2.9:

Theorem [B] Under the same hypothesis of Theorem [A], assume also that $H^0(X,\mathcal {L})\cong H^0(X,\mathcal {L}(D_W))$. It holds that if $\xi \in \ker (H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})\rightarrow H^1(X,\mathcal {F}^\vee \otimes \mathcal {L}(D_W)))$, then $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$.

In particular in the case $D_W=0$ Theorem [B] is a full characterization of the condition $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$.

Now by the Adjoint Theorem and by Theorem [B] we can study extension classes of sheaves via adjoint forms. Indeed even if $\mathcal {F}$ has no global sections we can always take the tensor product with a sufficiently ample linear system $\mathcal {M}$ such that $\mathcal {F}\otimes \mathcal {M}$ has enough global sections in order to apply the theory of adjoint forms. By applying the above idea to the case where $n>2$, $X\subset \mathbb {P}^n$ is an hypersurface of degree $d>3$ and $\mathcal {F}:=\Omega ^{1}_{X}\otimes _{\mathcal {O}_X}\mathcal {O}_X(2)$ we have a reformulation of the infinitesimal Torelli Theorem for X in the setting of generalized adjoint theory. In this paper we will not recall the theory concerning infinitesimal Torelli Theorems, for which a reference is [16], in any case a quick introduction to this topic is also given in [13]. Here we point out only that given a degree d form $F\in \mathbb {C}[\xi _0,\ldots ,\xi _n]$ the Jacobian ideal of F is the ideal $\mathcal {J}$ generated by the partial derivatives $\frac{\partial {F}}{\partial {\xi _i}}$ for $i=0,\ldots ,n$ and by [9][Theorem 9.8], any infinitesimal deformation $\xi \in H^1(X,\Theta _X)$, where $X=(F=0)$ and $\Theta _X$ is the sheaf of tangent vectors on X, is given by a class [R] in the quotient $\mathbb {C}[\xi _0,\ldots ,\xi _n]/\mathcal {J}$ where R is a homogeneous form of degree d.

Theorem [C] For a smooth hypersurface X of degree d in $\mathbb {P}^n$ with $n\ge 3$ and $d>3$ the following are equivalent:

• (1) the differential of the period map is zero on the infinitesimal deformation

  $$\begin{aligned}{}[R]\in ( \mathbb {C}[\xi _0,\ldots ,\xi _n]/\mathcal {J})_{d}\simeq H^1(X,\Theta _X) \end{aligned}$$

• (2) R is an element of the Jacobian ideal $\mathcal {J}$

• (3) $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {O}_X(2))\otimes \lambda ^nW \rightarrow H^0(X,\mathcal {O}_X(n+d-1)))$ for the generic generalized adjoint $\Omega$

• (4) The generic generalized adjoint $\Omega$ lies in $\mathcal {J}$.

Note that Theorem [C] has a different flavor with respect to the analogue [9, Theorem 9.8] since we essentially use meromorphic 1-forms over X; see Proposition 3.7. Finally we want to mention that in a forthcoming paper [14] we show how to recover also the Green’s infinitesimal Torelli Theorem for a sufficiently ample divisor of a smooth variety in terms of generalized adjoint theory.

2 The theory of generalized adjoint forms

2.1 Definition of generalized adjoint form

Let X be a smooth compact complex variety of dimension m and let $\mathcal {F}$ and $\mathcal {L}$ be two locally free sheaves on X of rank n and 1, respectively. Consider the exact sequence of locally free sheaves

$$\begin{aligned} 0\rightarrow \mathcal {L}\rightarrow \mathcal {E}\rightarrow \mathcal {F}\rightarrow 0 \end{aligned}$$

(2.1)

associated to an element $\xi \in \text {Ext}^1(\mathcal {F},\mathcal {L})\cong H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})$. Recall that the invertible sheaf $\det \mathcal {F}:=\bigwedge ^n\mathcal {F}$ fits into the exact sequence

$$\begin{aligned} 0\rightarrow \bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L}\rightarrow \bigwedge ^n\mathcal {E}\rightarrow \det \mathcal {F}\rightarrow 0, \end{aligned}$$

(2.2)

which still corresponds to $\xi$ under the isomorphism $\text {Ext}^1(\mathcal {F},\mathcal {L})\cong \text {Ext}^1(\det \mathcal {F},\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L})\cong H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})$. Furthermore $\det \mathcal {F}$ satisfies

$$\begin{aligned} \det \mathcal {F}\otimes \mathcal {L}\cong \det \mathcal {E}. \end{aligned}$$

(2.3)

Let $\partial _\xi :H^0(X,\mathcal {F})\rightarrow H^1(X,\mathcal {L})$ be the connecting homomorphism related to (2.1), and let $W\subset \ker (\partial _\xi )$ be a vector subspace of dimension $n+1$. Choose a basis $\mathcal {B}:=\{\eta _1,\ldots ,\eta _{n+1}\}$ of W. By definition we can take liftings $s_1,\ldots ,s_{n+1}\in H^0(X,\mathcal {E})$ of the sections $\eta _1,\ldots ,\eta _{n+1}$. If we consider the natural map

$$\begin{aligned} \Lambda ^n:\bigwedge ^{n}H^0(X,\mathcal {E})\rightarrow H^0(X,\bigwedge ^n\mathcal {E}) \end{aligned}$$

we can define the sections

$$\begin{aligned} \Omega _i:=\Lambda ^n(s_1\wedge \ldots \wedge \hat{s_i}\wedge \ldots \wedge s_{n+1}) \end{aligned}$$

(2.4)

for $i=1,\ldots ,n+1$. Denote by $\omega _i$, for $i=1,\ldots ,n+1$, the corresponding sections in $H^0(X,\det \mathcal {F})$. Obviously we have that $\omega _i=\lambda ^n(\eta _1\wedge \ldots \wedge \hat{\eta _i}\wedge \ldots \wedge \eta _{n+1})$, where $\lambda ^n$ is the natural morphism

$$\begin{aligned} \lambda ^n:\bigwedge ^{n}H^0(X,\mathcal {F})\rightarrow H^0(X,\det \mathcal {F}). \end{aligned}$$

The vector subspace of $H^0(X,\det \mathcal {F})$ generated by $\omega _1,\ldots ,\omega _{n+1}$ is denoted by $\lambda ^nW$.

Definition 2.1

If $\lambda ^nW$ is nontrivial, it induces a sublinear system $|\lambda ^n W|\subset \mathbb {P}(H^0(X,\det \mathcal {F}))$ that we will call adjoint sublinear system. We call $D_W$ its fixed divisor and $Z_W$ the base locus of its moving part $|M_W|\subset \mathbb {P}(H^0(X,\det \mathcal {F}(-D_W)))$.

Definition 2.2

The section $\Omega \in H^0(X,\det \mathcal {E})$ corresponding to $s_1\wedge \ldots \wedge s_{n+1}$ via

$$\begin{aligned} \Lambda ^{n+1}:\bigwedge ^{n+1}H^0(X,\mathcal {E})\rightarrow H^0(X,\det \mathcal {E}) \end{aligned}$$

(2.5)

is called generalized adjoint form.

Remark 2.3

It is easy to see by local computation that this section is in the image of the natural injection $\det \mathcal {E}(-D_W)\otimes \mathcal {I}_{Z_W}\rightarrow \det \mathcal {E}$.

We want to study the condition

$$\begin{aligned} \Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i\right\rangle \rightarrow H^0(X,\det \mathcal {E})) \end{aligned}$$

(2.6)

or, equivalently,

$$\begin{aligned} \Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW \rightarrow H^0(X,\det \mathcal {E})). \end{aligned}$$

(2.7)

The first map is given by the wedge product, the second one by (2.3). Note that if $H^0(X,\mathcal {L})=0$ this condition is equivalent to $\Omega =0$.

Remark 2.4

The choice of the liftings is not relevant for this purpose. Take different liftings $s_1',\ldots ,s_{n+1}'\in H^0(X,\mathcal {E})$ of $\eta _1,\ldots ,\eta _{n+1}$ and call $\Omega _i'\in H^0(X,\bigwedge ^n\mathcal {E})$ and $\Omega '\in H^0(X,\det \mathcal {E})$ the corresponding sections constructed as above. Obviously

$$\begin{aligned} \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i\right\rangle \rightarrow H^0(X,\det \mathcal {E}))=\mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i'\right\rangle \rightarrow H^0(X,\det \mathcal {E})), \end{aligned}$$

(2.8)

since they are both equal to $\mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW \rightarrow H^0(X,\det \mathcal {E}))$. It is also easy to see that $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i\right\rangle \rightarrow H^0(X,\det \mathcal {E}))$ iff $\Omega '\in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i'\right\rangle \rightarrow H^0(X,\det \mathcal {E}))$.

Remark 2.5

Consider another basis $\mathcal {B}':=\{\eta _1',\ldots ,\eta _{n+1}'\}$ of W and let A be the matrix of the basis change. The sections $s_1',\ldots ,s_{n+1}'$ obtained from $s_1,\ldots ,s_{n+1}$ through the matrix A are liftings of $\eta _1',\ldots ,\eta _{n+1}'$. The section $\Omega ':=\Lambda ^{n+1}(s_1'\wedge \ldots \wedge s_{n+1}')$ satisfies $\Omega '=\det A\cdot \Omega$. Moreover $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i\right\rangle \rightarrow H^0(X,\det \mathcal {E}))$ iff $\Omega '\in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i'\right\rangle \rightarrow H^0(X,\det \mathcal {E}))$.

Lemma 2.6

If $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \left\langle \Omega _i\right\rangle \rightarrow H^0(X,\det \mathcal {E}))$, then we can find liftings $\tilde{s_i}\in H^0(X,\mathcal {E})$, $i=1,\ldots ,n+1$, such that $\tilde{\Omega }:=\Lambda ^{n+1}(\tilde{s_1}\wedge \ldots \wedge \tilde{s}_{n+1})=0$.

Proof

By hypothesis there exist $\sigma _i\in H^0(X,\mathcal {L})$ such that

$$\begin{aligned} \Omega =\sum ^{n+1}_{i=1} \sigma _i\wedge \Omega _i \end{aligned}$$

(2.9)

We can define new liftings for the element $\eta _i$:

$$\begin{aligned} \tilde{s_i}:=s_i+(-1)^{n-i}\sigma _i. \end{aligned}$$

Now, since

$$\begin{aligned} \tilde{s_1}\wedge \ldots \wedge \tilde{s}_{n+1}=s_1\wedge \ldots \wedge s_{n+1}-\sum ^{n+1}_{i=1} s_1\wedge \ldots \wedge \hat{s_i}\wedge \ldots \wedge s_{n+1}\wedge \sigma _i, \end{aligned}$$

(2.10)

we immediately deduce $\tilde{\Omega }=0$. $\square$

From the natural map

$$\begin{aligned} \mathcal {F}^\vee \otimes \mathcal {L}\rightarrow \mathcal {F}^\vee \otimes \mathcal {L}(D_W) \end{aligned}$$

we have a homomorphism

$$\begin{aligned} H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})\mathop {\rightarrow }\limits ^{\rho } H^1(X,\mathcal {F}^\vee \otimes \mathcal {L}(D_W)); \end{aligned}$$

we call $\xi _{D_W}=\rho (\xi )$.

2.2 Castelnuovo’s free pencil trick

Consider the case where both $\mathcal {L}$ and $\mathcal {F}$ are of rank one, while X has dimension m. In this case $W=\left\langle \eta _1,\eta _2\right\rangle \subset H^0(X,\mathcal {F})$ has dimension two; as usual we choose liftings $s_1,s_2\in H^0(X,\mathcal {E})$ of $\eta _1,\eta _2$. Note also that $\omega _1=\eta _2$ and $\omega _2=\eta _1$, in particular $W=\lambda ^1W$ so $D_W$ is the fixed part of W and $Z_W$ is the base locus of its moving part. Call $\tilde{\eta }_i \in H^0(X,\mathcal {F}(-D_W))$ the sections corresponding to the $\eta _i$’s via $H^0(X,\mathcal {F}(-D_W))\rightarrow H^0(X,\mathcal {F})$. The following lemma is well known and it is the core of the Castelnuovo base point free pencil trick.

Lemma 2.7

We have an exact sequence

$$\begin{aligned} 0\rightarrow \mathcal {F}^\vee (D_W)\mathop {\rightarrow }\limits ^{i}\mathcal {O}_X\oplus \mathcal {O}_X\mathop {\rightarrow }\limits ^{\nu }\mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}\rightarrow 0 \end{aligned}$$

(2.11)

where the morphism i is given by contraction with $-\tilde{\eta }_1$ and $\tilde{\eta }_2$, while $\nu$ is given by evaluation with $\tilde{\eta }_2$ on the first component and $\tilde{\eta }_1$ on the second one.

It is easy to see by local computation that sequence (2.11) fits into the following commutative diagram

(2.12)

The morphism $\mathcal {E}^\vee \rightarrow \mathcal {O}_X\oplus \mathcal {O}_X$ is given by contraction with the sections $-s_1$ and $s_2$, the morphism $\mathcal {L}^\vee \rightarrow \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}$ by contraction with the adjoint $\Omega$. We can prove now the following

Theorem 2.8

Let X be an m-dimensional complex compact smooth variety. Let $\mathcal {F}$, $\mathcal {L}$ be invertible sheaves on X. Consider $\xi \in H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})$ associated to the extension (2.1). Define $W=\left\langle \eta _1,\eta _{2}\right\rangle \subset \ker ({\partial _\xi })\subset H^0(X,\mathcal {F})$ and $\Omega$ as above. We have that $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes W\rightarrow H^0(X,\det \mathcal {E}))$ if and only if $\xi _{D_W}=0$.

Proof

Tensoring (2.12) by $\mathcal {L}$ and passing to cohomology we have the following diagram

(2.13)

Obviously $\beta (1)=\Omega$ and, by commutativity, $\delta (\beta (1))=\xi _{D_W}$. We have then $\xi _{D_W}=0$ if and only if $\Omega \in \mathrm {Im}\,(H^0(\mathcal {L}\oplus \mathcal {L})\mathop {\rightarrow }\limits ^{\nu } H^0(\mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}\otimes \mathcal {L}))$. Since $\nu$ is given by the sections $\tilde{\eta }_2$ and $\tilde{\eta }_1$, this condition is equivalent to $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes W\rightarrow H^0(X,\det \mathcal {E}))$, since $\det \mathcal {E}=\mathcal {F}\otimes \mathcal {L}$. $\square$

2.3 The Adjoint Theorem

We go back now to the general case with $\mathcal {F}$ locally free of rank n. By obvious identifications the natural map

$$\begin{aligned} \text {Ext}^1(\det \mathcal {F},\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L})\rightarrow \text {Ext}^1(\det \mathcal {F}(-D_W),\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L}) \end{aligned}$$

gives an extension $\mathcal {E}^{(n)}$ and a commutative diagram:

(2.14)

2.3.1 The proof of the Adjoint Theorem

By the hypothesis $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$ and by lemma (2.6), we can choose liftings $s_i\in H^0(X,\mathcal {E})$ of $\eta _i$ with $\Omega =0$.

Since $D_W$ is the fixed divisor of the linear system $|\lambda ^n W|$ and the sections $\omega _i$ generate this linear system, then the $\omega _i$ are in the image of

$$\begin{aligned} \det \mathcal {F}(-D_W)\rightarrow \det \mathcal {F}, \end{aligned}$$

so we can find sections $\tilde{\omega }_i\in H^0(X,\det \mathcal {F}(-D_W))$ such that

$$\begin{aligned} \tilde{\omega }_i\cdot d=\omega _i, \end{aligned}$$

(2.15)

where d is a global section of $\mathcal {O}_X(D_W)$ with $(d)=D_W$. Hence, using the commutativity of (2.14), we can find liftings $\tilde{\Omega }_i\in H^0(X,\mathcal {E}^{(n)})$ of the sections $\Omega _i$. The evaluation map

$$\begin{aligned} \bigoplus _{i=1}^{n+1}\mathcal {O}_X\mathop {\rightarrow }\limits ^{\tilde{\mu }}\mathcal {E}^{(n)} \end{aligned}$$

given by the global sections $\tilde{\Omega }_i$, composed with the map $\alpha$ of (2.14), induces a map $\mu$ which fits into the following diagram

We point out that the morphism $\mu$ is given by multiplication by $\tilde{\omega }_i$ on the i-th component. The sheaf $\mathrm {Im}\,\tilde{\mu }$ is torsion free since it is a subsheaf of the locally free sheaf $\mathcal {E}^{(n)}$. Moreover, since $\Omega =0$, a local computation shows that $\mathrm {Im}\,\tilde{\mu }$ has rank one outside $Z_W$. On the other hand the sheaf $\mathrm {Im}\,\mu$ is by definition

$$\begin{aligned} \mathrm {Im}\,\mu =\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}. \end{aligned}$$

The morphism

$$\begin{aligned} \alpha :\mathcal {E}^{(n)}\rightarrow \det \mathcal {F}(-D_W) \end{aligned}$$

induces a surjective morphism, that we continue to call $\alpha$,

$$\begin{aligned} \mathrm {Im}\,\tilde{\mu }\mathop {\rightarrow }\limits ^{\alpha }\mathrm {Im}\,\mu \end{aligned}$$

between two sheaves that are locally free of rank one outside $Z_W$. This morphism is also injective, because its kernel is a torsion subsheaf of the torsion free sheaf $\mathrm {Im}\,\tilde{\mu }$, hence it is trivial.

We have proved that

$$\begin{aligned} \mathrm {Im}\,\tilde{\mu }\cong \det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}, \end{aligned}$$

so

$$\begin{aligned} \mathcal {E}^{(n)}\supset (\mathrm {Im}\,\tilde{\mu })^{\vee \vee }\cong \det \mathcal {F}(-D_W). \end{aligned}$$

This isomorphism gives the splitting

Since $\xi _{D_W}$ is the element of $H^1(X,\mathcal {F}^\vee \otimes \mathcal {L}(D_W))$ associated to this extension, we conclude that $\xi _{D_W}=0$.

We have proved the Adjoint Theorem.

2.3.2 An inverse of the Adjoint Theorem

We prove now an inverse of the Adjoint Theorem.

Theorem 2.9

Let X be an m-dimensional complex compact smooth variety. Let $\mathcal {F}$ be a rank n locally free sheaf on X and $\mathcal {L}$ an invertible sheaf. Consider an extension $0\rightarrow \mathcal {L}\rightarrow \mathcal {E}\rightarrow \mathcal {F}\rightarrow 0$ corresponding to $\xi \in {\mathrm{Ext}}^1(\mathcal {F},\mathcal {L})$. Let $W=\langle \eta _1,\ldots ,\eta _{n+1}\rangle$ be a $n+1$-dimensional sublinear system of $\ker ({\partial _\xi })\subset H^0(X,\mathcal {F})$. Let $\Omega \in H^0(X,\det \mathcal {E})$ be an adjoint form associated to W as above. Assume that $H^0(X,\mathcal {L})\cong H^0(X,\mathcal {L}(D_W))$. If $\xi \in \ker (H^1(X,\mathcal {F}^\vee \otimes \mathcal {L})\rightarrow H^1(X,\mathcal {F}^\vee \otimes \mathcal {L}(D_W)))$, then $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$.

Proof

If $\mathcal {F}$ is a rank one sheaf, then (2.8) gives the thesis without the extra assumption $H^0(X,\mathcal {L})\cong H^0(X,\mathcal {L}(D_W))$. We assume then rank $\mathcal {F}$ $\ge 2$.

By (2.3), we can write $(\Omega )=D_W+F$ with F effective. In the first step of the proof we want to find a global section

$$\begin{aligned} \Omega '\in H^0\left( X,\bigwedge ^n\mathcal {E}\otimes \mathcal {L}(-F)\right) \end{aligned}$$

which restricts, through the natural map

$$\begin{aligned} \bigwedge ^n\mathcal {E}\otimes \mathcal {L}(-F)\rightarrow \det \mathcal {E}(-F), \end{aligned}$$

to the section $d\in H^0(\det \mathcal {E}(-F))$, where $(d)=D_W$.

Consider the commutative diagram:

By the hypothesis $\xi _{D_W}=0$ it follows easily that there exists a lifting $\tilde{\Omega }\in H^0(X,\bigwedge ^{n}\mathcal {E}\otimes \mathcal {L})$ of $\Omega$. Indeed, tensor (2.14) by $\mathcal {L}$ and take a global lifting $f\in H^0(X, \det \mathcal {E}(-D_W))$ of $\Omega$. Since $\xi _{D_W}=0$, f can be lifted to a section $e\in H^0(X, \mathcal {E}^{(n)}\otimes \mathcal {L})$. Define $\tilde{\Omega }:=\psi (e)$. By commutativity, $H_3(\tilde{\Omega }|_F)=0$ hence we call $\bar{\mu }\in H^0(X,\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L}^{\otimes 2}|_F)$ the lifting of $\tilde{\Omega }|_F$. A local computation shows that the connecting homomorphism

$$\begin{aligned} \delta :H^0(X,\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L}^{\otimes 2}|_F)\rightarrow H^1(X,\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L}^{\otimes 2}(-F)) \end{aligned}$$

maps $\bar{\mu }$ to $\xi _{D_W}$, which is zero by hypothesis. Then there exists a global section

$$\begin{aligned} \mu \in H^0(X,\bigwedge ^{n-1}\mathcal {F}\otimes \mathcal {L}^{\otimes 2}) \end{aligned}$$

which is a lifting of $\bar{\mu }$. The section

$$\begin{aligned} \hat{\Omega }:=\Omega -\tau (\mu )\in H^0(X,\bigwedge ^{n}\mathcal {E}\otimes \mathcal {L}) \end{aligned}$$

is a new lifting of $\Omega$ that, by construction, vanishes when restricted to F. We call

$$\begin{aligned} \Omega '\in H^0(X,\bigwedge ^{n}\mathcal {E}\otimes \mathcal {L}(-F)) \end{aligned}$$

the global section which lifts $\hat{\Omega }$. It is easy to see that $G_2(\Omega ')=d$ so $\Omega '$ is the section we wanted.

In the second part of the proof we prove that $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$. The global sections

$$\begin{aligned} \omega _i:=\lambda ^n(\eta _1\wedge \ldots \wedge \hat{\eta _i}\wedge \ldots \wedge \eta _{n+1})\in H^0(X,\det \mathcal {F}) \end{aligned}$$

generate $\lambda ^nW$ and by definition they vanish on $D_W$, that is there exist global sections $\tilde{\omega }_i\in H^0(X,\det \mathcal {F}(-D_W))$ such that

$$\begin{aligned} \omega _i=\tilde{\omega }_i\cdot d. \end{aligned}$$

We consider the commutative diagram

(2.16)

The map $\beta$ is locally defined by

$$\begin{aligned} (f_1,\ldots ,f_{n+1})\mapsto (-1)^nf_1\cdot s_1+\cdots +f_{n+1}\cdot s_{n+1}. \end{aligned}$$

The map $\alpha$ is defined in the following way: if $f\in \mathcal {L}(-F)(U)$ is a local section, then, locally on U, $\alpha$ is given by

$$\begin{aligned} f\mapsto (\tilde{\omega }_1(f),\ldots ,\tilde{\omega }_{n+1}(f)), \end{aligned}$$

where we observe that the sections $\tilde{\omega }_i$ are global sections of the dual sheaf of $\mathcal {L}(-F)$. The sheaf $\bar{\mathcal {F}}$ is by definition the cokernel of the first row. Now, tensoring by $\mathcal {L}^\vee$, we have

(2.17)

Dualizing and tensoring again by $\mathcal {O}_X(D_W)$, we obtain the commutative square

where we have used the isomorphism of vector spaces $W^\vee \cong \bigwedge ^{n}W$, given by

$$\begin{aligned} \eta ^i\mapsto \eta _1\wedge \ldots \wedge \hat{\eta _i}\wedge \ldots \wedge \eta _{n+1}=:e_i \end{aligned}$$

where $\eta ^1,\ldots ,\eta ^{n+1}$ is the basis of $W^\vee$ dual to the basis $\eta _1,\ldots ,\eta _{n+1}$ of W. By definition of $\alpha$ we have that $\alpha ^\vee$ is the evaluation map given by the global sections $\tilde{\omega }_i$. Note that $\mathcal {E}^\vee \otimes \mathcal {L}(D_W)\cong \bigwedge ^{n}\mathcal {E}\otimes \mathcal {L}(-F)$. Taking global sections we have

The section $\Omega '\in H^0(X,\mathcal {E}^\vee \otimes \mathcal {L}(D_W))$ produces in $H^0(X,\det \mathcal {E})$ the adjoint $\Omega$, so by commutativity

$$\begin{aligned} \Omega =\overline{\alpha ^\vee }(\overline{\beta ^\vee }(\Omega ')). \end{aligned}$$

We have

$$\begin{aligned} \overline{\beta ^\vee }(\Omega ')=\sum _{i=1}^{n+1}c_i\cdot e_i\otimes \sigma _i, \end{aligned}$$

where $c_i\in \mathbb {C}$ and $\sigma _i\in H^0(X,\mathcal {L}(D_W))$. By our hypothesis $H^0(X,\mathcal {L})\cong H^0(X,\mathcal {L}(D_W))$, there exists sections $\tilde{\sigma _i}\in H^0(X, \mathcal {L})$ with $\sigma _i=\tilde{\sigma _i}\cdot d$. So

$$\begin{aligned} \Omega =\overline{\alpha ^\vee }(\overline{\beta ^\vee }(\Omega '))=\overline{\alpha ^\vee }(\sum _{i=1}^{n+1}c_i\cdot e_i\otimes \sigma _i)=\sum _{i=1}^{n+1}c_i\cdot \tilde{\omega }_i\cdot \sigma _i=\sum _{i=1}^{n+1} c_i\cdot \tilde{\omega }_i\cdot d\cdot \tilde{\sigma }_i=\sum _{i=1}^{n+1} c_i\cdot \omega _i\cdot \tilde{\sigma }_i. \end{aligned}$$

This is exactly our thesis. $\square$

By the Adjoint Theorem and (2.9) we deduce the following

Corollary 2.10

If $D_W=0$, then $\xi =0$ iff $\Omega \in \mathrm {Im}\,(H^0(X,\mathcal {L})\otimes \lambda ^nW\rightarrow H^0(X,\det \mathcal {E}))$.

3 Infinitesimal Torelli Theorem for projective hypersurfaces

In this section we want to study adjoint images in the case of smooth hypersurfaces of the projective space $\mathbb {P}^n$.

3.1 Meromorphic 1-forms on a smooth projective hypersurface

Let $V\subset \mathbb {P}^n$ be a smooth hypersurface defined by a homogeneous polynomial $F\in \mathbb {C}[\xi _0,\ldots ,\xi _n]$ of degree $\deg F=d$. An infinitesimal deformation $\xi \in \text {Ext}^1(\Omega ^1_V,\mathcal {O}_V)$ of V gives an exact sequence for the sheaf of differential forms $\Omega _V^1$:

$$\begin{aligned} 0\rightarrow \mathcal {O}_V\rightarrow \Omega ^1_{\mathcal {V}|V}\rightarrow \Omega ^1_V\rightarrow 0. \end{aligned}$$

(3.1)

We assume that $n\ge 3$, hence $H^0(V,\Omega ^1_V)=0$ and we can not construct the adjoint of this sequence directly, so we twist (3.1) by a suitable integer a such that $\Omega ^1_V(a)$ has at least $n=\mathrm {rank}\,(\Omega ^1_V)+1$ global sections. A standard computation shows that $a=2$ is enough for this purpose, so from now on we will consider the sequence

$$\begin{aligned} 0\rightarrow \mathcal {O}_V(2)\rightarrow \Omega ^1_{\mathcal {V}|V}(2)\rightarrow \Omega ^1_V(2)\rightarrow 0 \end{aligned}$$

(3.2)

which again corresponds to $\xi \in \text {Ext}^1(\Omega ^1_V(2),\mathcal {O}_V(2))\cong \text {Ext}^1(\Omega ^1_V,\mathcal {O}_V)\cong H^1(V,\Theta _V)$, where $\Theta _V$ denotes the sheaf of vector fields on V. Denote by $\mathcal {J}$ the Jacobian ideal of F, that is the ideal of $\mathbb {C}[\xi _0,\ldots ,\xi _n]$ generated by the partial derivatives $\frac{\partial {F}}{\partial {\xi _i}}$ for $i=0,\ldots ,n$. Following [9][Theorem 9.8], the deformation $\xi$ is given by a class [R] of degree d in the quotient $\mathbb {C}[\xi _0,\ldots ,\xi _n]/\mathcal {J}$. If we choose a representative R of degree d for this class, then $F~+~t R=0$, for small t, is the equation of the hypersurface that is the associated deformation of V.

Together with (3.2), we have the conormal exact sequence

$$\begin{aligned} 0\rightarrow \mathcal {O}_V(-d)\rightarrow \Omega ^1_{\mathbb {P}^n|V}\rightarrow \Omega ^1_V\rightarrow 0. \end{aligned}$$

(3.3)

If we put these sequences together we obtain the diagram

which can be completed as follows

(3.4)

By [9] the deformation $\xi$ of (3.2) comes from $R\in H^0(\mathbb {P}^n, \mathcal {O}_\mathbb {P}^n(d))$, then it gives the zero element in $H^0(V,\Theta _{ \mathbb {P}^n |V} )$, hence we have that the sheaf $\mathcal {G}$ in (3.4) is a direct sum $\mathcal {G}=\mathcal {O}_V(2)\oplus \Omega ^1_{\mathbb {P}^n|V}(2)$ and we have a natural morphism $\phi :\Omega ^1_{\mathbb {P}^n|V}(2)\rightarrow \Omega ^1_{\mathcal {V}|V}(2)$ which fits in the diagram

(3.5)

The morphism $\phi$ gives in a natural way a morphism

$$\begin{aligned}&\phi ^n:H^0(V,\det (\Omega ^1_{\mathbb {P}^n|V}(2)))\cong H^0(V,\mathcal {O}_V(n-1))\rightarrow H^0(V,\det (\Omega ^1_{\mathcal {V}|V}(2)))\\&\quad \cong H^0(V,\mathcal {O}_V(n+d-1)). \end{aligned}$$

We can write explicitly the isomorphism between $H^0(V,\det (\Omega ^1_{\mathbb {P}^n|V}(2)))=H^0(V,\Omega ^n_{\mathbb {P}^n|V}(2n))$ and $H^0(V,\mathcal {O}_V(n-1))$. Note that $H^0(\mathbb {P}^n,\Omega ^n_{\mathbb {P}^n}(2n))\rightarrow H^0(V,\Omega ^n_{\mathbb {P}^n|V}(2n))$ is surjective, so we will focus on the rational n-forms on $\mathbb {P}^n$. By [9][Corollary 2.11] this forms may be written as $\omega =\frac{P\Psi }{Q}$ where $\Psi =\sum _{i=0}^n(-1)^i\xi _i(d\xi _0\wedge \ldots \wedge d\widehat{\xi _i}\wedge \ldots \wedge d\xi _n)$ gives a generator of $H^0(\mathbb {P}^n,\Omega ^n_{\mathbb {P}^n}(n+1))$ and $\deg Q=\deg P +(n+1)$. In our case Q is a polynomial of degree 2n, hence P has degree $n-1$. This identification depends on the (noncanonical) choice of the polynomial Q and gives an isomorphism $H^0(V,\Omega ^n_{\mathbb {P}^n|V}(2n))\rightarrow H^0(V,\mathcal {O}_V(n-1))$ defined by $\omega |_V\mapsto P$.

Proposition 3.1

$\phi ^n$ is given via the multiplication by the polynomial R (modulo F).

Proof

Locally we can see $\mathcal {V}$ in the product $\Delta \times \mathbb {P}^n$ of the projective space with a disk; here $\mathcal {V}$ is defined by the equation $F+tR=0$. Hence $d(F+tR)=0$ in $\Omega ^1_{\mathcal {V}}$, that is $dF+dt\cdot R+dR\cdot t=0$.

Call $F_i:=\frac{\partial F}{\partial \xi _i}$. Since V is smooth, there exist i such that $U_i=(F_i\ne 0)$ is a nontrivial open subset; let for example $U_1$ be nontrivial. Take local coordinates $z_i=\frac{\xi _i}{\xi _0}$ in the open set $(\xi _0\ne 0)\cap U_1$. Then we have

$$\begin{aligned} d z_1=-\frac{Rdt}{F_1}-\frac{tdR}{F_1}-\sum _{i>1}\frac{F_i}{F_1}d z_i \end{aligned}$$

(3.6)

which gives in V (that is for $t=0$)

$$\begin{aligned} d z_1=-\frac{Rdt}{F_1}-\sum _{i>1}\frac{F_i}{F_1}d z_i \end{aligned}$$

(3.7)

The image $\phi ^n(\omega |_V)$ is then obtained by the substitution of (3.7) in $\frac{P(z)}{Q(z)}dz_1\wedge \ldots \wedge dz_n$, which is the local form of $\frac{P(\xi )\Psi }{Q(\xi )}$. Hence

$$\begin{aligned} \frac{P(z)}{Q(z)}dz_1\wedge \ldots \wedge dz_n=-\frac{P(z) R(z)}{Q(z) F_1(z)}dt\wedge dz_2\wedge \ldots \wedge dz_n. \end{aligned}$$

(3.8)

If we homogenize we obtain on $U_1$

$$\begin{aligned} \frac{P\Psi }{Q}=-\frac{PR}{QF_1}\sum _{i\ne 1}(-1)^{i-1}\text {sgn}(i-1)\xi _idt\wedge d\xi _0\wedge \widehat{d\xi _1}\ldots \wedge \widehat{d\xi _i}\wedge \ldots \wedge d\xi _n \end{aligned}$$

Hence

$$\begin{aligned} \phi ^n(\omega |_V)=-\frac{PR}{QF_1}\sum _{i\ne 1}(-1)^{i-1}\text {sgn}(i-1)\xi _idt\wedge d\xi _0\wedge \widehat{d\xi _1}\ldots \wedge \widehat{d\xi _i}\wedge \ldots \wedge d\xi _n \end{aligned}$$

(3.9)

and it is clear that $\phi ^n$ is given by multiplication with R. $\square$

3.2 A canonical choice of adjoints on a hypersurface of degree $d>2$

We want now to construct adjoint forms associated to the sequence (3.2).

Assume that $n\ge 3$, so that $H^1(V,\mathcal {O}_V(2))=H^1(V,\mathcal {O}_V(2-d))=0$, and we can lift all the global sections of $H^0(V,\Omega ^1_V(2))$ both in the horizontal and in the vertical sequence of (3.5).

We take $\eta _1,\ldots ,\eta _n\in H^0(V,\Omega ^1_V(2))$ global forms and we want to find liftings $s_1,\ldots ,s_n\in H^0(V,\Omega ^1_{\mathcal {V}|V})$. This can be done since $H^1(V,\mathcal {O}_V(2))$ is zero. A generalized adjoint is then the global section of the sheaf $\det (\Omega ^1_{\mathcal {V}|V}(2))=\mathcal {O}_V(n+d-1)$ given by $\Omega :=\Lambda ^n(s_1\wedge \ldots \wedge s_n)\in H^0(V,\det (\Omega ^1_{\mathcal {V}|V}(2)))$.

We point out another interesting way to compute this generalized adjoint form using Proposition (3.1).

Consider the sequence (3.3), that is the vertical sequence in (3.5). Since $H^1(V,\mathcal {O}_V(2-d))=0$, we can find liftings $\tilde{s_1},\ldots ,\tilde{s_n}\in H^0(V, \Omega ^1_{\mathbb {P}^n|V}(2))$ of the sections $\eta _1,\ldots ,\eta _n$. Furthermore they are unique if $d>2$. We can thus consider the adjoint form associated to (3.3) given by $\widetilde{\Omega }:=\Lambda ^n (\tilde{s_1}\wedge \ldots \wedge \tilde{s_n})$. This adjoint is independent from the deformation $\xi$; it depends only on V and its embedding in $\mathbb {P}^n$. If $d>2$, then $\widetilde{\Omega }$ is unique.

To describe $\widetilde{\Omega }$ explicitly we first consider the exact sequence

$$\begin{aligned} 0\rightarrow \Omega ^1_{\mathbb {P}^n}(2-d)\rightarrow \Omega ^1_{\mathbb {P}^n}(2)\rightarrow \Omega ^1_{\mathbb {P}^n|V}(2)\rightarrow 0. \end{aligned}$$

(3.10)

If $d>2$, the vanishing of $H^0(\mathbb {P}^n,\Omega ^1_{\mathbb {P}^n}(2-d))$ and $H^1(\mathbb {P}^n,\Omega ^1_{\mathbb {P}^n}(2-d))$ (c.f. Bott Formulas) gives the isomorphism $H^0(\mathbb {P}^n,\Omega ^1_{\mathbb {P}^n}(2))=H^0(V,\Omega ^1_{\mathbb {P}^n|V}(2))$. Hence, the forms $\tilde{s_i}$ are the restriction on V of global rational 1-forms. By [9][Theorem 2.9] we can write

$$\begin{aligned} \tilde{s_i}=\frac{1}{Q}\sum _{j=0}^n L^i_j d\xi _j \end{aligned}$$

(3.11)

where $\deg Q=2$ and $L^i_j$ is a homogeneous polynomial of degree 1 which does not contain $\xi _j$ in its expression. Hence

$$\begin{aligned} \widetilde{\Omega }= \Lambda ^n(\tilde{s_1}\wedge \ldots \wedge \tilde{s_n})=\frac{1}{Q^n}\sum _{i=0}^n M_i d\xi _0\wedge \ldots \wedge \widehat{d\xi _i}\wedge \ldots \wedge d\xi _n \end{aligned}$$

(3.12)

where $M_i$ is the determinant of the matrix obtained by

$$\begin{aligned} \begin{pmatrix} L^1_0&{}\ldots &{}L^n_0\\ \vdots &{}&{}\vdots \\ L^1_n&{}\ldots &{}L^n_n \end{pmatrix} \end{aligned}$$

(3.13)

removing the i-th row. Since $\widetilde{\Omega }$ is a rational n-form on $\mathbb {P}^n$, following [9][Corollary 2.11] it can be written as $\frac{P\Psi }{Q^n}$, and we deduce that

$$\begin{aligned} \frac{M_i}{(-1)^i\xi _i}=P \end{aligned}$$

(3.14)

for all $i=0,\ldots ,n$. P is a polynomial of degree $n-1$ and it corresponds to $\widetilde{\Omega }$ via the isomorphism $H^0(V,\Omega ^n_{\mathbb {P}^n|V}(2n))\cong H^0(V,\mathcal {O}_V(n-1))$. Hence by (3.1) we have that the form $\Omega \in H^0(V,\mathcal {O}_V(n+d-1))$ given by PR is a canonical choice of adjoint form for $W=\langle \eta _1,\ldots , \eta _n\rangle$ and $\xi$.

Remark 3.2

Alternatively this can be seen using the Euler sequence on V:

$$\begin{aligned} 0\rightarrow \mathcal {O}_V\rightarrow \bigoplus ^{n+1}\mathcal {O}_V(1)\rightarrow \Theta _{\mathbb {P}^n|V}\rightarrow 0. \end{aligned}$$

(3.15)

This sequence, dualized and conveniently tensorized gives

$$\begin{aligned} 0\rightarrow \Omega _{\mathbb {P}^n|V}^1(2)\rightarrow \bigoplus _{i=1}^{n+1}\mathcal {O}_V(1)\rightarrow \mathcal {O}_{V}(2)\rightarrow 0. \end{aligned}$$

(3.16)

The sections $\tilde{s}_i$ are associated via the first morphism to an $n+1$-uple of linear polynomials $(L_i^0,\ldots ,L_i^n)$. Then, taking the wedge product of (3.16) we obtain an exact sequence

$$\begin{aligned} 0\rightarrow \Omega ^n_{\mathbb {P}^n|V}(2n)\cong \mathcal {O}_V(n-1)\rightarrow \bigwedge ^n\mathcal {O}_V(1)=\bigoplus ^{n+1}\mathcal {O}_V(n)\rightarrow \Omega ^{n-1}_{\mathbb {P}^n|V}(2n)\rightarrow 0 \end{aligned}$$

(3.17)

where the morphism $\mathcal {O}_V(n-1)\rightarrow \bigoplus ^{n+1}\mathcal {O}_V(n)$ is given by

$$\begin{aligned} G\mapsto (G\xi _0,\ldots ,(-1)^nG\xi _n). \end{aligned}$$

(3.18)

Since $\widetilde{\Omega }=\Lambda ^n (\tilde{s_1}\wedge \ldots \wedge \tilde{s_n})\in H^0(V,\Omega ^n_{\mathbb {P}^n|V}(2n))$ is sent exactly to $(L_0^0,\ldots ,L_0^n)\wedge \ldots \wedge (L_n^0,\ldots ,L_n^n)=(M_0,\ldots ,M_n)$ (using the same notation as above), then we conclude that $\widetilde{\Omega }$ corresponds in $H^0(V,\mathcal {O}_V(n-1))$ to a polynomial P which satisfies

$$\begin{aligned} \frac{M_i}{(-1)^i\xi _i}=P. \end{aligned}$$

(3.19)

3.3 The adjoint sublinear systems obtained by meromorphic 1-forms

To study the conditions given in (2.6) and (2.7), we need to describe the sections

$$\begin{aligned} \widetilde{\Omega _i}:=\Lambda ^{n-1}(\tilde{s_1}\wedge \ldots \wedge \hat{\tilde{s_i}}\wedge \ldots \wedge \tilde{s_n})\in H^0(V,\Omega ^{n-1}_{\mathbb {P}^n|V}(2n-2)) \end{aligned}$$

(c.f. (2.4)) and their images in $H^0(V,\Omega ^{n-1}_{V}(2(n-1)))=H^0(V,\mathcal {O}_V(n+d-3))$ that we have denoted by $\omega _i$.

A computation similar to the above shows that

$$\begin{aligned} \widetilde{\Omega _i}= \Lambda ^{n-1}(\tilde{s_1}\wedge \ldots \wedge \hat{\tilde{s_i}}\wedge \ldots \wedge \tilde{s_n})=\frac{1}{Q^{n-1}}\sum _{j<k}M^i_{jk}d\xi _0\wedge \ldots \wedge \hat{d\xi _j}\wedge \ldots \wedge \hat{d\xi _k}\wedge \ldots \wedge d\xi _n \end{aligned}$$

(3.20)

where $M^i_{jk}$ is the determinant of the matrix obtained by (3.13) removing the i-th column and the j-th and k-th rows. On the other hand, rearranging the expression of [9][Theorem 2.9] we can write

$$\begin{aligned} \widetilde{\Omega _i}=\frac{1}{Q^{n-1}}\sum _{j} A^i_j(\sum _{k\ne j} (-1)^{k+j}\text {sgn}(k-j)\xi _k d\xi _0\wedge \ldots \wedge \hat{d\xi _j}\wedge \ldots \wedge \hat{d\xi _k}\wedge \ldots \wedge d\xi _n) \end{aligned}$$

(3.21)

with $\deg A^i_j=n-2$.

Comparing (3.20) and (3.21) gives

$$\begin{aligned} M^i_{jk}=(-1)^{j+k}(A^i_j\xi _k-\xi _j A_k^i). \end{aligned}$$

(3.22)

As before this can be computed also via the Euler sequence.

We call $\Xi _j:=\sum _{k\ne j} (-1)^{k+j}\text {sgn}(k-j)\xi _k d\xi _0\wedge \ldots \wedge \hat{d\xi _j}\wedge \ldots \wedge \hat{d\xi _k}\wedge \ldots \wedge d\xi _n$. Note that the sections $\Xi _j$, for $j=0,\ldots ,n$ give a basis of $H^0(V,\Omega ^{n-1}_{\mathbb {P}^n|V}(n))$.

Proposition 3.3

$\omega _i=\sum _{j} A^i_j\cdot F_j$ in $H^0(V,\mathcal {O}_V(n+d-3))$

Proof

It is enough to show that the image of $\Xi _j$ through the morphism $\Omega ^{n-1}_{\mathbb {P}^n|V}(n)\rightarrow \mathcal {O}_V(d-1)$ is $F_j$. Consider the exact sequence of the tangent sheaf of V:

$$\begin{aligned} 0\rightarrow \Theta _V\rightarrow \Theta _{\mathbb {P}^n|V}\rightarrow \mathcal {O}_V(d)\rightarrow 0. \end{aligned}$$

(3.23)

The beginning of the Koszul complex is

$$\begin{aligned} \bigwedge ^n\Theta _{\mathbb {P}^n|V}\otimes \mathcal {O}_V(-d)\rightarrow \bigwedge ^{n-1}\Theta _{\mathbb {P}^n|V} \end{aligned}$$

(3.24)

which, tensored by $\mathcal {O}_V(-n)$, gives

$$\begin{aligned} \bigwedge ^n\Theta _{\mathbb {P}^n|V}\otimes \mathcal {O}_V(-n-d)\rightarrow \bigwedge ^{n-1}\Theta _{\mathbb {P}^n|V}\otimes \mathcal {O}_V(-n). \end{aligned}$$

(3.25)

This is exactly the dual of $\Omega ^{n-1}_{\mathbb {P}^n|V}(n)\rightarrow \mathcal {O}_V(d-1)$. Hence we only need to show that the morphism (3.25) composed with the contraction by $\Xi _i$

$$\begin{aligned} \bigwedge ^{n-1}\Theta _{\mathbb {P}^n|V}\otimes \mathcal {O}_V(-n)\mathop {\rightarrow }\limits ^{\Xi _i}\mathcal {O}_V \end{aligned}$$

(3.26)

is the multiplication by $F_i$. This is easy to see by a standard local computation. $\square$

Remark 3.4

We immediately have that the polynomials associated to the sections $\omega _i$ are in the Jacobian ideal of V.

Condition (2.7), that is

$$\begin{aligned} \Omega \in \mathrm {Im}\,(H^0(V,\mathcal {O}_V(2))\otimes \lambda ^nW \rightarrow H^0(V,\mathcal {O}_V(n+d-1))), \end{aligned}$$

(3.27)

can be written, modulo F, as

$$\begin{aligned} RP=\sum \omega _i\cdot S_i=\sum _{i,j} A^i_j\cdot F_j\cdot S_i, \end{aligned}$$

(3.28)

where $\deg S_i=2$. In particular this implies that RP is in the Jacobian ideal of V.

Proposition 3.5

The base locus $D_W$ of the linear system $|\lambda ^nW|$ is zero for the generic W.

Proof

By [11][Proposition 3.1.6] it is enough to prove that $H^0(V,\Omega ^1_V(2))$ generically generates the sheaf $\Omega ^1_V(2)$ and that $D_{H^0(V,\Omega ^1_V(2))}=0$. We have an explicit basis for $H^0(V,\Omega ^1_V(2))$ given by

$$\begin{aligned} \frac{\xi _id\xi _j-\xi _jd\xi _i}{Q} \end{aligned}$$

(3.29)

where $i<j$ and $\deg Q=2$. The vector space $\lambda ^nH^0(V,\Omega ^1_V(2))\subset H^0(V,\mathcal {O}_V(n+d-3))$ is obviously nonzero, hence $H^0(V,\Omega ^1_V(2))$ generically generates the sheaf $\Omega ^1_V(2)$.

It remains to prove that $D_{H^0(V,\Omega ^1_V(2))}=0$. An easy computation (for example by induction) shows that $\lambda ^nH^0(V,\Omega ^1_V(2))$ contains all the polynomials of the form

$$\begin{aligned} \xi _{i_{1}}\xi _{i_{2}}\ldots \xi _{i_{n-2}} \frac{\partial F}{\partial {\xi _j}} \end{aligned}$$

(3.30)

where $\{i_1,\ldots ,i_{n-2}\}\subset \{1,\ldots ,n+1\}$ and $j\notin \{i_1,\ldots ,i_{n-2}\}$. Since V is smooth, these polynomials do not vanish simultaneously on a divisor, hence $D_{H^0(V,\Omega ^1_V(2))}=0$, and we are done. $\square$

3.4 On Griffiths’s proof of infinitesimal Torelli Theorem

In this section we will prove Theorem [C] of the Introduction.

It is well known by [9] that the deformation $\xi$ is trivial if and only if R lies in the Jacobian ideal $\mathcal {J}$ of the variety V. The following lemma gives a translation of this condition in the setting of adjoint forms.

Lemma 3.6

R is in the Jacobian ideal $\mathcal {J}$ if and only if $\Omega \in \mathrm {Im}\,(H^0(V,\mathcal {O}_V(2))\otimes \lambda ^nW \rightarrow H^0(V,\mathcal {O}_V(n+d-1)))$ for the generic $\Omega$.

Proof

If $\Omega \in \mathrm {Im}\,(H^0(V,\mathcal {O}_V(2))\otimes \lambda ^nW \rightarrow H^0(V,\mathcal {O}_V(n+d-1)))$, then by the Adjoint Theorem, $\xi _{D_W}=0$. Since $D_W=0$, the deformation is trivial, hence R lies in the Jacobian ideal.

Viceversa if $R\in \mathcal {J}$, the deformation is trivial and by theorem (2.9), we have that $\Omega \in \mathrm {Im}\,(H^0(V,\mathcal {O}_V(2))\otimes \lambda ^nW \rightarrow H^0(V,\mathcal {O}_V(n+d-1)))$. $\square$

Our theory gives another characterization for $[R]\in ( \mathbb {C}[\xi _0,\ldots ,\xi _n]/\mathcal {J})_{d}\simeq H^1(V,\Theta _V)$ to be trivial.

Proposition 3.7

Assume that $\deg R=d>3$. Then R is in the Jacobian ideal $\mathcal {J}$ if and only if $RP\in \mathcal {J}$ for every polynomial $P\in H^0(V,\mathcal {O}_V(n-1))$ corresponding to a generalized adjoint $\widetilde{\Omega }\in H^0(V,\Omega ^n_{\mathbb {P}^n|V}(2n))$.

Proof

One implication is trivial.

To prove the other one the idea is to show that every monomial of $H^0(V,\mathcal {O}_V(n-1))$ corresponds to a suitable generalized adjoint. Hence, if $RP\in \mathcal {J}$ for every polynomial $P\in H^0(V,\mathcal {O}_V(n-1))$ corresponding to a generalized adjoint, we have that $R\cdot H^0(V,\mathcal {O}_V(n-1))\subset \mathcal {J}$ and we are done by Macaulay Theorem (c.f. [16] Theorem 6.19 and Corollary 6.20).

We work by induction at the level of $\mathbb {P}^n$, since $H^0(\mathbb {P}^n,\mathcal {O}_{\mathbb {P}^n}(n-1))\rightarrow H^0(V,\mathcal {O}_V(n-1))$ is surjective. The base of the induction is for $n=2$. A simple computation shows that the map

$$\begin{aligned} \bigwedge ^2H^0(\mathbb {P}^2,\Omega ^1_{\mathbb {P}^2}(2))\rightarrow H^0(\mathbb {P}^2,\mathcal {O}_{\mathbb {P}^2}(1)) \end{aligned}$$

(3.31)

is surjective because its image contains the canonical basis of degree one monomials.

For the general case we show that every monomial of degree $n-1$ is given by a generalized adjoint. Consider the natural homomorphism:

$$\begin{aligned} \bigwedge ^nH^0(\mathbb {P}^n,\Omega ^1_{\mathbb {P}^n}(2))\rightarrow H^0(\mathbb {P}^n,\mathcal {O}_{\mathbb {P}^n}(n-1)) \end{aligned}$$

(3.32)

and take a monomial M with $\deg M=n-1$. There is a variable $\xi _i$ which does not appear in M. We restrict to the hyperplane $\xi _i=0$ and we use induction on $\frac{M}{\xi _j}$, where $\xi _j$ appears in M. There exist $s_1,\ldots ,s_{n-1}\in H^0(\mathbb {P}^{n-1},\Omega ^1_{\mathbb {P}^{n-1}}(2))$ with $s_1\wedge \ldots \wedge s_{n-1}$ which corresponds to $\frac{M}{\xi _j}$, that is

$$\begin{aligned} s_1\wedge \ldots \wedge s_{n-1}=\frac{M\Psi '}{\xi _j\cdot Q^{n-1}} \end{aligned}$$

(3.33)

where $\Psi '=\sum _{k=0,k\ne i}^n(-1)^k\xi _k(d\xi _0\wedge \ldots \wedge \hat{d\xi _i}\ldots \wedge \hat{d\xi _k}\ldots \wedge d\xi _n)$ gives a basis of $H^0(\mathbb {P}^{n-1},\Omega _{\mathbb {P}^{n-1}}^{n-1}(n))$. It is easy to see that

$$\begin{aligned} s_1\wedge \ldots \wedge s_{n-1}\wedge \frac{(\xi _jd\xi _i-\xi _id\xi _j)}{Q}=\frac{M\Psi }{Q^{n}}, \end{aligned}$$

(3.34)

i.e. M corresponds to a generalized adjoint, which is exactly our thesis. $\square$

From the previous results we deduce immediately Theorem [C] of the Introduction.