Proof. Let $\mathcal {B}=\{\eta _{1},\,\ldots,\, \eta _{n+1}\}$ be a basis of $W$. Set $\omega _i$ for $i=1,\,\dots,\,n+1$ as above and denote by $\tilde {\omega }_i\in H^{0}(\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W})$ the corresponding sections via

\[ 0\to H^{0}(X, \det\mathcal{F}({-}D_W)\otimes \mathcal{I}_{Z_W})\to H^{0}(X,\det\mathcal{F}). \]

Recall that $\lambda ^{n}W:=\left \langle \omega _1,\,\dots,\,\omega _{n+1}\right \rangle \subset H^{0}(X,\,\det \mathcal {F})$ is the vector space generated by the sections $\omega _i$. Note also that the sheaf $\bigwedge ^{n}W\otimes \mathcal {O}_{X}$ is trivial and choosing $\eta _1\wedge \ldots \wedge \widehat {\eta _i}\wedge \ldots \wedge \eta _{n+1}$, $i=1,\,\dots,\,n+1$, as a basis for $\bigwedge ^{n}W$ we obtain an isomorphism to $\mathcal {O}_X^{n+1}$.

The standard evaluation map $\bigwedge ^{n}W\otimes \mathcal {O}_{X}\to \det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_{W}}$ given by $\tilde {\omega }_1,\,\dots, \tilde {\omega }_{n+1}$ gives the following exact sequence

(2.4)

which is associated with a class $\xi '\in \text {Ext}^{1}(\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W},\,\mathcal {K})$. The sequence (2.4) fits into the following commutative diagram

(2.5)

where $f$ is the map given by the contraction by the sections $(-1)^{n+1-i}s_i$, for $i=1,\,\dots,\,n+1$, and $g$ is given by the global section $\sigma \in H^{0}(X,\,\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W})$ corresponding to the adjoint form $\omega$. We have the standard factorization

(2.6)

where the sequence in the middle is associated with the class $\xi ''\in H^{1}(X,\,\mathcal {K})$ which is the image of $\xi \in H^{1}(X,\,\mathcal {F}^{\vee })$ through the map $H^{1}(X,\,\mathcal {F}^{\vee })\to H^{1}(X,\,\mathcal {K})$. In particular, we obtain the commutative square:

(2.7)

By commutativity, we immediately have that the image of $\sigma \in H^{0}(X,\,\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W})$ through the coboundary map $H^{0}(X,\,\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W})\to H^{1}(X,\,\mathcal {K})$ is $\xi ''$. Tensoring by $\det \mathcal {F}$, the map $\mathcal {F}^{\vee }\to \mathcal {K}$ gives

(2.8)

and, since $\xi \cdot \omega \in H^{1}(X,\,\mathcal {F}^{\vee }\otimes \det \mathcal {F})$ is sent to $\xi ''\cdot \omega \in H^{1}(X,\,\mathcal {K}\otimes \det \mathcal {F})$, we have that

(2.9)\begin{equation} H^{1}(\Gamma)(\xi\cdot \omega)=\xi''\cdot\omega,\end{equation}

where $\xi \cdot \omega$ is the cup product.

By hypothesis $\partial _\xi ^{n}(\omega )=\xi \cdot \omega =0\in H^{1}(X,\,\bigwedge ^{n-1}\mathcal {F})$, so also $\xi ''\cdot \omega =0\in H^{1}(X,\,\mathcal {K}\otimes \det \mathcal {F})$, hence, the global section $\sigma \cdot \omega \in H^{0}(X,\,\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}\otimes \det \mathcal {F})$ is in the kernel of the coboundary map $H^{0}(X,\,\det \mathcal {F}(-D_W)\otimes \mathcal {I}_{Z_W}\otimes \det \mathcal {F})\to H^{1}(X,\, \mathcal {K}\otimes \det \mathcal {F})$ associated to the sequence

(2.10)

This occurs if there exist $L^{\sigma }_{i}\in H^{0}(X,\,\det \mathcal {F})$, $i=1,\,\ldots,\, n+1$ such that

(2.11)\begin{equation} \sigma\cdot\omega=\displaystyle\sum_{i=1}^{n+1}\tilde{\omega}_{i}\cdot L^{\sigma}_{i}. \end{equation}

This relation gives the following relation in $H^{0}(X,\,\det \mathcal {F}^{\otimes 2})$:

(2.12)\begin{equation} \omega\cdot\omega=\displaystyle\sum_{i=1}^{n+1}L^{\sigma}_{i}\cdot\omega_{i}. \end{equation}

Then, equation (2.12) gives an adjoint quadric. By contradiction, the claim follows.

Proof. The first claim follows by Theorem 2.7 and by Theorem 2.5. To show the second claim, we recall that by [Reference Pirola and Zucconi23, Proposition 3.1.6] $D_{\mathcal {F}}=D_W$ since $W$ is a generic $n+1$-dimensional subspace of $H^{0}(X,\,\mathcal {F})$. Then, the claim follows.

Proof. Let $\mu _b\in {{\rm Ext}}^{1}(\varOmega ^{1}_{A_b},\,\mathcal {O}_{A_b})$ be the class given by the family $p\colon \mathcal {A}\to B$, that is the class of the following extension:

\[ 0\to \mathcal{O}_{A_{b}}\to\varOmega^{1}_{\mathcal{A}|A_{b}}\to\varOmega^{1}_{A_{b}}\to 0. \]

Now $\phi _b^{*}\mathcal {O}_{A_{b}}=\mathcal {O}_{X_b}$ and the map $\phi _b^{*}\mathcal {O}_{A_{b}}\to \phi _b^{*}\varOmega ^{1}_{\mathcal {A}|A_{b}}$ is generically injective; hence, it is injective because otherwise the kernel would be a torsion subsheaf of $\mathcal {O}_{X_b}$. Thus, we have the following exact sequence

\[ 0\to \phi_b^{*}\mathcal{O}_{A_{b}}\to\phi_b^{*}\varOmega^{1}_{\mathcal{A}|A_{b}}\to\phi_b^{*}\varOmega^{1}_{A_{b}}\to 0 \]

which fits into the following diagram

In cohomology, we have

so, by commutativity and by the hypothesis $H^{0}({\mathcal {X}},\,\varOmega _{{\mathcal {X}}}^{1}) \twoheadrightarrow H^{0}(X_{b},\,\varOmega _{X_{b}}^{1})$, we immediately obtain $H^{0}(X_b,\,\phi _b^{*}\varOmega ^{1}_{\mathcal {A}|A_{b}}) \twoheadrightarrow H^{0}(X_{b},\,\phi _b^{*}\varOmega ^{1}_{A_{b}})$ and hence the coboundary $\partial _{\mu _{b}}\colon H^{0}(A_b ,\,\varOmega ^{1}_{A_{b}})\to H^{1}(A_b,\, \mathcal {O}_{A_{b}})$ is trivial. Then we can apply the argument in cf. [Reference Collino and Pirola7, Page 77, 78] on a conveniently small $B$.

Proof. By Proposition, 3.4 we can assume that $p\colon \mathcal {A}\to B$ is trivial, that is $\mathcal {A}\simeq A\times B$ and $p\colon \mathcal {A}\to B$ is the first projection. Up to base change, the Albanese family ${{\rm alb}}(\mathcal {X})\colon \mathcal {X}\to {{\rm Alb}}(\mathcal {X})$ exists and by Proposition 3.2, our claim is equivalent to show that the Albanese family ${{\rm alb}} (\mathcal {X}) \colon \mathcal {X}\to {{\rm Alb}} (\mathcal {X})$ is locally translation equivalent to the trivial family. Hence, it is not restrictive to assume that ${{\rm Alb}}(\mathcal {X})=A\times B$ too. In particular, we can restrict to consider only the case where ${{\rm Alb}}(X_b)=A$ and the map $\phi _b\colon X_b\to A$ is of degree one for every $b\in B$.

Denote by $\xi _b\in H^{1}(X_b,\,\varTheta _{X_b})$ a class associated with an infinitesimal deformation of $X_b$ induced by the fibration $\pi \colon \mathcal {X}\rightarrow B$. We know that $q\geq n+1$ where $q={{\rm dim}}_\mathbb {C} A$. Let $\mathcal {B}:=\{dz_1,\,\ldots,\,dz_{n+1}\}$ be a basis of an $n+1$-dimensional generic subspace $W$ of $H^{0}( A,\,\varOmega ^{1}_{A})$, (if $q=n+1$ we can take $H^{0}( A,\,\varOmega ^{1}_{A})=W$). For every $b\in B$ let $\eta _i(b):= {{\rm alb}}(X_b)^{*}dz_i$, $i=1,\,\ldots,\, n+1$. By standard theory of the Albanese morphism, it holds that $\mathcal {B}_b:=\{ \eta _1(b),\,\ldots,\, \eta _{n+1}(b)\}$ is a basis of the pull-back $W_{b}$ of $W$ inside $H^{0}(X_b,\,\varOmega _{X_{b}}^{1})$. Let

\[ \omega_i (b):= \lambda^{n}(\eta_1(b)\wedge\ldots\wedge\eta_{i-1}(b)\wedge{\widehat{\eta_i(b)}}\wedge\ldots\wedge\eta_{n+1}(b)) \]

for $i=1,\,\ldots,\, n+1$. Note that if $\omega '_i:=dz_1\wedge \ldots \wedge dz_{i-1} \wedge {\widehat {dz_i}} \wedge \ldots \wedge dz_{n+1}$ then $\omega _i (b):=\phi _{b}^{}\omega _i'$, $i=1,\,\ldots,\,n$. Since $\varPhi \colon \mathcal {X}\to \mathcal {A}$ is a family of Albanese type, ${{\rm dim}}\lambda ^{n}W_{b}\geq 1$, (actually if $q>n+1$ by [Reference Pirola and Zucconi23, Theorem 1.3.3] it follows that $\lambda ^{n}W_{b}$ has dimension $n+1$), and we can write: $\lambda ^{n}W_{b}=\langle \omega _1(b),\,\ldots,\,\omega _{n+1}(b)\rangle$.

By extremal liftability assumptions, we can form the Massey class for every $[W]\in \mathbb {G}(n+1,\,q)$ where we denote by $\mathbb {G}(n+1,\,q)$ the Grassmannian of $n+1$-dimensional subspaces of $H^{0}(X_b,\,\varOmega ^{1}_{X_b})$.

Consider the following diagram in Dolbeaut's cohomology:

(3.2)

By the identification $H^{0}(X_b,\,\phi _{b}^{*}(\varOmega ^{n}_{A})\otimes \omega _{X_b})=H^{n,0}(A)\otimes H^{n,0}(X_b)$, it follows that

\[ j:=\phi_b^{*}\otimes {{\rm id}}\colon H^{n,0}(A)\otimes H^{n,0}(X_b)\to H^{n,0}(X_b)\otimes H^{n,0}(X_b). \]

Now, we can consider the induced diagram of the symmetric part:

(3.3)

We set $V_b:= {{\rm Im}}\phi _b^{*}\subset H^{n,0}(X_b)$. Recall the decomposition (1.4) in § 1

\[ {{\rm Sym}}^{2}H^{n,0}(X_b)=V_b\odot H^{n,0}(X_b)\oplus {{\rm Sym}}^{2}(\overline{\text{Ann}(V_b)}) \]

and that this direct sum induces homomorphisms

(3.4)\begin{equation} \nu_{X_b}\colon {{\rm Sym}}^{2}{\overline{{\rm Ann}}(V_b)}\to H^{0}(X,\omega_{X_b}^{{\otimes} 2}) \end{equation}

and

(3.5)\begin{equation} \gamma_{X_b}\colon V_b\odot H^{n,0}(X_b)\to H^{0}(X,\omega_{X_b}^{{\otimes} 2}). \end{equation}

Now assume that for the generic $W_b$, the generic adjoint form $\omega$ has an adjoint quadric

(3.6)\begin{equation} Q:=\omega\odot\omega -\displaystyle\sum_{i=1}^{n+1} \omega_i(b)\odot L_i\in {{\rm Sym}}^{2}H^{n,0}(X_b). \end{equation}

By Definition 2.6, $Q$ is in ${{\rm Ker}}(\mu )$.

By the Transversality Theorem 3.6, $\omega$ vanishes on $\overline {V_b}$, hence $\omega \odot \omega$ is an element of ${{\rm Sym}}^{2}(\overline {\text {Ann}(V_b)})$. On the other hand, recall that by definition the forms $\omega _i(b)$ are in $V$, hence $\sum \nolimits _{i=1}^{n+1} \omega _i(b)\odot L_i$ is an element of $V_b\odot H^{n,0}(X_b)$. By the hypothesis that for each element $\eta$ contained in $\overline {{{\rm Ann}}(V_b)}$, its square is not contained in the image of $\gamma _{X_b}$, we get that $\omega \odot \omega =0$. By the Volumetric Theorem 3.7, the claim follows easily.

Proof. By [Reference Pirola and Zucconi23, Proposition 6.2.2], we know that $\varPhi :\mathcal {X}\rightarrow \mathcal {A}$ is equivalent to a Nori trivial family. By Theorem [A], the claim follows.

Proof. We assume that $\mathcal {D}$ is the zero locus of a holomorphic section $s$ of $\mathcal {L}$. The complex space $V$ can be naturally identified with the space of holomorphic vector fields $\varTheta _{{{A}}}$ on ${{A}}$. Fixed $\omega$ a non-zero $(a,\,0)$-form on ${{A}}$, recall that the map

(4.5)\begin{equation} V \cong H^{0}({{A}}, \varTheta_{{{A}}}) \longrightarrow H^{0}({{A}}, \varOmega_{{{A}}}^{a-1})\end{equation}

sends under this identification a vector $v$ of $V$ to the holomorphic $(a-1)$-form $w$ obtained by contracting the $(a,\,0)$-form $\omega$ with the vector field $\frac {\partial }{\partial v}$.

The holomorphic function $\frac {\partial s}{\partial v}$ can be seen by adjunction as a holomorphic section of the canonical bundle of $\mathcal {D}$, which coincides with the restriction to $\mathcal {D}$ of the $(a-1)$-form $w$ defined above. On the other side, the connecting homomorphism can be computed by using the fact that there is a canonical isomorphism of cohomology groups sequences

\[ H^{p}(\varLambda, H^{0}({V}, \pi^{*}({\cdot})) \cong H^{p}(A, \cdot) \]

where $\pi$ denotes the projection of ${V}$ onto ${{A}}$, and it holds (see also: [Reference Cesarano6] Proposition 1.1, p. 4):

(4.6)\begin{equation} f\left(\displaystyle\frac{\partial s}{\partial v}\right) = [{\pi H(v,\lambda)}_{\lambda \in \varLambda}] \end{equation}

It remains to compute $d_0\phi _{\mathcal {L}}(\tfrac {\partial }{\partial v} )$. Let us consider $S := Spec(\mathbb {C}[\epsilon]/{\epsilon^{2}})$ the scheme of dual numbers over $\mathbb {C}$ and ${{A}}_S$ the base change. We have the exact sequence of sheaves

\[ 0 \longrightarrow \mathcal{O}_{{{A}}} \longrightarrow \mathcal{O}^{*}_{{{A}}_S} \longrightarrow \mathcal{O}^{*}_{{{A}}} \longrightarrow 0 \]

Its long cohomology sequence identifies $H^{1}({{A}},\, \mathcal {O}_{{{A}}})$ with the kernel of the map ${{\rm Pic}}({{A}}_S) \longrightarrow {{\rm Pic}}({{A}})$, which to a line bundle on ${{A}}_S$ whose transition functions $g_{\alpha \beta }=g'_{\alpha \beta } + \epsilon g''_{\alpha \beta }$ associates the line bundle on ${{A}}$ with transition functions $g'_{\alpha \beta }$. Moreover, under the identification $H^{1}({{A}},\, \mathcal {O}_{{{A}}}^{*}) \cong H^{1}(\varLambda,\, H^{0}({V},\, \mathcal {O}_{{V}}^{*}))$

(4.7)\begin{equation} {{\rm Pic}}({{A}}_S) = H^{1}(\varLambda, H^{0}({V},\mathcal{O}_{{V}}^{*})\otimes_{\mathbb{C}} \mathbb{C}[\epsilon]) \end{equation}

since ${{A}}_S$ is defined through a flat base change. Now, for every $z$ on $A$, $\phi _{\mathcal {L}}(z)$ is the line bundle of degree $0$ with cocycles $[\{e^{2 \pi i E(z, \lambda )}\}_{\lambda }]$. Hence, $d_0\phi _{\mathcal {L}}(\tfrac {\partial }{\partial v} )$ is the line bundle on ${{A}}_S$ whose cocycles, according to identification 4.7, are precisely

\[ [\{e^{2 \pi i \epsilon E(v, \lambda)}\}_{\lambda}] = [\{1 + 2\pi i E(v,\lambda)\epsilon\}_{\lambda}] \in H^{1}(\varLambda, H^{0}({V},\mathcal{O}_{{V}}^{*})\otimes_{\mathbb{C}} \mathbb{C}[\epsilon]) \]

In conclusion, we have

(4.8)\begin{equation} d_0\phi_{\mathcal{L}}\left(\displaystyle\frac{\partial}{\partial v}\right) = [\{2\pi i E(v,\lambda)\}_{\lambda}] \end{equation}

It is now easy to see that the two elements in the cohomology group $H^{1}(\varLambda,\, H^{0}({V},\,\mathcal {O}_{{V}}^{*}))$ are the same. Indeed, it is enough to show, by the definitions of group cohomology, that there exists a holomorphic function $F$ on $V$ such that, for every $z$ on $V$ and every $\lambda$ on $\varLambda$, it holds that

(4.9)\begin{equation} \pi H(v,\lambda) = 2\pi i E(v,\lambda) + F(z+\lambda)-F(z) \end{equation}

But $E$ is defined as the imaginary part of $H$, which is an alternating $\mathbb {R}$-bilinear form on ${V}$, and $H$ can be recovered by $E$. Indeed, for every $z$ and $w$ on ${V}$ it holds:

\[ H(z,w) = iE(z,w)+E(iz,w) \]

In conclusion, with $F(z) := -\pi (i E(v,\,z) - E(iv,\, z))$, it is easily seen that $F$ is $\mathbb {C}$-linear on ${V}$ and that 4.9 holds true.

Proof. Let us begin by the case in which the polarization type is $(2,\,\cdots 2)$. For such a polarization, the matrix $M_{\rho }$ in 4.18 with $\rho \in Hom(\mathbb {Z}_2^{g},\, \mathbb {C}^{*})$ is just the scalar:

\[ C_{\rho} = \displaystyle\sum_{z \in A[2]}\rho(z){\theta_{z}^{\mathcal{L}^{2}}}(0) \]

Now inside the moduli space of $(2,\,2,\, \cdots 2)$-polarized abelian variety it is easy to conclude that $C_{\rho } \neq 0$ holds for the general abelian variety since it holds for the product of $2$-polarized elliptic curves. If we now take a general $(1,\,1,\,\cdots 1,\, 2,\, \cdots 2)$-polarization we can consider $(A',\, \mathcal {L}')$ a $(2,\,\cdots,\,2)$-polarized abelian variety with an isogeny $h \colon A' \longrightarrow A$ such that $h^{*}\mathcal {L} = \mathcal {L}'$. Then the multiplication map $\mu$ on the sections of $\mathcal {L}$ is just the restriction of the multiplication map

(4.19)\begin{equation} \mu_{A'} \colon {{\rm Sym}}^{2}H^{0}(A', \mathcal{L}') \longrightarrow H^{0}(A', \mathcal{L}'^{2})\end{equation}

to the symmetric product of the subvector space of the $Ker(h)$-invariant sections of $\mathcal {L}'$.

Proof. This follows immediately by the above theorem and Diagram (4.11).

Proof. As in the proof of Theorem 4.3, it is enough to work on $(2,\,\dots,\,2)$-polarizations. First, we prove that the locus in the moduli space of polarized abelian varieties $({{A}},\, \mathcal {D})$ such that there is no element $\eta$ in $\overline {{{\rm Ann}}(V)} \cong H^{0}({{A}},\, \mathcal {D})$ whose square is contained in the image of $V \odot H^{0}(\mathcal {D},\, \omega _{\mathcal {D}})$ is open.

Indeed, the moduli space $U$ of smooth divisors in the polarization of an $a$-dimensional $(2,\, \cdots,\,2)$-polarized abelian variety is an open subset of a $\mathbb {P}^{d - 1}$-bundle on the moduli space $\mathcal {A}_a$ of $(2,\, \cdots,\,2)$-polarized abelian varieties of dimension $a$, where $d = 2^{a}$. Note that $U$ is a smooth Kuranishi family, in fact

\[ \dim U = \dim (\mathcal{A}_a) + d - 1 \]

and for every element $[{{A}},\,\mathcal {D}]$ of $U$, it is known that

(4.22)\begin{equation} \begin{aligned} h^{1}(\mathcal{D}, \mathcal{T}_{\mathcal{D}}) & = \dim Ext^{1}_{\mathcal{O}_{\mathcal{D}}}(\varOmega^{1}_{\mathcal{D}}, \mathcal{O}_{\mathcal{D}}) \\ & = \dim (\mathcal{A}_a) + d - 1 \\ & = \dim U. \end{aligned} \end{equation}

Now for every element $[{{A}},\,\mathcal {D}]$ of $U$, the canonical bundle of $\mathcal {D}$ is very ample and we can choose uniformizing coordinates $[X_1,\, \cdots,\, X_a,\, Y_0,\, Y_1,\, \cdots,\,Y_{d-1}]$ on $\mathbb {P}^{N} \cong \mathbb {P}(H^{0}(\mathcal {D},\, \omega _{\mathcal {D}}))$, where $N = d + a$. Let $H$ be the Hilbert scheme of closed subvarieties of $\mathbb {P}^{N}$ with Hilbert polynomial $p(n) = (2n)^{a} - (2n-1)^{a}$. The natural morphism of schemes $\phi \colon U\to H$ is smooth and finite onto a locally closed subscheme $Z\subset H$ by (4.22).

The decomposition of $H^{0}(\mathcal {D},\, \omega _{\mathcal {D}})$ for every divisor $\mathcal {D}$ into the direct sum $V \oplus \overline {{{\rm Ann}}(V)}$ induces a decomposition $\mathbb {P}^{N} = \mathbb {P}(V' \oplus W')$, where $V$ is generated by the forms $X_1 \cdots X_a$, and $W$ by the remaining forms $Y_0 \cdots Y_{d-1}$.

The set $Q$ of points of $Z$ which are contained in a quadric of the form $w^{2} - \sum \nolimits _{i} v_i z_i$ in $H^{0}(\mathbb {P}^{N},\, \mathcal {O}_{\mathbb {P}}(2))$, where $w \in W'$ and the elements $v_i$ are forms of $V'$ not all equal to $0$, is a proper locally closed subset of $Z$. Since $\phi$ is flat, the scheme theoretic counterimage of $Q$ is also a proper locally closed subset of $U$.

We have shown that the locus in the moduli space of polarized abelian varieties $({{A}},\, \mathcal {D})$ such that there is no element $\eta$ in $\overline {{{\rm Ann}}(V)} \cong H^{0}({{A}},\, \mathcal {D})$ whose square is contained in the image of $V \odot H^{0}(\mathcal {D},\, \omega _{\mathcal {D}})$ is open. Therefore, it suffices to prove that the claim of the Theorem holds true in the case of a smooth divisor $\mathcal {D}$ in a product of $a$ $(2)$-polarized elliptic curves.

To this purpose, we denote by $A$ the product of $E_1,\, \cdots,\, E_a$ elliptic curves, each of them considered as the quotient of the complex plane by the lattice ${\tau _i \mathbb {Z} \oplus 2\mathbb {Z}}$, where $\tau _i$ denotes a certain element in the Siegel upper half-space, and equipped with the polarization $\mathcal {L}_i$ of type $(2)$ induced by the divisor $2\cdot 0$. For each $i$, we denote moreover by $\theta ^{(i)}_{0}$ and $\theta ^{(i)}_{1}$ the canonical theta functions which span the vector space of the global holomorphic sections of the polarization $\mathcal {L}_i$ on $E_i$.

Considering $\mathcal {L}$ to be the induced product polarization on $A$, we first fix a basis for the vector space of the global holomorphic sections of $\mathcal {L}$ and a convenient notation for its elements. Since $\mathcal {L}$ is the product of all the polarizations on the factors $E_i$, a basis for the global sections of $\mathcal {L}$ can be easily defined by considering all the possible products of sections on each factor $E_i$, arising by selecting $\theta ^{(i)}_{0}$ or $\theta ^{(i)}_{1}$ for each index $i$. Every global section of this basis corresponds to a unique subset of $\{1,\, \cdots,\, a\}$ containing the indices $i$ of the factors on which $\theta ^{(i)}_{1}$ has been selected. Hence, if we consider $\mathcal {P}$ to be the power set of $\{1,\, \cdots,\, a\}$, we can denote for every $S \in \mathcal {P}$ the following global section of $\mathcal {L}$:

\[ \theta_S(z) := \prod_{i=1}^{a} \theta^{(i)}_{\chi_S(i)}(z_i) \]

where $\chi _S(i)$ is the characteristic function of $S$. It can be now easily seen that $\{\theta _S\}_{S \in \mathcal {P}}$ is a basis for $H^{0}(A,\,\mathcal {L})$. For the reader's convenience, we remark that $\theta _{\emptyset }$ is the element of the latter basis obtained by multiplying on every factor $E_i$ the section $\theta ^{(i)}_{0}$:

\[ \theta_{\emptyset}(z) = \prod_{i=1}^{a} \theta^{(i)}_{0}(z_i) \]

Let us consider now a general non-zero section

(4.23)\begin{equation} s := \displaystyle\sum_{S \in \mathcal{P}}a_S \theta_S \in H^{0}(A,\mathcal{L}) \end{equation}

for some complex coefficients $a_S$ such that its zero locus $\mathcal {D}$ is smooth.

We prove that the kernel of the multiplication map (4.21) is generated by elements of $\text {Sym}^{2}\overline {{{\rm Ann}}(V)}$ of the form:

(4.24)\begin{equation} \theta_S \otimes \theta_T - \theta_A \otimes \theta_B \end{equation}

with $S \cup T = A \cup B$ and $S \cap T = A \cap B$. This will immediately imply our thesis that for each element $\eta$ contained in $\overline {{{\rm Ann}}(V)}$, its square is not contained in the image of $V \odot H^{0}(\mathcal {D},\, \omega _{\mathcal {D}})$, since we recall that the intersection of this space with $\text {Sym}^{2}\overline {{{\rm Ann}}(V)}$ is trivial.

This statement can be proven on a suitable affine open subset of $\mathcal {D}$.

On each factor, the canonical theta functions $\theta ^{(i)}_{0}$ and $\theta ^{(i)}_{1}$ induce a covering $x_i: E_i \longrightarrow \mathbb {P}^{1}$, where:

(4.25)\begin{equation} x_{i}(z_i) = \displaystyle\frac{\theta^{(i)}_0(0)}{\theta^{(i)}_1(0)} \frac{\theta^{(i)}_1(z_i)}{\theta^{(i)}_0(z_i)} \end{equation}

This covering is of degree $2$, branched over four distinct points $1$, $-1$, $\delta _{i}$, $-\delta _{i}$, which are, respectively, the images of the points $0$, $1$, $\frac {\tau _i}{2}$, $1 + \frac {\tau _i}{2}$.

Hence, the elliptic curve $E_i$ is the Riemann surface which on an affine neighbourhood $\mathcal {U}_{i}$ with local coordinates ($x_{i}$,$y_{i}$) is defined by the equation:

(4.26)\begin{equation} h_i(x_i, y_i) := y_i^{2} - (x_{i}^{2} - 1)(x_{i}^{2} - \delta_{i}^{2}) \end{equation}

The affine model in (4.26) is called the Legendre normal form of $E_{i}$ (see also [Reference Catanese5]).

Let us consider the affine neighbourhood of $\mathcal {U}$ of $\mathcal {D}$ defined as:

\[ \mathcal{U} := \mathcal{D} - div(\theta_{\emptyset}) \]

Then $\mathcal {U}$ can be described as the Zariski closed subsed of the affine space $\mathbb {A}^{2a}$ with coordinates $x_1 \cdots,\, x_a,\, y_1,\, \cdots,\, y_a$ with defining equations (4.26) together with the local equation of $\mathcal {D}$, which can we obtain from (4.23) by dividing by $\theta ^{(0)}_i$ for each $i$ (according to the definition of $x_i$ in (4.25)). Hence, the local equation of $\mathcal {D}$ on $\mathcal {U}$ is $f(x_1,\, \cdots,\, x_a) = 0$, where

(4.27)\begin{equation} \begin{aligned} f(x_1, \cdots, x_a) & := \displaystyle\frac{s}{\theta_{\emptyset}} = \displaystyle\sum_{S \in \mathcal{P}}b_S X_S \\ X_S & := \prod_{j \in S} x_j \end{aligned} \end{equation}

for some complex coefficients $b_S$. For every $j$ in $\{1,\, \cdots,\, a\}$, the tangent vector $\frac {\partial }{\partial z_j}$ is naturally identified with the holomorphic $a-1$-form

\[ \omega_j := \displaystyle\frac{dx_1}{y_1} \wedge \cdots \wedge \widehat{\frac{dx_j}{y_j}} \wedge \cdots \wedge \frac{dx_a}{y_a} \]

(see (4.4) in the proof of Proposition 4.1).

Hence, on the affine subset $\mathcal {V}$ of $\mathcal {U}$ where $f_a:= \frac {\partial f}{\partial x_a}$ does not vanish, we obtain, up to a sign:

(4.28)\begin{equation} \omega_j = \displaystyle\frac{f_j}{f_a \cdot y_1 \cdot y_2 \cdots \widehat{y_j} \cdots y_a} d x_{1} \wedge \cdots \wedge d x_{a-1} \end{equation}

where $f_j$ denotes the derivative of $f$ with respect to $x_j$.

The global holomorphic differentials on $\mathcal {D}$ obtained by restricting the global sections of the polarization of $A$ to $\mathcal {D}$ can be computed by applying the residue map $H^{0}(A,\, \mathcal {O}_{A}(\mathcal {D})) = H^{0}(A,\, \omega _{A}(\mathcal {D})) \longrightarrow H^{0}(\mathcal {D},\, \omega _{\mathcal {D}})$. When restricted to $\mathcal {V}$ this gives, for each element $S$ of $\mathcal {P}$,

(4.29)\begin{equation} \psi_{S} := (\theta_{S} \cdot d z_{1} \wedge \cdots \wedge d z_{a}) \neg \left(\displaystyle\frac{1}{\theta_{\emptyset}f_a} \frac{\partial}{\partial x_a}\right) \end{equation}

where $\neg$ is the contraction operator. We have in conclusion

(4.30)\begin{equation} \begin{aligned}\psi_{S} & := (\theta_{S} \cdot d z_{1} \wedge \cdots \wedge d z_{a}) \neg \left( \displaystyle\frac{1}{\theta_{\emptyset}f_a} \frac{\partial}{\partial x_a}\right)\\ & =\left(\theta_{S} \cdot \frac{d x_{1}}{y_1} \wedge \cdots \wedge \frac{d x_{a}}{y_a}\right) \neg \left(\frac{1}{\theta_{\emptyset}f_a} \frac{\partial}{\partial x_a}\right)\\ & =\frac{\theta_S}{\theta_{\emptyset}}\frac{1}{f_ay_1\dots y_a}d x_{1} \wedge \cdots \wedge d x_{a-1} \end{aligned} \end{equation}

and since $X_S$ as in (4.27) is equal to $\frac {\theta _S}{\theta _{\emptyset }}$ (up to a non-zero constant):

(4.31)\begin{equation} \psi_{S} = \displaystyle\frac{X_{S}}{f_a y_1 \cdots y_a} d x_{1} \wedge \cdots \wedge d x_{a-1}. \end{equation}

Hence, if we multiply the expressions (4.28) and (4.31) by $f_a y_1 \cdots y_a$, we see that the elements in (4.24) become, up to a constant

(4.32)\begin{equation} X_S \otimes X_T - X_A \otimes X_B \end{equation}

which are mapped by the multiplication map to

\[ X_S\cdot X_T - X_A \cdot X_B = X_{S\cap T}X_{S\cup T} - X_{A\cap B}X_{A\cup B} = 0 \]

by the assumptions that $S \cup T = A \cup B$ and $S \cap T = A \cap B$.

On the other side, if a linear combination of tensor elements $X_S\otimes X_T$, say

\[ K = \displaystyle\sum_{S,T}a_{ST}X_S\otimes X_T \]

is mapped to zero by the multiplication map, then we have

\begin{align*} 0 & = \displaystyle\sum_{S,T} a_{ST}X_SX_T \\ & = \sum_{U \subseteq V} \bigg(\sum_{\substack{S,T\\S \cap T = U \\ S \cup T = V}} a_{ST} \bigg) X_U \cdot X_V \end{align*}

This implies that for every couple of subsets $U$ and $V$ of $\{1,\, \cdots,\, a\}$ with $U \subseteq V$, we have:

\[ \displaystyle\sum_{\substack{S,T\\S \cap T = U \\ S \cup T = V}} a_{ST} = 0 \]

since $x_j$ appears in every term $X_U \cdot X_V$ with degree $0$, $1$ or $2$, according to whether $j \notin V$, $j \in V-U$ or $j \in U$, which implies that the tensor element

\[ K_{UV}:=\displaystyle\sum_{\substack{S,T\\S \cap T = U \\ S \cup T = V}} a_{ST} X_S \otimes X_T \]

is linear combination of elements of the form (4.32) with $S \cap T = U = A \cap B$ and $S \cup T = V = A \cup B$. Finally, also $K$ must be linear combination of such elements, since clearly $K = \sum \nolimits _{U \subseteq V}K_{UV}$.

Hence, our claim that the kernel of the multiplication map is generated by elements of the form as in (4.24) holds true once we prove that there are no other quadratic relations between the polynomials $X_S$ for all $S \in \mathcal {P}$ and $Q_j := f_j y_j$ for $j \in \{1,\, \cdots,\, a\}$ in the quotient ring:

We stress here that $V$ corresponds to the vector space spanned by all polynomials $Q_j$, while the polynomials $X_S$ span a vector subspace which, according to (4.10), corresponds precisely to the subspace $\overline {{{\rm Ann}}(V)}$. In particular, $V$ is generated by elements which involve the letters $y_j$.

Since $X_{\emptyset } = 1$, we can define also the vector space $W$ inside $R$ generated as $\mathbb {C}$-vector space by all monomials in the letters $x_j$ of degree at most $2$ in each letter $x_j$.

A quadratic relation among the polynomials $X_S$ and $Q_j$ would give in $R$ the following relation:

(4.33)\begin{equation} \beta + \displaystyle\sum_{j = 1}^{a} \gamma_j y_j + \sum_{i,j = 1}^{a} \eta_{ij}y_i y_j = 0 \end{equation}

where $\beta,\, \gamma _j,\, \eta _{ij}\in W$ since for $\eta _{ij}$ we have the form

\[ \eta_{ij} = c_{ij}f_if_j \]

where $c_{ij}$ are constants depending on the definition of the polynomials $Q_j$.

If we rearrange the terms of the equation (4.33) and we express it in the form of an algebraic relation between $y_a$ and the remaining generators of $R$, we obtain:

(4.34)\begin{equation} \left[ \beta + \displaystyle\sum_{j = 1}^{a-1} \gamma_j y_j + \sum_{i,j = 1}^{a-1} \eta_{ij}y_i y_j \right] + \left[\sum_{i= 1}^{a-1}\eta_{ia} y_i + \gamma_a \right] y_a + \eta_{aa} y_a^{2} = 0\end{equation}

However, since by (4.26), the equality $y_a^{2} = (x_{a}^{2} - 1)(x_{a}^{2} - \delta _{a}^{2})$ holds in $R$, we obtain:

(4.35)\begin{equation} \left[ \beta - \eta_{aa}(x_{a}^{2} - 1)(x_{a}^{2} - \delta_{a}^{2}) + \displaystyle\sum_{j = 1}^{a-1} \gamma_j y_j + \sum_{i,j = 1}^{a-1} \eta_{ij}y_i y_j \right] + \left[\sum_{i= 1}^{a-1}\eta_{ia} y_i + \gamma_a \right] y_a = 0 \end{equation}

Since this holds in $R$ and the polynomial which in the latter equation (4.35) multiplies $y_a$ is of degree at most $1$ in $x_a$, it must be $0$ in $R$, since the only relation involving $y_a$ is $h_a$, which is of degree $2$ in $y_a$ and $4$ in $x_a$. Hence, $\eta _{ia} = 0$ for every $i \neq a$ and $\gamma _{a} = 0$. By applying the same procedure to each index, it follows that $\eta _{ij} = 0$ for every $i$ and $j$ with $i \neq j$, and $\gamma _{i} = 0$ for every $i$.

It follows that the quadratic relation (4.33) can be written in the following form:

\[ \beta + \displaystyle\sum_{j=1}^{a} \eta_{jj}y_j^{2} = 0. \]

Using again the relations $h_1,\, \cdots,\, h_a$, we can write:

\[ \beta + \displaystyle\sum_{j=1}^{a} c_{jj}f_j^{2}(x_{j}^{2} - 1)(x_{j}^{2} - \delta_{j}^{2}) = 0. \]

If at least one of the coefficients $c_{jj}$ is non-zero, say $c_{aa}$, then there exists a polynomial $u = u(x_1 \cdots x_a)$ such that, the following relation holds in the polynomial ring $\mathbb {C}[x_1,\, \cdots,\, x_a]$:

(4.36)\begin{equation} \beta + \displaystyle\sum_{j=1}^{a} c_{jj}f_j^{2}(x_{j}^{2} - 1)(x_{j}^{2} - \delta_{j}^{2}) = uf. \end{equation}

We recall that the variable $x_a$ occurs in $\beta$ with an exponent at most 2, hence it occurs in degree 4 in the left side of (4.36). Clearly, if $u$ is equal to zero, then $c_{aa}=0$ is a contradiction. Hence, assume $u\neq 0$. By definition, we can write $f = p + qx_a$, where $x_a$ does not occur neither in $p$ nor in $q$. This forces that $u$ must have degree 3 in $x_a$. On the other hand, the left side of (4.36) does not contain monomials of degree 3 in $x_a$. This implies $p=0$. Hence, $x_a$ divides $f$ and $\mathcal {D}$ is reducible, which contradicts our hypothesis.