Infinitesimal Torelli for elliptic surfaces revisited

Abstract

In this article we give a new proof for the infinitesimal Torelli theorem for minimal elliptic surfaces without multiple fibers with Euler number at least 24 for nonconstant j-invariant. In the case of constant j-invariant we find a new proof in the case of Euler number at least 72. We also discuss several new counterexamples.

Introduction

Let $\pi :X\to C$ be a minimal elliptic surface without multiple fibers. We say that X satisfies infinitesimal Torelli if the differential of the period map

$${\delta }_X:H^1(X,{\mathrm{\Theta }}_X)\to \mathrm{Hom}(H^0({\mathrm{\Omega }}_X^2),H^1({\mathrm{\Omega }}_X^1))$$

is injective. (See e.g., [12, Chapter 10].)

There have been various results as to whether the infinitesimal Torelli property holds for X. If the geometric genus $p_g(X)$ vanishes then obviously the infinitesimal Torelli property does not hold. If X is an elliptic K3 surface then it holds by the results on K3 surfaces.

The case $g(C)=0$, $p_g(X)\ge 2$ can be studied by techniques developed by Lieberman–Wilsker–Peters [9] and Kii [7]. These papers give a sufficient criterion for infinitesimal Torelli for varieties with divisible canonical bundle. In the latter paper Kii proved infinitesimal Torelli for elliptic surfaces in case $g(C)=0$, $p_g(X)\ge 2$ and the j-invariant is nonconstant. In [8] the author used a similar method to show that infinitesimal Torelli holds if again $g(C)=0$, $p_g(X)\ge 2$ holds, but the j-invariant is constant, and π has at least $p_g(X)+3$ singular fibers. It is well known that an elliptic surface with $p_g(X)\ge 1,g(C)=0$ has at least $p_g(X)+2$ singular fibers. In [8] it is also shown that elliptic surfaces with $p_g(X)\ge 2,g(C)=0$ and $p_g(X)+2$ singular fibers do not satisfy infinitesimal Torelli. Chakiris [2], in his proof of the generic Torelli theorem for elliptic surfaces with a section and $g(C)=0$ obtained a generic infinitesimal Torelli result. On the other hand, it is known that variational Torelli does not hold for elliptic surfaces [3].

One easily checks that Kii's criterion cannot be applied in the case where the genus of C is positive. M.-H. Saito claimed in [11] that infinitesimal Torelli holds for elliptic surfaces without multiple fibers such that $p_g(X)\ge 1$ and either the j-invariant is nonconstant or the j-invariant is constant, different from $0,1728$ and $\chi ({\mathcal{O}}_X)\ge 3$ hold, but no constraints on the base curve. However, Ikeda [6] recently obtained an example of an elliptic surface without multiple fibers, with nonconstant j-invariant and $p_g(X)=g(C)=1$, for which infinitesimal Torelli does not hold.

The original purpose of this paper was to give a new proof for infinitesimal Torelli for elliptic surfaces, by methods different from Saito's. However, when preparing this paper, we learned that each of the steps in Saito's proof hold under the additional assumption that ${\mathrm{\Omega }}_X^2$ is base point free.

Our basic idea is to use Green's version of Kii's criterion [5, Corollary 4.d.3]. By doing so, we can reproduce Saito's result under the same hypothesis that ${\mathrm{\Omega }}_X^2$ is base point free. The main difference with Saito's proof is that In the case of nonconstant j-invariant, the infinitesimal Torelli result is almost an immediate corollary from the Green-Kii criterion. In the constant j-invariant case our method covers many cases left out by Saito. In particular, we obtain results for elliptic surfaces with constant j-invariant 0 and 1728, and for elliptic fiber bundles which are not principal. Moreover, our method yields a series of new counterexamples to infinitesimal Torelli.

To formulate our statement, we need a further invariant of the elliptic fibration $\pi :X\to C$. Let $\mathcal{L}$ be the dual of the line bundle $R^1{\pi }_{⁎}{\mathcal{O}}_X$ on C and let $d=\mathrm{deg}(\mathcal{L})$. It is well known that $d\ge 0$ and $d=0$ corresponds to the case of an elliptic fiber bundle. To apply Green's version of Kii's criterion, we need to check that ${\mathrm{\Omega }}_X^2$ is base point free. It is easy to check that this happens if $d>1$ or $d=1$ and $h^0(\mathcal{L})=0$.

Moreover let Δ be the reduced effective divisor on C whose support coincides with the support of the discriminant. In this paper we prove two results on classes of elliptic surfaces for which infinitesimal Torelli holds, one in the nonconstant j-invariant case and one in the constant j-invariant case.

Theorem 1.1

Let $\pi :X\to C$ be an elliptic surface without multiple fibers and with nonconstant j-invariant. If $d\ne 1$ or $d=1$ and $h^0(\mathcal{L})=0$ hold then X satisfies infinitesimal Torelli.

Theorem 1.2

Let $\pi :X\to C$ be an elliptic surface without multiple fibers, and with constant j-invariant. Suppose that

• if $g=0$ then $d>2$;

• if $d=1$ then $h^0(\mathcal{L})=0$;

• if $h^1(X)$ is odd then $\mathcal{L}\ncong {\mathcal{O}}_C$.

Then X satisfies infinitesimal Torelli if and only if the multiplication map

$${\mu }_{\pi }:H^0({\mathit{\Omega }}_C^1\otimes \mathcal{L})\otimes H^0({\mathit{\Omega }}_C^1\otimes {\mathcal{L}}^{-1}(\mathit{\Delta }))\to H^0({({\mathit{\Omega }}_C^1)}^2(\mathit{\Delta }))$$

is surjective.

We will comment a bit one the cases excluded. If $g=0$ and $d\le 2$ then, depending on d, we have product surfaces $E\times {\mathbf{P}}^1$ $(d=0)$, rational elliptic surfaces $(d=1)$ or elliptic K3 surfaces $(d=2)$. For each of these cases it is well known whether infinitesimal Torelli does hold (K3) or does not hold (products and rational surfaces). If $h^1(X)$ is odd and $\mathcal{L}\cong {\mathcal{O}}_C$ then it is known that X does not satisfy infinitesimal Torelli [11, Section 8] and we will come back to this in Section 5. Hence the only cases not covered by the above two theorems are the cases $d=1$ and $h^0(\mathcal{L})>0$, i.e., when ${\mathrm{\Omega }}_X^2$ is not base point free.

The second theorem does not give a conclusive answer whether infinitesimal Torelli holds, but the map ${\mu }_{\pi }$ is studied extensively in the literature. For many cases we know that ${\mu }_{\pi }$ is surjective, which yields to the following corollary. Recall that if $d\ge 1$ then $s\ge d+1$.

Corollary 1.3

Let $\pi :X\to C$ be an elliptic surface without multiple fibers and with constant j-invariant. Let s be the number of singular fibers. Suppose that if $d=1$ then $h^0(\mathcal{L})=0$ holds. Moreover, suppose that one of the following holds

• $d\ge 6$;

• $3\le d\le 5$ and $s\ge d+2$;

• $d\in \{1,2\}$ and $s\ge d+3$;

• $d\in \{4,5\}$, $s=d+1$; $h^0({\mathcal{L}}^{-1}(\mathit{\Delta }))=0$ and $\mathrm{Cliff}(C)\ge 2$;

• $d\in \{1,2,3\}$, $s=d+1$ and $h^0({\mathcal{L}}^{-1}(\mathit{\Delta }))=0$, either one of ${\mathit{\Omega }}_C^1\otimes \mathcal{L}$ or ${\mathit{\Omega }}_C^1\otimes \mathcal{L}(-\mathit{\Delta })$ is very ample and $\mathrm{Cliff}(C)\ge 2$;

• $d\in \{1,2\}$, $s=d+2$, $h^0({\mathcal{L}}^{-2}(\mathit{\Delta }))=0$;

• $d=0$; $h^1(X)$ is even; $\mathcal{L}\cong {\mathcal{O}}_C$ and C is not hyperelliptic.

Then X satisfies infinitesimal Torelli.

In some cases one can show that ${\mu }_{\pi }$ is not surjective.

Theorem 1.4

Suppose $\pi :X\to C$ is an elliptic surface, with constant j-invariant and $d+1$ singular fibers. If $g=0$ then suppose additionally that $d\ge 3$ and if $d=1$ then suppose additionally that $h^0(\mathcal{L})=0$. If $h^0({\mathcal{L}}^{-1}(\mathit{\Delta }))>0$ then X does not satisfy infinitesimal Torelli

If $g=0$, $d\ge 2$ and $s=d+1$ then $h^0({\mathcal{L}}^{-1}(\mathrm{\Delta }))=2$ for degree reasons. Hence X does not satisfy infinitesimal Torelli in this case. In this way we recover the counterexamples from [8]. However, the result in that paper is much stronger. Namely, there we proved for $d\ge 3$ that the period map is constant on the locus of elliptic surfaces with constant j-invariant and $d+1$ singular fibers. However, the above result yields new counterexamples if the base curve has genus 1:

Theorem 1.5

Suppose $\pi :X\to C$ is an elliptic surface, with $g(C)=1$, $d\in \{1,2\}$ and constant j-invariant different from $0,1728$. Then X does not satisfy infinitesimal Torelli.

There are a few cases not covered by our results. For a certain number of classes of elliptic surfaces with $d\le 5$ and constant j-invariant we do not know whether ${\mu }_{\pi }$ is surjective or not. This is an extensively researched problem and we do not aim to elaborate on this.

A second class of surfaces excluded are the surfaces with $d=1$ and $h^0(\mathcal{L})>0$, i.e., the case where ${\mathrm{\Omega }}_X^2$ has a one-dimensional base locus and our method breaks down. We expect that infinitesimal Torelli does not hold in this case and we will present evidence for this. Note that if moreover the j-invariant is constant then ${\mu }_{\pi }$ is not surjective in this case. Also, Ikeda's counterexample is of this type.

Our strategy is to use Green's version of Kii's criterion. If ${\mathrm{\Omega }}_X^n$ is base point free then this criterion reduces infinitesimal Torelli to a problem on the vanishing of a certain Koszul cohomology group. In the case of elliptic surfaces we can relate this Koszul cohomology group with a Koszul cohomology group on the base curve. If the j-invariant is nonconstant then it is easy to prove that this group vanishes, whereas if the j-invariant is constant then this group vanishes if and only if ${\mu }_{\pi }$ is surjective. This strategy leaves out the cases where ${\mathrm{\Omega }}_X^2$ has a base locus, i.e., the case where $d=1,h^0(\mathcal{L})>0$; the case $g=d=0$ and some particular cases (K3 surfaces, nonalgebraic principal elliptic fiber bundles), because of some technicalities in the proof.

The paper is organized as follows. In Section 2 we recall some preliminaries on elliptic surfaces and on Koszul cohomology. In Section 3 we prove the Torelli result for elliptic fibrations with nonconstant j-invariant. In Section 4 we prove the results for constant j-invariant such that $d>0$. In Section 5 we discuss the case $d=0$. Finally, in Section 6 we discuss what happens if $d=1$ and $h^0(\mathcal{L})>0$ hold.