The Infinitesimal Torelli Theorem for hypersurfaces in abelian varieties

TL;DR

This paper proves the Infinitesimal Torelli Theorem for smooth hypersurfaces in simple abelian varieties with high self-intersection, extending the understanding of period map injectivity in complex geometry.

Contribution

It establishes the Infinitesimal Torelli Theorem for a new class of manifolds, providing an effective bound on Green's previous results for hypersurfaces in abelian varieties.

Findings

Proves injectivity of the period map for hypersurfaces in simple abelian varieties.

Provides an effective bound on self-intersection for the theorem to hold.

Extends the class of manifolds satisfying the Infinitesimal Torelli property.

Abstract

Given a compact K\"ahler manifold, the Infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example for hyperelliptic curves of genus greater than $2$. Nevertheless the Infinitesimal Torelli Theorem holds for many other classes of manifolds. We will prove it for smooth hypersurfaces in simple abelian varieties with sufficiently high self-intersection giving an effective bound on a result by Green in this particular case.