Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds

TL;DR

This paper extends classical methods to weighted projective spaces, proving the infinitesimal Torelli theorem for specific Fano threefolds and analyzing automorphism actions, with implications for Lang-Vojta's conjecture.

Contribution

It generalizes the infinitesimal Torelli map to weighted complete intersections and establishes Torelli for certain Fano threefolds.

Findings

Proves infinitesimal Torelli theorem for hyperelliptic Fano threefolds

Describes automorphism group action on cohomology

Lays groundwork for Lang-Vojta's conjecture proof

Abstract

We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove the infinitesimal Torelli theorem for hyperelliptic Fano threefolds of Picard rank 1, index 1, degree 4 and study the action of the automorphism group on cohomology. The results of this paper are used to prove Lang-Vojta's conjecture for the moduli of such Fano threefolds in a follow-up paper.