Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives

TL;DR

This paper studies the infinitesimal Torelli problem for special Gushel-Mukai and related Fano threefolds, demonstrating injectivity of the period map's invariant part using Hodge theory and categorical methods, with implications for moduli space descriptions.

Contribution

It proves the injectivity of the invariant part of the infinitesimal period map for special Gushel-Mukai threefolds using novel Hodge theoretical and categorical approaches.

Findings

Injectivity of the invariant part of the period map for special Gushel-Mukai threefolds.

Description of the kernel of the period map differential via Bridgeland moduli spaces.

Extension of results to other prime Fano threefolds and Verra threefolds.

Abstract

We investigate infinitesimal Torelli problems for some of the Fano threefolds of the following two types: (a) those which can be described as zero loci of sections of vector bundles on Grassmannians (for instance, ordinary Gushel-Mukai threefolds), and (b) double covers of rigid Fano threefolds branched along a $K3$ surface (such as, special Gushel-Mukai threefolds). The differential of the period map for ordinary Gushel-Mukai threefolds has been studied by Debarre, Iliev and Manivel; in particular, it has a $2$-dimensional kernel. The main result of this paper is that the invariant part of the infinitesimal period map for a special Gushel-Mukai threefold is injective. We prove this result using a Hodge theoretical argument as well as a categorical method. Through similar approaches, we also study infinitesimal Torelli problems for prime Fano threefolds with genus $7$, $8$, $9$, $10$, $12$ (type (a)) and for special Verra threefolds (type (b)). Furthermore, a geometric description of the kernel of the differential of the period maps for Gushel-Mukai threefolds (and for prime Fano threefolds of genus $8$) is given via a Bridgeland moduli space in the Kuznetsov components.