Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models

TL;DR

This paper establishes the smoothness and unobstructedness of the moduli space of Landau-Ginzburg models, extending classical deformation theorems and exploring their role in mirror symmetry and non-commutative Hodge structures.

Contribution

It formulates and proves a Tian-Todorov theorem for Landau-Ginzburg models, developing Hodge theory and analyzing moduli space properties in this context.

Findings

Proved smoothness of the Landau-Ginzburg moduli space.

Established a Tian-Todorov type unobstructedness theorem.

Connected Hodge structures of Landau-Ginzburg models to mirror symmetry.

Abstract

In this paper we prove the smoothness of the moduli space of Landau-Ginzburg models. We formulate and prove a Tian-Todorov theorem for the deformations of Landau-Ginzburg models, develop the necessary Hodge theory for varieties with potentials, and prove a double degeneration statement needed for the unobstructedness result. We discuss the various definitions of Hodge numbers for non-commutative Hodge structures of Landau-Ginzburg type and the role they play in mirror symmetry. We also interpret the resulting families of de Rham complexes attacted to a potential in terms of mirror symmetry for one parameter families of symplectic Fano manifolds and argue that modulo a natural triviality property the moduli spaces of Landau-Ginzburg models posses canonical special coordinates.