[Paper Review] Tian-Todorov theorems for Landau-Ginzburg models

Ludmil Katzarkov, Maxim Kontsevich+1 more|arXiv (Cornell University)|Sep 21, 2014

Advanced Algebra and Geometry1 citations

TL;DR

This paper establishes the smoothness of the moduli space of Landau-Ginzburg models by proving a Tian-Todorov theorem for their deformations, developing Hodge theory for varieties with potentials, and proving a double degeneration statement. It shows that modulo a triviality condition, these moduli spaces admit canonical special coordinates, and clarifies the role of Hodge numbers in mirror symmetry for one-parameter families of symplectic Fano manifolds.

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.

Motivation & Objective

• To establish the unobstructedness of deformations of Landau-Ginzburg models by proving a Tian-Todorov-type theorem in this context.
• To develop a Hodge theory framework for algebraic varieties equipped with a potential function, extending classical Hodge theory to non-compact and singular settings.
• To clarify the definition and role of Hodge numbers in non-commutative Hodge structures of Landau-Ginzburg type, particularly in mirror symmetry.
• To interpret families of de Rham complexes associated to a potential in terms of mirror symmetry for one-parameter families of symplectic Fano manifolds.
• To show that, modulo a natural triviality condition, the moduli spaces of Landau-Ginzburg models carry canonical special coordinates.

Proposed method

• Formulate and prove a Tian-Todorov theorem for Landau-Ginzburg models by analyzing the cohomology of the Koszul complex associated to the potential.
• Introduce and study the Hodge theory of the Hodge-de Rham complex associated to a potential on a variety, generalizing classical Hodge theory.
• Prove a double degeneration statement for the Hodge filtration on the de Rham complex of the potential, which is essential for unobstructedness.
• Use the formalism of non-commutative Hodge structures to define and compare various notions of Hodge numbers in the Landau-Ginzburg setting.
• Construct families of de Rham complexes parametrized by the moduli space of potentials and interpret them via mirror symmetry for Fano manifolds.
• Establish the existence of canonical special coordinates on the moduli space by analyzing the period map and its triviality condition.

Experimental results

Research questions

• RQ1 How can the Tian-Todorov theorem be generalized to the setting of Landau-Ginzburg models?
• RQ2 What is the appropriate Hodge theory for varieties equipped with a potential function, and how do Hodge numbers behave in this context?
• RQ3 How do families of de Rham complexes attached to a potential relate to mirror symmetry for one-parameter families of symplectic Fano manifolds?
• RQ4 Under what conditions do moduli spaces of Landau-Ginzburg models admit canonical special coordinates?
• RQ5 What is the role of the triviality condition in the construction of special coordinates on the moduli space?

Key findings

• The moduli space of Landau-Ginzburg models is smooth, as the deformation theory is unobstructed due to the double degeneration statement for the Hodge filtration.
• A Tian-Todorov theorem for Landau-Ginzburg models is established, proving unobstructed deformations via the vanishing of the second cohomology of the Koszul complex.
• Hodge numbers in the non-commutative Hodge structure of Landau-Ginzburg type are well-defined and play a central role in mirror symmetry, particularly in the context of Fano manifolds.
• Families of de Rham complexes associated to a potential are interpreted as mirror families for one-parameter families of symplectic Fano manifolds.
• Modulo a natural triviality condition on the period map, the moduli space of Landau-Ginzburg models carries canonical special coordinates, which are essential for mirror symmetry constructions.
• The Hodge-de Rham complex associated to a potential admits a well-behaved Hodge filtration, and its degeneration at the E2-page is key to the unobstructedness result.