Quantitative approach to Mirror Symmerty: weak Landau--Ginzburg models.
Talk by Dr. Victor Przyjalkowski
Speaker invited by: Prof. Dr. Andrew Kresch
Date: 11.04.11 Time: 13.15 - 14.45 Room: Y27H25
This will be the first of a series of three talks. Given a smooth Fano variety, Mirror Symmetry predicts the existence of a so called Landau--Ginzburg model --- a pencil, whose symplectic geometry reflects the algebraic geometry of the Fano variety, and viceversa. The most deep Mirror Symmetry conjecture, Homological Mirror Symmetry conjecture (HMS), is due to Kontsevich. It states the mirror correspondence in terms of derived categories. Unfortunately it is very complicated to prove such a correspondence in higher dimensions in general. We discuss a quantitative reflection of (a part of) HMS --- Mirror Symmetry conjecture of variations of Hodge structures. Roughly speaking it states that the solutions of a particular differential equation constructed using Gromov--Witten invariants of a Fano variety are periods of the mirror Landau--Ginzburg model. This conjecture enables one to construct explicitly mirror Landau--Ginzburg models for a large class of varieties. Under some conditions on these models we assume that they (or their minimal compactifications) are dual models for HMS. Studying them from this point of view one can predict some (numerical) invariants of initial Fano variety that can be extracted from Landau--Ginzburg model. In the first talk we observe a Mirror Symmetry conjecture of Hodge structure variations and consider some examples. In the second talk we discuss Gromov--Witten theory of Fano varieties and their quantum differential operators (Dubrovin's connections). We also discuss some basic properties of weak Landau--Ginzburg models. We give an optimistic picture of a relation between weak Landau--Ginzburg models for Fano variety and its toric degenerations. In the last talk we discuss more deep properties of weak Landau--Ginzburg models. We discuss some of their applications. Finally we discuss how based on weak Landau--Ginzburg model for Fano variety predict its invariants: characteristic numbers, Hodge numbers, birational type.