Modularity of Landau-Ginzburg models

Abstract: In the past decades, there were proposed many different inter-related approaches to Mirror Symmetry for Fano varieties. The goal of this talk is to show that in the case of Fano threefolds these approaches are in harmony with each other. General anticanonical sections of a Fano threefold and general fibres of its Landau-Ginbzurg model are K3 surfaces, so it is natural to consider Mirror Symmetry for K3 surfaces as well. One of its most interesting forms is so called Dolgachev-Nikulin duality: for a lattice $L$ it corresponds to a complete family of $L$-polarized K3 surfaces a complete family of $L^*$-polarized K3 surfaces, where $L^*$ is a dual lattice. For any smooth Fano threefold $X$ we show that the polarization of its general anticanonical section by $\mathrm{Pic}(X)$ is Dolgachev-Nikulin dual to the polarization of a general fibre $F$ of its tame compactified toric Landau-Ginzburg model $Z\rightarrow\mathbb{P}^1$ by the (explicitly constructed) lattice of monodromy invariants. Moreover, if the anticanonical class of $X$ is very ample, we prove that the deformation space of pairs $(Z, F)$ form a complete family of $\mathrm{Pic}(X)^*$-polarized K3 surfaces. As a consequence, we obtain that for any such Fano threefold $X$ the corresponding moduli space of $\mathrm{Pic}(X)^*$-polarized K3 surfaces is uniruled. This is a joint work with Charles Doran, Andrew Harder, Ludmil Katzarkov, and Victor Przyjalkowski.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html