Mirror symmetry in the non-Kähler setting

Speaker: Dan Popovici
Institution: Université Paul Sabatier, Institut de Mathématiques de Toulouse (France)

Abstract

Compact Kähler three-dimensional complex manifolds whose canonical bundle is trivial are predicted to come in pairs such that the small deformations of the complex structure (namely the Kuranishi family) of one member correspond to the metric structures (namely the complexified Kähler cone) of the other member and vice versa via a local biholomorphism that is called the mirror map.

It has long been known that this mirror duality cannot hold entirely within the Kähler realm. An extension of the mirror symmetry conjecture to the non-Kähler setting was proposed in 2015 by Lau, Tseng and Yau from the standpoint of SYZ theory. We proposed a completely different non-Kähler mirror symmetry project in 2018 from a cohomological point of view that replaces the first page of the Frölicher spectral sequence with the second page and the Kâhler cone with the Gauduchon cone we had introduced a few years earlier. Our main result of 2018 stipulates that the Iwasawa manifold, a well-known compact complex non-Kähler manifold, is its own mirror dual. The notion of essential small deformations of this manifold was introduced on that occasion.

In this talk, after a brief overview of the main ideas involved in this generalised approach to mirror symmetry, we will outline the main subsequent developments, especially:
-the theory of page-r-ddbar-manifolds that we introduced in a series of three joint papers with J. Stelzig and L. Ugarte;
-the theory of holomorphic p-contact structures, generalising the classical holomorphic contact structures, that we proposed very recently in joint work with H. Kasuya and L. Ugarte.