This year we will focus on some aspects of two different topics:
(a) Hodge-theoretic mirror symmetry for projective toric manifolds;
(b) K-stability for polarised manifolds and its generalisations.
In the last part of the course we will explain some nontrivial connections between (a) and (b).
Plan of the course:
1) Background
Classical residue pairing (Griffiths-Harris, Chapter 5.1-5.2). Higher residue pairing (e.g. Matsumoto; Matsubara-Heo). Deligne pairings (e.g. Dervan, Section 3.1). Brief review of toric geometry (e.g. facts from Fulton, Chapters 1-3).
2) Overview of toric Hodge-Theoretic mirror symmetry
We will cover enough of the work of Corti, Coates, Iritani and Tseng in order to understand their general mirror theorem in the case of smooth projective toric manifolds.
3) K-stability for polarised manifolds
We will introduce the main stability notion for projective manifolds, K-stability, in the more general context of stability conditions for polarised manifolds in the sense of Dervan.
4) Mirrors of a K-semistable toric manifold
We will explain how Hodge-theoretic mirror symmetry allows to express the K-semistability of a projective toric polarised manifold as a condition on the mirror Landau-Ginzburg model, and provide some nontrivial applications.
References:
Coates, Corti, Iritani and Tseng, Hodge-theoretic mirror symmetry for toric stacks.
Dervan, Stability conditions for polarised varieties.
Fulton, Introduction to toric varieties.
Griffiths-Harris, Principles of algebraic geometry.
Matsubara-Heo, Localization formulas of cohomology intersection numbers.
Matsumoto, Intersection numbers for logarithmic k-forms.