Conference-presentation "Newly Matured Birational invariants"

I'll talk about a recent advancement in theory of atoms for vartieties with a finite group action, and its applications to questions of equivariant birationality. The main tool is the quantum product on Chen-Ruan cohomology of the global quotient stack. There are many new results, some are valid in arbitrary dimensions. This is a joint work with L.Cavenaghi and L.Katzarkov.

In this talk, we aim to introduce the concept of G-equivariant atoms, where G is a finite group. Then we explain how they can be used to answer questions in G-equivariant birational geometry. This talk is based on joint work with L. Katzarkov and M. Kontsevich.

We revisit the classical McKay correspondence through the lens of homological mirror symmetry. In particular, we show that this correspondence can be formulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. Building on this local picture, we introduce a global version of the spectrum of singularities as a stringy invariant of singular varieties. For varieties with quotient singularities, we establish a McKay-type correspondence between the stringy spectrum and a suitably defined orbifold spectrum. Finally, we discuss possible connections between the stringy spectrum and the elliptic genus of singular varieties.

We will give a brief introduction to the theory of atoms. The basic examples are going to be discussed.