tbranzov

This two talks explore Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate several families of such manifolds, including (Quasi) Hopf, (Quasi) Calabi-Eckmann, and LVM manifolds, alongside a class of definable complex manifolds that encompasses both algebraic varieties and the aforementioned special cases. Our main result establishes the preservation of the motivic nature of Hopf polynomials inside this broader context.

In this talk I'll present a geometric relation between the Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization.

Anti-canonical pairs (Y, D) are logarithmic K3 surfaces. It is well known that they have a rich geometry. A recent result, whose proof was motivated by mirror-symmetry, establishes a conjecture by Looijenga giving conditions for smoothability of the cusp obtained by contracting D. A central ingredient in the proof is a global Torelli theorem using the mixed Hodge structure on H2(Y −D). In this talk we will formulate and sketch the proof of this result.

I will describe an extension, proposed by Hitchin and Gualtieri, of the notion of a Calabi-Yau structure to generalized Kähler geometry. I will then discuss a conjectural classification of the generalized Kähler Calabi-Yau geometries, expressed in terms of Bogomolov-Beauville decomposition, and present a partial resolution.