Algebraic and Symplectic Geometry Seminar

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to show that a large class of homogeneous varieties have a supposedly rare behaviour: over a field of positive characteristic, they admit no deformation to any ring of characteristic greater than p.

In the 1990s, physicists introduced an ideal way to count curves inside a Calabi-Yau 3-fold, called the Gopakumar-Vafa (GV) theory. Building on several previous attempts, Maulik-Toda recently gave a mathematical rigorous definition of the GV invariants. We expect that the GV invariants and the Gromov-Witten (GW) invariants are related by an explicit formula, but this stands as a challenging open problem. In this talk, I will explain recent mathematical developments on the GV theory, especially for local curves, including the cohomological chi-independence theorem and the GV/GW correspondence in a special case.