Equivariant geometry and Calabi-Yau manifolds

Equivariant geometry and Calabi-Yau manifolds - Daniel Halpern-Leistner

Daniel Halpern-Leistner
Columbia University
March 16, 2017
Abstract: Mirror symmetry has led to deep conjectures regarding the geometry of Calabi-Yau
manifolds. One of the most intriguing of these conjectures states that various geometric
invariants, some classical and some more homological in nature, agree for any two Calabi-Yau
manifolds which are birationally equivalent to one another. I will discuss how new methods in
equivariant geometry have shed light on this conjecture over the past few years, ultimately
leading to a proof of the conjecture for compact Calabi-Yau manifolds which are birationally
equivalent to a moduli space of sheaves on a K3 surface. This represents the first substantial
progress on the conjecture in dimension > 3 in several years. The key technique is the new
theory of ``Theta-stratifications," which allows one to bring ideas from equivariant Morse theory
into the setting of algebraic geometry.