Virasoro constraints and moduli of twisted sheaves

Abstract

This thesis is about the enumerative geometry of sheaves and twisted sheaves. In the first part of the thesis, we study Virasoro constraints on toric surfaces. Even though Virasoro constraints are conjectured to exist for all surfaces, toric surfaces are of a combinatorial nature and particularly easy to understand. For moduli of sheaves, we use work of Klyachko, Perling and Kool to explicitely verify Virasoro constraints by explicitely computing all relevant descendent invariants. In this way we provide strong evidence for the conjecture. In the second part of the thesis, we shift our attention to twisted sheaves. Twisted sheaves are sheaf-like objects that do not have a natural notion of global sections. We first provide a detailed construction of the moduli space of twisted sheaves on an arbitrary smooth projective variety, along with the corresponding obstruction theory. Next, we study the deformation of twists. Using Hodge theory, we provide a general criterion in which a twist can be deformed to a trivial one. This creates a powerful link between the enumerative theory of twisted and untwisted sheaves. We apply this to concrete problems, such as S-duality and the Period-Index conjecture.