This talk will be a general introduction to perverse sheaves and their applications to the study of algebraic varieties, with a view towards enumerative geometry. It is aimed at non-experts.
We will start by considering constructible sheaves and local systems, and how they relate to the notion of stratification: this offers some insight in the relationship with intersection cohomology, which perverse sheaves generalise in a precise sense.
We will then introduce some technical notions, like t-structures, perversities, and intermediate extensions, in order to define perverse sheaves and explore their properties.
Time permitting, we will consider the relevant example of nearby and vanishing cycle functors associated with a critical locus, their relationship with the (hyper)-cohomology of the Milnor fibre and how this is exploited to define refined enumerative invariants in Donaldson-Thomas theory.
- Junior Geometry and Topology Seminar