GV sheaves, Fourier-Mukai, and generic vanishing theorems -- Mihnea Popa

The classical Kodaira and Kawamata-Viehweg vanishing theorems have very useful partial analogues, called Generic Vanishing Theorems (first discovered by Green and Lazarsfeld), when the positivity hypotheses on line bundles are weakened. I will explain how abstract Fourier-Mukai functors and homological algebra allow one to relate in a formal sense generic vanishing theorems to classical vanishing theorems. In particular I will generalize (and provide algebraic proofs of) the previously known generic vanishing results to obtain a natural weakening of Kodaira vanishing. I will also show how the same techniques produce higher rank generic vanishing on moduli spaces of sheaves considered by Mukai and Bridgeland (among others).