Generic vanishing and compact Kähler manifolds

In the late 1980s, Green and Lazarsfeld studied the cohomology of topologically trivial line bundles on compact Kähler manifolds and proved, among many other things, that the cohomology groups of a generically chosen line bundle vanish below a certain degree. In the projective case, their results have been refined by Chen, Hacon, Pareschi, Popa, and others, and have found a surprising number of applications, for example to the optimal birational characterization of abelian varieties. In the talk, I will review this story, and then explain how one can extend everything back to arbitrary compact Kähler manifolds. (Based on joint work with Pareschi and Popa.)