SPEAKER: Bhargav Bhatt

TITLE: A local Lefschetz theorem

ABSTRACT: A classical theorem of Lefschetz asserts that non-trivial
line bundles on a smooth projective variety of dimension >= 3 remain
non-trivial upon restriction to an ample divisor. In SGA2,
Grothendieck recast this result in purely local algebraic terms. Answering a
question raised recently by Koll\'ar, we will explain how this local
reformulation remains true under much milder hypotheses than those in
SGA2. Our method uses a vanishing theorem in characteristic p, and
formal geometry over certain very large (non-noetherian) schemes. This
is joint work with Johan de Jong.