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.