Alladi Ramakrishnan Hall

Lefschetz theorems for higher rank bundles.

GV Ravindra

University of Missouri, St. Loius, USA

A conjecture of Hailong Dao, subsequently proved by Kestutis Cesnavicius,
states (among other things) that a vector bundle on a smooth complete intersection of
dimension at least three splits into a sum of line bundles if its endomorphism bundle
satisfies certain vanishing conditions. This generalizes the Grothendieck-Lefschetz theorem
to arbitrary rank bundles. In this talk, which is based on joint work with Amit Tripathi (IIT Hyderabad),
we will describe a geometric proof of this theorem by relating it to a vanishing theorem of
Kempf (which was sharpened subsequently by Mohan Kumar and I. Biswas). We will also talk
about a version of this theorem for complete intersection surfaces which generalizes the
Noether-Lefschetz theorem.