#### 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.

Done