Explicit symmetric differential forms on complete intersection varieties and applications

TL;DR

This paper investigates the cohomology of symmetric powers of cotangent bundles on complete intersection varieties, providing explicit descriptions, non-vanishing results, and constructing examples with ample cotangent bundles related to Debarre's conjecture.

Contribution

It offers explicit cohomology descriptions, demonstrates non-invariance under deformation, and constructs new varieties with ample cotangent bundles, advancing understanding in algebraic geometry.

Findings

Explicit cohomology descriptions in terms of defining equations

Non-vanishing of certain symmetric differential forms

Construction of varieties with ample cotangent bundle

Abstract

In this paper we study the cohomology of tensor products of symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. We give several applications. First we prove a non-vanishing result, then we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Our main application is the construction of varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.