The algebra of symmetric tensors on smooth projective varieties

TL;DR

This paper explores the algebra of symmetric tensors on smooth projective varieties, providing computations, bounds, and conjectures related to the structure and properties of tangent bundles.

Contribution

It offers explicit calculations, bounds on Krull dimension, and a conjecture characterizing certain projective manifolds with pseudo-effective tangent bundles.

Findings

Computed algebra H^0(X, Sym*TX) in simple cases

Established a sharp bound on its Krull dimension

Proposed a conjecture for non-uniruled projective manifolds

Abstract

We discuss in this note the algebra H^0(X, Sym*TX) for a smooth complex projective variety X . We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.