Projective manifolds whose tangent bundle is Ulrich

TL;DR

This paper characterizes projective manifolds with Ulrich tangent bundles, showing only the twisted cubic and Veronese surface qualify, and establishes that cotangent bundles cannot be Ulrich.

Contribution

It provides the first classification of projective manifolds with Ulrich tangent bundles and introduces new numerical restrictions on Chern classes.

Findings

Only twisted cubic and Veronese surface have Ulrich tangent bundles.

Cotangent bundles are never Ulrich.

Numerical restrictions on Chern classes of Ulrich bundles.

Abstract

In this article, we give numerical restrictions on the Chern classes of Ulrich bundles on higher-dimensional manifolds, which are inspired by the results of Casnati in the case of surfaces. As a by-product, we prove that the only projective manifolds whose tangent bundle is Ulrich are the twisted cubic and the Veronese surface. Moreover, we prove that the cotangent bundle is never Ulrich.