Smooth projective surfaces with pseudo-effective tangent bundles

TL;DR

This paper characterizes smooth projective surfaces with pseudo-effective tangent bundles, showing they are precisely those with nef canonical divisor and zero second Chern class, and explores blow-ups of certain ruled surfaces.

Contribution

It provides a complete characterization of non-uniruled surfaces with pseudo-effective tangent bundles and analyzes their behavior under blow-ups.

Findings

Tangent bundle is pseudo-effective iff $K_S$ is nef and $c_2(S)=0$

Characterization applies to non-uniruled surfaces

Studied blow-ups of non-rational ruled surfaces

Abstract

Let $S$ be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle $T_S$ is pseudo-effective if and only if the canonical divisor $K_S$ is nef and the second Chern class vanishes, i.e., $c_2(S)=0$. Moreover, we study the blow-up of a non-rational ruled surface with pseudo-effective tangent bundle.