Semistability and restrictions of tangent bundle to curves

TL;DR

This paper explores conditions under which the tangent bundle of complex projective manifolds fails to be semistable, proposing a conjecture that all such manifolds fall into specific classes, and proves this for several cases.

Contribution

It introduces a conjecture classifying all complex projective manifolds based on tangent bundle semistability and proves it for manifolds with finite fundamental group, certain morphisms, and specific canonical bundle properties.

Findings

Manifolds with finite fundamental group are of the above type.

Manifolds admitting a nonconstant morphism from the projective line are of the above type.

Manifolds with positive, negative, or trivial canonical bundle are of the above type.

Abstract

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,\varphi)$, where $C$ is a compact connected Riemann surface and $\varphi : C\to X$ a holomorphic map, such that the pull back $\varphi^*TX$ is not semistable. 2. The variety $X$ admits an \'etale covering by an abelian variety. 3. The dimension $\dim X \leq 1$. We conjecture that all complex projective manifolds are of the above type, and prove that the following classes are among those that are of the above type. i) All $X$ with a finite fundamental group. ii) All $X$ such that there is a nonconstant morphism from the projective line to $X$. iii) All $X$ such that the canonical line bundle $K_X$ is either positive or negative or $c_1(K_X) \in H^2(X, {\mathbb Q})$ vanishes. iv) All projective surfaces.