Semistability and restrictions of tangent bundle to curves

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.