Ample tangent bundle

Question

I am looking for definition of ample tangent bundle or positive tangent bundle and why on a complex manifold , positive bisectional curvature means ample tangent bundle?

Answer 1

An ample tangent bundle is a tangent bundle associated to a complex manifold that is an ample vector bundle.

There are several equivalent definitions for ampleness of a vector bundle, here is one we will use here: Given a holomorphic vector bundle $E$ on a complex manifold $M$, let $\mathbb{P}(E)$ be the projectivized vector bundle constructed by taking the one dimensional quotients of every fiber $E_x$ at points $x\in{M}$. Now over $\mathbb{P}(E)$ take the twisting sheaf $L(E):=\mathcal{O}_{\mathbb{P}(E)}(1)$. The bundle $E$ is said to be ample if $L(E)$ is ample over $\mathbb{P}(E)$. For equivalent definitions see Robin Hartshorne's article, and this question on stackexchange.

Now, Kobayashi and Ochiai (1970) defined positivity of a vector bundle $E$ by the positivity of the line bundle $L(E)$ as defined above. By the Kodaira embedding theorem we can deduce that positivity of $E$ is equivalent to ampleness of $E$.

Kobayashi and Ochiai proved the following theorem (Theorem 6.4 in their paper cited above): If a complex manifold has positive holomorphic bisectional curvature then the tangent bundle $TM$ is positive, and is therefore ample. The idea is that pushing the metric to the quotient increases curvature, and it therefore remains positive for the line bundle $L(TM)$.

Some final comments: Mori (1979) proved Hartshorne's conjecture that every projective variety with ample tangent bundle must be isomorphic to projective space, and thus by discussion the above the Frankel conjecture that any compact Kaehler manifold with positive holomorphic bisectional curvature is biholomorphic to complex projective space.