All Questions

The total space of cotangent bundle of any manifold $M$ is a symplectic manifold. Is it true/false/unknown that for any $M$, $T^*M$ has Kähler structure? Please support your claim with reference or ...

Let $X$ be a smooth projective connected variety over $\mathbb{C}$ and let $T_X$ be its tangent bundle. If $T_X$ is ample, then $X$ is isomorphic to a projective space by Mori's theorem. If $T_X$ is ...