Demailly Campana Peternell Conjecture for isolated singularities I have asked this question on StackExchange but didn't get an answer, therefore I am asking again here.
If $M$ is smooth, and $T^*M\to$ Spec$H^0(T^*M,\mathcal{O}(T^*M))$ is a projective birational map, then the conjecture predicts that $M=G/P$ for some semisimple group G and a parabolic $P$. If in addition that Spec$H^0(T^*M,\mathcal{O}(T^*M))$ has as isolated singularity then according to here Conj 1.3 one can prove that $M$ must be a projective space – this is S. Mori’s famous theorem on smooth varieties with ample tangent bundle. However I don't really see how this is related to Mori's theorem. Can someone explain why this is true for someone like me who is very ignorant in birational geometry.
 A: This is related to Mori's theorem through
Grauert's ampleness criterion in
Hartshorne's "Ample vector bundles" (Proposition 3.5).
Let's assume that $M$ is projective and $\dim M  \ge 2$.
Let $\alpha : T^*M \to Y$ denote the affinization of $T^*M$.
To show that $TM$ is ample, according to the criterion
it suffices to show that
the zero section $M \subset T^*M$ is the only subvariety
contracted by $\alpha$.
Assume to the contrary that $\alpha$ contains a fiber $F$ such that $F \ne M$.
Since $\alpha$ is projective, necessarily
the fibers of $\alpha$ have dimension $\le \dim M$.
As we assume that $\dim M \ge 2$, $\alpha$ is a small contraction, so
the image of the exceptional locus of $\alpha$ in $Y$ is the singular locus of $Y$.
Now as $F \ne M$ and the intersection of $F$ with each fiber of $T^*M \to M$
is finite, $F$ is not stable under the $\mathbf{C}^*$-action on $T^*M$.
Since the $\mathbf{C}^*$-action descends to $Y$,
it implies that the singular locus of $Y$ contains $\alpha(\mathbf{C}^* \cdot F)$,
which contradicts the assumption that $Y$ has only isolated singularities.
