Demailly Campana Peternell Conjecture for isolated singularities

Question

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.

Answer 1

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.