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](https://arxiv.org/pdf/math/0608143.pdf) 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.



Demailly Campana Peternell Conjecture for isolated singularities
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.


