Sections of symmetric powers of the tangent bundle
Establish whether every Fano manifold of Picard number one admitting a non-isomorphic surjective endomorphism has a positive integer $m$ for which $H^0(X,\operatorname{Sym}^m T_X)\neq 0$, and determine whether its tangent bundle is big.
References
It is a folklore conjecture that a Fano manifold of Picard number one admitting a non-isomorphic surjective endomorphism is isomorphic to projective space. The conjecture was partially solved in under the assumption that $\mathcal{O}_{\mathbb{P}(T_X)}(1)$ is big. Building on this result, Theorem \ref{t:pos} may be viewed as a first step toward the following expectation.
\begin{conjecture}[{cf.~Remark \ref{r:pos-cur} (3)}]\label{Conj-big} Let $X$ be a Fano manifold of Picard number one admitting a non-isomorphic surjective endomorphism. Then there is a positive integer $m$ such that $H0(X,SymmT_X)\neq 0$. More optimistically, the tangent bundle $T_X$ is big. \end{conjecture}