Characterization of projective spaces by positivity of Chern characters

Let $X$ be an $n$-dimensional Fano manifold satisfying condition ${\mathfrak {F}_k}$, where

$$k=\left \lceil{\log_2(n+1)}\right \rceil.$$

The projective-space characterization conjecture. Then $X\cong \mathbb{P}^n$.

This conjecture proposes that sufficiently strong positivity of the Chern characters characterizes projective space among Fano manifolds. It strengthens the previously posed question concerning Fano $n$-folds satisfying ${\mathfrak {F}_n}$; the general statement remains open.