The positive Chern character conjecture for Fano manifolds

Let $X$ be a Fano manifold of dimension $n$. Here ${\rm ch}_k(X)$ denotes the degree-$k$ component of the Chern character, and positivity is understood in the sense used for Chern characters of Fano manifolds.

Positive Chern character conjecture. If ${\rm ch}_k(X)$ is positive for every $k\le n$, then

$$X\simeq \mathbb{P}^n.$$

More strongly, if ${\rm ch}_k(X)$ is positive for every

$$k\le \left\lceil \log_2(n+2)\right\rceil,$$

then $X\simeq\mathbb{P}^n$.

Positivity conditions on Chern characters are known to be restrictive, with classification results in several cases involving large index or low-degree Chern characters. The conjecture proposes that positivity through degree $\lceil\log_2(n+2)\rceil$ already characterizes projective space.