4 problems

Let $X$ be a compact complex manifold endowed with a Hermitian metric $g$, and let $K$ denote its holomorphic sectional curvature for the Chern connection. Constant holomorphic sec…

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$, and let $H$ denote the Chern holomorphic sectional curvature, with $R$ the curvature of the Chern conn…

Let $(M^n,g)$ be a compact Hermitian manifold with $n\geq 2$. Denote by $R$ and $R^c$ the curvature tensors of the Levi-Civita and Chern connections, and by $H$ and $H^c$ their hol…

Constant holomorphic sectional curvature conjecture. If $H$ is equal to a constant $c$, then $g$ is Kähler when $c\neq 0$, and $g$ is Chern flat, namely $R=0$, when $c=0$.