5 problems

Holomorphic Markus conjecture. A compact complex manifold with a flat holomorphic Riemannian metric should be a quotient of

Local homogeneity conjecture. Holomorphic Riemannian metrics on compact complex manifolds are locally homogeneous.

Local homogeneity conjecture. Any such structure $\phi$ is locally homogeneous. This implies that if $\phi$ is rigid, then the fundamental group of $X$ is infinite.

Local homogeneity conjecture. Any holomorphic Riemannian metric on $X$ is locally homogeneous.

Let $M$ be a compact complex manifold endowed with a holomorphic Riemannian metric of constant non-vanishing curvature. The dimension of $M$ is the complex dimension. Dumitrescu–Ze…