3 problems

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

Local homogeneity conjecture. Holomorphic $\operatorname{SL}(2,\mathbb C)$-structures on compact complex manifolds of odd dimension are always locally homogeneous.

Local homogeneity conjecture. For higher complex odd dimensions, holomorphic ${\rm SL}_2({\mathbb C})$-geometries on compact complex manifolds are always locally homogeneous.