2 problems

Let $M$ be a compact complex manifold homeomorphic to a torus, and suppose that $M$ admits a holomorphic $(X,G)$-structure. Complex-torus characterization conjecture. Then $M$ is b…

Let $P$ be a complex Lie group and let $\boldsymbol{\pi}$ be a discrete cocompact subgroup, so that $P/\pi$ is a compact quotient. Invariant-structure conjecture. Every holomorphic…