7 problems

Let $X$ be a complex projective manifold and let $E$ be a holomorphic vector bundle on $X$ that admits a holomorphic connection. Atiyah conjecture. The bundle $E$ always admits a f…

Generalized connection conjecture. The conclusion of Corollary 1.6 should hold for $M$: both $E$ and its holomorphic linear connection are trivial.

Torus-covering conjecture. If $M$ bears a holomorphic connection on $TM$, then $M$ admits a finite unramified covering by a compact complex torus.

Let $M$ be a simply connected compact complex manifold with trivial canonical bundle. Trivial-canonical-bundle nonexistence conjecture. Then $M$ does not admit any holomorphic proj…

Let $M$ be a simply connected compact complex manifold bearing a holomorphic projective connection. Simply connected holomorphic projective connection conjecture. Then $M$ is isomo…

Let $M$ be a compact complex manifold bearing a holomorphic projective connection $\phi$. Assume that $M$ does not admit any global holomorphic torsionfree affine connection projec…

Cartan geometry model conjecture. Every compact complex simply connected manifold admitting a holomorphic Cartan geometry with model $(G,H)$ is biholomorphic to the model $G/H$.