The Hadamard theorem states that： any complete and simply connected manifold M with nonpositive sectional curvature is diffeomorphic to the euclidean space.
Is it also true if we assume that $M$ is a Hermitian manifold and replace sectional curvature with holomorphic sectional curvature？ If it’s not enough，how about changing nonpositive sectional curvature to semi-negative curvature form？
Thank you for your answer！
Best Answer
This is very much false. There used to be Yau's conjecture (dating to 1982-1983) about such manifolds, but it was disproven few years ago in
Jean-Paul Mohsen, Construction of negatively curved complete intersections.
In particular, there are compact simply connected Kahler manifolds of negative holomorphic sectional curvature (and even bisectional curvature). These manifolds are certain complete intersections in complex-projective spaces, the Kahler metric is the one induced from the ambient FS-metric. The difficulty is to ensure negative holomorphic curvature: The latter goes down for submanifolds but the ambient metric has positive holomorphic curvature.