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Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.

Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces of genus $\geq 1$) ?

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    $\begingroup$ What if you take the quotient of the product of two higher genus Rieman surfaces by the free action of a finite group? You still have nonpositive curvature, right? $\endgroup$ – Francesco Polizzi Jun 5 '18 at 9:34
  • $\begingroup$ I see. Indeed...Thanks! I suppose the action can be chosen so that the resulting beast is not the product of two surfaces eh ? A follow up question - do you think such a beast will have non-trivial deformations of complex structure ? $\endgroup$ – Vamsi Jun 5 '18 at 9:43
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    $\begingroup$ There is a huge literature on the subject, starting with F. Catanese's article Fibred surfaces, varieties isogenous to a product and related moduli spaces, American Journal of Mathematics, Volume 122. Have a look at this article and at the papers citing it on MathSciNet. $\endgroup$ – Francesco Polizzi Jun 5 '18 at 9:47
  • $\begingroup$ Which kind of curvature are you looking at? Holomorphic sectional? Holomorphic bisectional? Kähler-Ricci? Scalar? Riemannian sectional? $\endgroup$ – diverietti Jun 5 '18 at 10:24
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    $\begingroup$ Anyway, since curvature in complex geometry decreases when passing to submanifolds, any surface which is a closed submanifold of a complex torus is an example of compact Kähler surface with non positive (whatever, except Riemannian sectional) curvature. $\endgroup$ – diverietti Jun 5 '18 at 10:27
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Examples of compact Kähler manifolds with non positive holomorphic bisectional curvature are given by:

  1. Closed submanifolds of complex tori.
  2. Smooth compact quotients of bounded symmetric domains.

Moreover, the second class has strictly negative holomorphic sectional curvature as well as strictly negative (constant, indeed) Ricci curvature (the Bergman metric is Kähler-Einstein as soon as the domain is homogeneous).

Also, very recently (see this paper on the ArXiv) Jean-Paul Mohsen has constructed lot of examples of complete intersections in projective space with, depending on the numerology, negative holomorphic bisectional curvature, or negative holomorphic sectional curvature, or negative Ricci curvature, or negative scalar curvature.

This is certainly not an exhaustive list.

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