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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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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Examples of compact Kähler manifolds with non positive holomorphic bisectional curvature are given by:

- Closed submanifolds of complex tori.
- 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.

Fibred surfaces, varieties isogenous to a product and related moduli spaces, American Journal of Mathematics, Volume122. Have a look at this article and at the papers citing it on MathSciNet. $\endgroup$ – Francesco Polizzi Jun 5 '18 at 9:47