# Examples of surfaces with negative Kahler curvature operator

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 $\geq 1$) ?

• 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? Jun 5 '18 at 9:34
• 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 ? Jun 5 '18 at 9:43
• 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. Jun 5 '18 at 9:47
• Which kind of curvature are you looking at? Holomorphic sectional? Holomorphic bisectional? Kähler-Ricci? Scalar? Riemannian sectional? Jun 5 '18 at 10:24
• 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. Jun 5 '18 at 10:27