I'm looking for some fairly explicit varieties to use as (counter?-)examples for my thesis and I'd appreciate any suggestions. I need a smooth projective variety $X$ of general type that satisfies:

- The Hodge number $h^{1,1}$ is at least 2.
- The cohomology ring of $X$ (or at least the subring generated by $(1,1)$-classes) is explicit.
- The Kahler cone of $X$ is known(-ish).

I want to calculate the sectional curvatures of the Riemannian metric on the Kahler cone of $X$ which is defined by the intersection product. These curvatures may be expressed by the intersection product on the subring $A$ of $H^*(X)$ genereated by degree $(1,1)$ classes, which explains the conditions I impose.

The condition on the Hodge number is necessary, since when $h^{1,1} = 1$ one ends up with the metric $g(x,y)(t) = xy/t$ on the half-line $\mathbb R_+$ and not many interesting things remain unsaid about this case. This excludes most hypersurfaces in $\mathbb P^n$, except perhaps for those in $\mathbb P^3$.

I must also exclude the example of a blowup of several points of a variety $X$ with $h^{1,1} = 1$, since one can calculate explicitly what happens in this case. Are there other relatively easy examples?