Example of a variety with explicit cohomology ring and Kahler cone

Question

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?

Answer 1

You could look at surfaces with maximal Picard rank. A surface is said to have maximal Picard rank if $H^{1,1}(X) \cap H^1(X, \mathbb{R})$ is spanned by curve classes. So the Kahler cone is the same as the ample cone, and you can compute the intersection pairing on $H^{1,1}$ by just intersecting curves with each other.

Now, this raises the question of whether there is a surface of general type with maximal Picard rank. Some googling found me the following:

Section 3 of Quintic surfaces with maximum and other Picard numbers shows that the Fermat sextic $w^6+x^6+y^6+z^6=0$ has maximal Picard rank. Apparently, this was originally computed by Beauville.

Roulleau shows that there are infinitely many Fano surfaces with maximal Picard rank, and gives a basis for the Neron-Severi group. (Confusingly, the Fano surface is not a Fano variety, but rather is of general type.)

A warning: If you look at this paper or this one, they compute a lot of surfaces with maximal Picard. However, their method is to make a surface with an $A_n$ singularity and prove that its resolution has maximal Picard. Since this resolution has a $-2$ curve, it is automatically not of general type. Sorry. As Donu says, I was confused. $K$ will not be ample, but it will be big, and the varieties are general type.

I was coming back to edit in one more idea -- if $C$ is the Klein quartic, then I think that $C \times C$ has maximal Picard rank. But Jason Starr's comment is simpler than any of my suggestions.