I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
- Possibly singular (but still normal) toric varieties
- Relative dR cohomology of a (toric) morphism of toric varieties
- Characteristic $p$
- Calculation described in fan-theoretic terms (polytope is second-best)
For the relative cohomology of a morphism, by far the most important cases are where the morphism is relative dimension 1, or where the base is dimension 1.
In addition to ordinary de Rham, I would also be interested in logarithmic de Rham cohomology.
The singularities and the positive characteristic usually throw a big wrench into de Rham calculations, but I was really hoping that things would be nicer in the toric situation.
I would be really grateful for some references!
