Fix a Weil cohomology theory. If I give you a presentation of a smooth projective scheme over an algebraically closed field, do you have an explicit algorithm for computing its cohomology groups? Presentations can be either
- the defining equations in the projective space, or
- an affine open cover and the glueing maps (because of separatedness, we should be able to give a cover where intersections are also affine). I am not sure you can you check given only this data if the resulting scheme is in fact smooth projective, but you have my word that it is.
If working with random Weil cohomology theories is hard, you can give an answer where you treat only the famous ones ($l$-adic, crystalline, de Rham, maybe something else).
P.S. Bonus points if you can also compute the integral structure.