# ag.algebraic geometry – nonvanishing higher cohomology of a very ample divisor

I am looking for smooth projective varieties $$X$$, with $$h^i(X, mathcal{O}_X) = 0$$ for $$i > 0$$, with a very ample line bundle $$L$$ with some nonvanishing higher cohomology.

What is clear:

(1) Curves will not work (because $$mathbf{P}^1$$ is the only such curve)

(2) Fano varieties will not work, by Kodaira vanishing (in char. 0)

(3) Hypersurfaces and complete intersections will not work, because the only ones with $$h^i(X, mathcal{O}_X) = 0$$ for $$i > 0$$ will be Fano

Maybe $$X$$ can be a surface of non-negative Kodaira dimension with $$p_g = q = 0$$, e.g. general type or Enriques?

(Note that the question is easy if I just wanted ample $$L$$, not very ample, as $$L = mathcal{O}(K_X)$$ on a Godeaux surface or a fake projective plane would be such an example.)