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.)