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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.)

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    $\begingroup$ Take a hypersurface in $\mathbb{P}^2\times\mathbb{P}^1$ of bidegree $(d,1)$ for $d>3$ with the very ample invertible sheaf of bidegree $(1,1)$. $\endgroup$
    – Jason Starr
    Commented Oct 19, 2020 at 0:51
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    $\begingroup$ Thanks, Jason, that's great! I see that the projection of $X$ to the the first factor $\mathbf{P}^2$ identifies $X$ with a blow up of $\mathbf{P}^2$ in $d^2$ points obtained from a pair of intersecting degree $d$ plane curves, in particular $X$ is rational and so $p_g(X) = q(X) = 0$. Furthermore, I see how for large $d$, $\chi(L) < 0$, hence $L$ has nonvanishing $h^1$. $\endgroup$
    – Evgeny Shinder
    Commented Oct 19, 2020 at 11:12
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    $\begingroup$ You are welcome. $\endgroup$
    – Jason Starr
    Commented Oct 19, 2020 at 20:21

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