Non-vanishing cohomology of line bundles on projective varieties in prime characteristic?

Question

This is a somewhat naive question about the expected non-vanishing behavior of sheaf cohomology groups $H^i(X, \mathcal{L})$, where $X$ is a smooth projective variety of dimension $d$ over an algebraically closed field of characteristic $p>0$ and $\mathcal{L}$ is a line bundle on $X$. Here one starts with a few general principles: cohomology is finite-dimensional and vanishes in degrees outside the interval $0 \leq i \leq d$, while Serre duality holds.

Though my knowledge of algebraic geometry is sketchy at best, I am motivated by the case when $X$ is a flag variety for a connected reductive algebraic group $G$ and the line bundle comes from a character of some maximal torus of $G$ (lifted to a Borel subgroup). Unlike the nice classical theory in characteristic 0 where cohomology can be nonzero in at most one degree and then affords an irreducible representation of $G$, the situation in prime characteristic becomes intricate. Results here, due mainly to Kempf and Andersen, provide a few important general theorems as well as some low rank examples. But the associated representation theory is poorly understood and seems to be determined somehow in terms of Kazhdan-Lusztig theory for the affine Weyl group of Langlands dual type relative to $p$ in place of the usual Weyl group.

Two features occur frequently in low ranks and may hold in general, so I wonder if they are seen elsewhere in the world of smooth projective varieties. Here it's just a question of nonvanishing cohomology rather than more subtle module structure:

(A) Connectedness of the interval of degrees between 0 and $d$ in which non-vanishing occurs.

(B) Occurrence of non-vanishing in no more than $\frac{d}{2} +1$ degrees.

Is either (A) or (B) usual/unusual in other cases of non-vanishing cohomology in prime characteristic? (And are there relevant examples, references?)

ADDED: The responses show that in some respects my question is too broad and therefore naive, since projective varieties come in so many shapes and sizes. Flag varieties are homogeneous spaces for a nice algebraic group (in any characteristic), though the cohomology of line bundles mysteriously gets far more complicated when the characteristic is prime. It's hard to trace the breakdown of most of the classical Borel-Weil-Bott theory, which motivates my question. At the same time, it's hard to see the role of geometry here apart from the related group theory.

By the way, (A) above is more speculative than (B) due to possible over-reliance on low rank examples. But neither property of non-vanishing has an obvious source in the geometry of the situation. I'm still interested in seeing similar examples elsewhere involving constraints on degrees where non-vanishing can occur.

Answer 1

The following generalization of Kodaira vanishing theorem is due to Deligne and Illusie and can be found in the book of Esnault-Viehweg "Lectures on vanishing theorems", p. 43.

Theorem.

Let $X$ be a smooth projective variety defined over an algebraically closed field $k$ and $\mathcal{L}$ an ample invertible sheaf. If $char(k)=p >0$ assume in addition that $X$ and $\mathcal{L}$ admit a lift to the ring $W_2(k)$ of the Witt vectors and $\dim(X) \leq p$. Then

$H^i(X, \mathcal{L}^{-1})=0$ for $i < \dim(X)$.

Therefore, if $\mathcal{L}$ is ample, one has non-vanishing of the cohomology of $\mathcal{L}^{-1}$ at most in degree $\dim(X)$.

Answer 2

I am not sure to which extent your questions are really related to positive characteristic.

The obvious difference between positive characteristic and characteristic zero related to the question is of course failure of Kodaira vanishing in positive characteristic. However, that does not come up for flag varieties as Kodaira vanishing is true for them, either by the Deligne-Illusie result mentioned by Francesco or by Frobenius splitting (though interestingly enough one get examples of Kodaira non-vanishing by looking at "inseparable flag spaces"). The examples that the OP mentions are non-ample line bundles.

If one drops (or never introduces it in the first place) the ampleness condition that almost anything can happen but that is true in all characteristics.

To answer questions A and B one can take the product $X:=X_1\times X_2\times\cdots\times X_k$, with $n_i:=\dim X_i$, of smooth hypersurfaces in projective spaces of high enough degree so that $H^{n_i}(X_i,\mathcal O_{X_i})\ne0$, we always have $H^j(X_i,\mathcal O_{X_i})=0$ for $0<j<n_i$. The Künneth formula then gives that $H^*(X,\mathcal O_X)=\bigotimes_iH^*(X_i,\mathcal O_{X_i})$ and hence the degrees of non-vanishing are all sums $n_{i_1}+\cdots+n_{i_r}$ for $i_1<i_2<\cdots<i_r$. We can do the same trick for various other line bundles on the $X_i$ than the trivial ones.

Of course one expects to have worse behaviour in positive, we can for instance probably get examples as above with anti-ample line bundles.

Answer 3

For a line bundle $L$ on an Abelian variety, assuming that the group $K(L)$ is finite, there is a unique degree in which the cohomology is not zero, see Mumford, Abelian varieties, p. 150.

(The condition that the group $K(L)$ is finite is an algebraic variant of the non-degeneracy of the Riemann form for an Abelian variety over the complex numbers ; in that case, the above degree is the number of negative signs. In any characteristic, this condition can also be translated into a non-degeneracy on the $\ell$-adic Weil form.)