For sure answers to my questions are well known – but I never saw them anywhere.
Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the subset of $\text{Pic }X$ of all line bundles $L$ with nonzero $H^i(X, L)$.
General question. What does $(\text{Pic }X, A_0, \ldots, A_d)$ ($d=\dim X$) look like when seen from far far away?
Here are some more specific questions:
Question 1. Does the property $L\in A_i$ depend only on the numerical class of $L$, for $L$ ,,large enough''? Precisely: does there exist a bounded region $C$ in $NS(X)$ such that for all $L\in \text{Pic }X$, $M\in \text{Pic}^\tau X$ we have $\dim H^i(X, L) = \dim H^i (X, L\otimes M)$ when $L\notin C$?
Let $B_i$ be the image of $A_i$ in $NS(X)$.
Question 2. Does $B_i$ look like a union of finitely many ,,translated strictly convex cones''?
For example, when $X=G/B$ then $B_i=A_i$ is a union of the interiors of the Weyl chambers corresponding to the length $i$ elements of the Weyl group, shifted by half the canonical class.
Question 3. What can one say about the intersections of $B_i$ (again far away from zero)?
E.g. in the above example of $G/B$, there is at most one non-vanishing cohomology group. This seems to hold for many varieties as soon as we are ,,far away from zero''. So in addition to ,,ample directions'' and ,,anti-ample directions'' (Serre duality) there seem to be ,,$H^i$-directions'' as well… As far as I remember, something similar holds for abelian varieties.
Motivation. The only examples I know pretty well are curves, abelian varieties and $G/B$ and in a sense they look similar.
Note. I'm sure MO users will quickly post counterexamples or comment on how I could make the questions more precise or reasonable. If that is okay with MO policy, I plan to edit the question to make it more complete and less silly.
Best Answer
Here's an example where $B_1$ is not a finite union of cones: Let $X$ be a K3 surface of Picard number 3, such that the cone of effective divisors, $Eff(X)=Nef(X)$ is one of the components of the $[D\in NS(X) | D^2\ge 0]$ (for example, a K3 surface without $(-2)$-curves, such surfaces can be constructed as in this paper). In this case $Eff(X)$ is non-rational polyhedral. Then it is easy to see that $$ B_0=Eff(X), B_2=-Eff(X) \mbox{ and } B_1= (B_0\cup B_2)^c $$In particular, $B_1$ is not a finite union of cones.
As for the the 'General question', I think the shapes of the $B_i$ are related to Alex Kuronya's asymptotic cohomological functions. These are basically higher cohomology versions of the volume function of a big line bundle and measure the asymptotic growth of cohomology. The definition is $$ \hat{h}^{i}(X,D) = \limsup_{m}\frac{h^i(X,O_X(mD))}{m^n/n!} . $$One of the main theorems in his paper is that the $\hat{h}^i$ define continuous functions on the Neron-Severi space $NS(X)=A^1(C)\otimes \mathbb{R}/\equiv$. The vanishing of these functions should be related to your question.
See his paper for a lot of examples of asymptotic cohomology vanishing (flag varieties, abelian varieites,..). In most of these examples it is clear that the regions of vanishing cohomology are unions of convex cones.
These functions have been used to study certain positivity conditions of line bundles. For example, in the paper "Higher cohomology of divisors on a projective variety" by T. de Fernex, A. Kuronya, R. Lazarsfeld, the authors show that a divisor $D$ is ample if and only if the higher asymptotic cohomological functions vanish in a neighbourhood of $D$ in $NS(X)$.
There is also the concept of $q-$ampleness, introduced by Demailly-Peternell-Schneider, Arapura, and Totaro among others. This is a generalization of the notion of an ample line bundle in the sense that high tensor powers of a line bundle are required to kill cohomology of coherent sheaves in degrees $>q$ (so $0$-ampleness coincides with ordinary ampleness.). This is related to $\hat{h}^i(X,D)$ in the sense that it is expected that the local vanishing of the $\hat{h}^i(X,D)$ in degrees $>q$ is equivalent to $q$-ampleness of $D$. In general it is known (and easy to prove) that the cones of $q$-ample line bundles are star-shaped.