3 Question 2 edited: now B_i is a asked to be a "union of cones"

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 cone''cones''? Specifically

For example, do there exist $f_i: NS(X)\to \mathbb{Z}$, when $c_i\in \mathbb{Z}$ such that X=G/B$then$H^i(X, L) = 0$whenever B_i=A_i$ is a union of the interiors of the Weyl chambers corresponding to the length $f([L]) < c_i$?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$and in a sense they look similar. Note. I'm sure MO users will quickly post counterexamples or comment on how I could make the questions more precise or reasonable. If that is okay with MO policy, I plan to edit the question to make it more complete and less silly. 2 f in Question 2 now depends on i For sure answers to my questions are well known - but I never saw them anywhere. Let$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$look like a ,,translated strictly convex cone''? Specifically, does do there exist a$f: f_i: NS(X)\to \mathbb{Z}$,$c_i\in \mathbb{Z}$such that$H^i(X, L) = 0$whenever$f([L]) < c_i$? Question 3. What can one say about the intersections of$B_i$(again far away from zero)? Motivation. The only examples I know pretty well are curves, abelian varieties and$G/B$and in a sense they look similar. Note. I'm sure MO users will quickly post counterexamples or comment on how I could make the questions more precise or reasonable. If that is okay with MO policy, I plan to edit the question to make it more complete and less silly. 1 # Cohomology of line bundles For sure answers to my questions are well known - but I never saw them anywhere. Let$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$look like a ,,translated strictly convex cone''? Specifically, does there exist a$f: NS(X)\to \mathbb{Z}$,$c_i\in \mathbb{Z}$such that$H^i(X, L) = 0$whenever$f([L]) < c_i$? Question 3. What can one say about the intersections of$B_i$(again far away from zero)? Motivation. The only examples I know pretty well are curves, abelian varieties and$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.