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, do there exist $f_i: NS(X)\to \mathbb{Z}$, $c_i\in \mathbb{Z}$ 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.

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.

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.

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, do there exist $f_i: NS(X)\to \mathbb{Z}$, $c_i\in \mathbb{Z}$ 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.