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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 effective cone of effective divisors, $Eff(X)$ 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$NS(X)$. 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.

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If I'm not mistaken, your

Here's an example where $B_{\dim X}$ looks asymptotically like the complement B_1$is not a finite union of the negative cones: Let$X$be a K3 surface of Picard number 3, such that the effective cone$Eff(X)$is one of the components of the$X$[D\in NS(X) | D^2\ge 0]$ (by Serre duality)such surfaces can be constructed as in this paper). So if you take a variety with non-rational polyhedral effective cone Then it is easy to see that $B_{\dim X}$ will $B_0=Eff(X), B_2=-Eff(X) \mbox{ and } B_1= (B_0\cup B_2)^c $$In particular, B_1 is not be a finite union of convex cones, right?. As for the the 'General question', I think the shapes of the B_i are related to Alex Kuronya's asymptotic cohomological functions. These are basically higher cohomology versions of the volume function of a big line bundle and measure the asymptotic growth of cohomology. The definition is$$ \hat{h}^{i}(X,D) = \limsup_{m}\frac{h^i(X,O_X(mD))}{m^n/n!} . $$One of the main theorems in his paper is that the \hat{h}^i define continuous functions on the Neron-Severi space NS(X). The vanishing of these functions should be related to your question. See his paper for a lot of examples of asymptotic cohomology vanishing (flag varieties, abelian varieites,..). In most of these examples it is clear that the regions of vanishing cohomology are unions of convex cones. These functions have been used to study certain positivity conditions of line bundles. For example, in the paper "Higher cohomology of divisors on a projective variety" by T. de Fernex, A. Kuronya, R. Lazarsfeld, the authors show that a divisor D is ample if and only if the higher asymptotic cohomological functions vanish in a neighbourhood of D in NS(X). There is also the concept of q-ampleness, introduced by Demailly-Peternell-Schneider, Arapura, and Totaro among others. This is a generalization of the notion of an ample line bundle in the sense that high tensor powers of a line bundle are required to kill cohomology of coherent sheaves in degrees >q (so 0-ampleness coincides with ordinary ampleness.). This is related to \hat{h}^i(X,D) in the sense that it is expected that the local vanishing of the \hat{h}^i(X,D) in degrees >q is equivalent to q-ampleness of D. In general it is known (and easy to prove) that the cones of q-ample line bundles are star-shaped. 2 added 101 characters in body If I'm not mistaken, your B_{\dim X} looks asymptotically like the complement of the negative of the effective cone of X (by Serre duality). So if you take a variety with non-rational polyhedral effective cone B_{\dim X} will not be a finite union of convex cones, right? As for the the 'General question', I think the shapes of the B_i are related to Alex Kuronya's asymptotic cohomological functions. These are basically higher cohomology versions of the volume function of a big line bundle and measure the asymptotic growth of cohomology. The definition is$$ \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$NS(X)$. 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.

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