Bounding patterns for the cohomology of vector bundles

Abstract:

Let $t \in \mathbb {N}$, let $K$ be a field and let $\mathcal {V}^t_K$ denote the class of all algebraic vector bundles over the projective space $\mathbb {P}^t_K$.

The cohomology table of a bundle $\mathcal {E} \in \mathcal {V}^t_K$ is defined as the family of non-negative integers $h_{\mathcal {E}}:= \big (h^i(\mathbb {P}^t_K,\mathcal {E}(n))\big )_{(i,n) \in \mathbb {N}_0 \times \mathbb {Z}}$.

A set $\mathbb {S} \subseteq \{0,\ldots ,t\}\times \mathbb {Z}$ is said to be a bounding pattern for the cohomology of vector bundles over $\mathbb {P}^t_K$ if for each family $(h^{(i,n)})_{(i,n) \in \mathbb {S}}$ of non-negative integers, the set of cohomology tables \[ \{h_{\mathcal {E}} \mid \mathcal {E}\in \mathcal {V}^t_K : \ h^i_{\mathcal {E}}(n) \leq h^{(i,n)} \mbox { for all} \ (i,n) \in \mathbb {S}\}\] is finite. Our main result says that this is the case if and only if $\mathbb {S}$ contains a quasi-diagonal of width $t$, that is, a set of the form \[ \{(i,n_i)| \ i=0,\ldots ,t\} \mbox { with integers } n_0> n_1 > \cdots > n_t.\]