Date & Time: Tuesday, July 05, 2016, 16:00-17:00.
Venue: Ramanujan Hall

Speaker: Markus Brodmann,
Institut für Mathematik, Universität Zürich

Title: Bounding cohomology of coherent sheaves over projective schemes

Abstract: Let $X = (X,\mathcal{O}_X)$ be a projective scheme over a field and with twisting sheaf $\mathcal{O}_X(1).$ Let $\mathcal{F}$ be a coherent sheaf of $\mathcal{O}_X$-modules. The \textit{cohomology table} of $\mathcal{F}$ is defined as the family $$h_{\mathcal{F}} := \big(h^i(X,\mathcal{F}(n))\big)_{(i,n) \in \mathbb{N}_0 \times \mathbb{Z}}.$$ We give a survey on results about the set of cohomology tables $$h_{\mathcal{C}} = \{h_{\mathcal{F}} \mid (X,\mathcal{F}) \in \mathcal{C}\},$$ for certain classes $\mathcal{C}$ of pairs $(X,\mathcal{F}).$ We particularly look at the following questions: \begin{itemize} \item[\rm{(1)}] Under which conditions is the set $h_{\mathcal{C}}$ finite, if $\mathcal{C}$ is the class of all pairs $(X,\mathcal{F})$ for which $\mathcal{F}$ has a given dimension? \item[\rm{(2)}] Under which conditions is the set $h_{\mathcal{C}}$ finite, if $\mathcal{C}$ is the class of all pairs $(X,\mathcal{F})$ in which $X = \mathbb{P}`r$ is a given projective space and $\mathcal{F}$ is an algebraic vector bundle over $X$ ? \item[\rm{(3)}] What can be said if $X$ runs throught all smooth complex projective surfaces and $\mathcal{F} =\mathcal{O}_X$ is the structure sheaf of $X$?