Cohomology Vanishing, Koszul Cohomology and Multigraded Regularity on Projective Varieties

Abstract

In this article, we prove a general cohomology vanishing theorem on arbitrary projective varieties within the framework of multigraded Castelnuovo--Mumford regularity. In particular, we apply this vaishing theorem to prove the vanishing of Koszul cohomology groups $K_{p,q}(X;F,L)$, where $L=B_1^{w_1}\otimes\cdots\otimes B_t^{w_t}$, the line bundles $B_1,\ldots,B_t$ are globally generated, and $F$ is a vector bundle. We also introduce the $K_{p,q}$-hierarchy, providing a unified perspective on the vanishing criteria for Properties $(N_{p})$ and $(M_{q})$ while shedding light on the vanishing of mixed-weight syzygies. These results generalize earlier work in \cite{HeringSchenckSmith}, \cite{GallegoPurnaprajnaII}, and \cite{Basu}. Furthermore, we give a complete description of minimal multigraded regularities of line bundles on arbitrary products of projective spaces. We also recover Green's vanishing theorem for projective space via a direct regularity argument, giving an alternative proof. Finally, we compute the complete graded Betti table of the canonical image of a hyperelliptic curve by combining our vanishing theorem with Green's duality.