Equivariant Koszul Cohomology of Canonical Curves
Kostas Karagiannis, Aristides Kontogeorgis, Konstantia Manousou Sotiropoulou
TL;DR
This work studies the representation-theoretic structure of Koszul cohomology under a finite group action on a smooth projective variety, with a focus on canonical curves. It develops a general framework of $G$-equivariant quasicoherent sheaves and $G$-functors, showing the Koszul complex is a complex of $kG$-modules and expressing its cohomology via $G$-equivariant global sections. For canonical curves, it derives explicit decompositions using equivariant Euler characteristics and Riemann–Roch, and uses Schur functors and generating functions to describe irreducible components and their characters, including Molien-type formulas in the representation ring. The results yield concrete, computable descriptions of equivariant Koszul representations and provide tools for calculating symmetric and wedge-power contributions under group actions, with potential applications to equivariant syzygies and symmetry-restricted geometric problems.
Abstract
This paper investigates the representation-theoretic structure of the Koszul cohomology of a smooth projective variety $X$ over an algebraically closed field $k$, admitting an action of a finite group $G$ of order coprime to ${\rm char}(k)$. Properties of $G$-equivariant functors are employed to show that the associated Koszul complex is a complex of $kG$-modules, and to generalize known dimension formulas to identities between virtual representations. In the case of canonical curves, explicit formulas are obtained by combining the theory of equivariant Euler characteristics and equivariant Riemann-Roch theorems with that of generating functions for Schur functors.