Moduli space of principal sheaves over projective varieties


Let $G$ be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal $G$-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan’s notion and construction to higher dimension, allowing also objects which we call semistable principal $G$-sheaves, in order to obtain a projective moduli space: a principal $G$-sheaf on a projective variety $X$ is a triple $(P,E,\psi)$, where $E$ is a torsion free sheaf on $X$, $P$ is a principal $G$-bundle on the open set $U$ where $E$ is locally free and $\psi$ is an isomorphism between $E|_U$ and the vector bundle associated to $P$ by the adjoint representation.

We say it is (semi)stable if all filtrations $E_\bullet$ of $E$ as sheaf of (Killing) orthogonal algebras, i.e. filtrations with $E^\perp_i=E_{-i-1}$ and $[E_i,E_j]\subset E_{i+j}^{\quad\vee\vee}$, have \[ \sum (P_{E_i}\operatorname{rk} E – P_E \operatorname{rk} E_i)\,(\preceq)\,0 , \] where $P_{E_i}$ is the Hilbert polynomial of $E_i$. After fixing the Chern classes of $E$ and of the line bundles associated to the principal bundle $P$ and characters of $G$, we obtain a projective moduli space of semistable principal $G$-sheaves. We prove that, in case $\dim X=1$, our notion of (semi)stability is equivalent to Ramanathan’s notion.


Tomás Gómez

Serrano 113 bis
28006 Madrid

Ignacio Sols

Departamento de Algebra
Facultad de Ciencias Matemáticas
Universidad Complutense of Madrid
28040 Madrid