Abstract
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.
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