The derived moduli space of stable sheaves
type: Item
instance of: scholarly article
title: The derived moduli space of stable sheaves
zbMATH Open document ID: 1327.14058
published in: Algebra \& Number Theory
publication date: 2014-08-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.1884
reviewed by: Oleksandr Iena
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 6331622
zbMATH Keywords: differential graded schemes
zbMATH Keywords: curved differential graded Lie algebras
zbMATH Keywords: stable sheaves
MaRDI profile type: Publication
OpenAlex ID: W3100167648
arXiv ID: 1004.1884
DOI: 10.2140/ANT.2014.8.781
author: Kai Behrend
author: Ionuţ Ciocan-Fontanine
author: Junho Hwang
author: Michael Rose
review text: The authors construct the derived scheme of stable sheaves on a smooth projective variety via derived moduli of finite graded modules over a graded ring. This is done by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov by a suitable algebraic gauge group. It is shown that the natural notion of GIT stability for graded modules reproduces stability for sheaves.NEWLINENEWLINELet \(Y\) be a smooth projective variety over \(\mathbb C\). Let \(A\) be the homogeneous coordinate ring of \(Y\). One describes a coherent sheaf \(\mathcal F\) on \(Y\) with Hilbert polynomial \(\alpha(t)\in \mathbb Q[t]\) using a finite-dimensional graded \(A\)-module \(\Gamma_{[p, q]}\mathcal F=\bigoplus_{i=p}^q\Gamma(Y, \mathcal F(i))\) with dimension vector \(\alpha|_{[p, q]}=(\alpha(p),\dots, \alpha(q))\). For big \(p\) and \(q\) this describes an open embedding of moduli functors. The authors construct a finite-dimensional differential graded Lie algebra \(L=\bigoplus_{n\geq 0} L^n\) with an algebraic gauge group \(G\) linearly acting on \(L\) such that the quotient \(MC(L)/G\) of the solutions \(MC(L)\) of the Maurer-Cartan equation \(dx+\frac{1}{2}[x, x]\), \(x\in L^1\), by the gauge group \(G\) is equal to the stack of graded \(A\)-modules concentrated in degrees between \(p\) and \(q\). The embedding of functors mentioned above automatically gives a derived or differential graded structure on the moduli space of sheaves. The GIT stability for the algebraic gauge group action reproduces the standard notion of stability for sheaves.NEWLINENEWLINEThe paper under review consists of an introduction and four sections. In Section~1 the derived scheme of finite-dimensional graded \(A\)-modules with fixed dimension vector is constructed for a unital graded \(\mathbb C\)-algebra \(A\). Various differential graded schemes and stacks are described here. In Section~2 the authors study stability. They construct quasi-projective derived moduli spaces of equivalence classes of st…
description: scientific article; zbMATH DE number 6331622