The derived moduli space of stable sheaves.

*(English)*Zbl 1327.14058The 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.

Let \(Y\) 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.

The 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 stable graded \(A\)-modules of given dimension vector. In Section 3 the moduli of sheaves are studied. \(A\) is taken to be the homogeneous coordinate ring of a projective scheme \(Y\). The stability notions for sheaves on \(Y\) and for graded \(A\)-modules are compared here. Section 4 presents a construction of the differential graded moduli scheme of stable sheaves on a projective variety \(Y\).

Let \(Y\) 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.

The 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 stable graded \(A\)-modules of given dimension vector. In Section 3 the moduli of sheaves are studied. \(A\) is taken to be the homogeneous coordinate ring of a projective scheme \(Y\). The stability notions for sheaves on \(Y\) and for graded \(A\)-modules are compared here. Section 4 presents a construction of the differential graded moduli scheme of stable sheaves on a projective variety \(Y\).

Reviewer: Oleksandr Iena (Luxembourg)

##### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14L24 | Geometric invariant theory |

17B70 | Graded Lie (super)algebras |

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\textit{K. Behrend} et al., Algebra Number Theory 8, No. 4, 781--812 (2014; Zbl 1327.14058)

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