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**An action of a Lie algebra on the homology groups of moduli spaces of stable sheaves.**
*(English)*
Zbl 1219.14013

Nakamura, Iku (ed.) et al., Algebraic and arithmetic structures of moduli spaces. Proceedings of the conference, Sapporo, Japan, September 2007. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-931469-59-4/hbk). Advanced Studies in Pure Mathematics 58, 403-459 (2010).

Let \(X\) be a smooth projective surface over \(\mathbb C\) and let \(H\) be an ample divisor on \(X\). Assume that \(X\) is a \(K3\) surface. Let \(M_H(v)\) be the moduli space of \(H\)-stable sheaves \(E\) with the Mukai vector \(v(E)=v\).

While studying in [J. Reine Angew. Math. 515, 97–123 (1999; Zbl 0940.14026)] a special type of Fourier-Mukai transforms, the author introduced the Brill-Noether locus of the moduli space. Similar results were obtained by E. Markman in [J. Algebr. Geom. 10, No. 4, 623–694 (2001; Zbl 1074.14525)]. The Brill-Noether locus turns out to be a Grassmannian bundle over a smooth manifold. A similar Grassmannian structure appears in the paper by H. Nakajima [Duke Math. J. 91, No. 3, 515–560 (1998; Zbl 0970.17017)].

Based on the desciption of the Brill-Noether locus mentioned above, N. Nakajima constructed in [Contemp. Math. 322, 75–87 (2003; Zbl 1064.14043)] an \(\mathfrak{sl}_2\)-action on \(\bigoplus_v H_*(M_H(v), \mathbb C)\), where \(v\) runs through a suitable set of Mukai vectors satisfying some minimality conditions, and \(H_*\) denotes the Borel-Moore homology group.

The paper under review aims to generalize Nakajima’s result. The author constructs a Lie algebra action on the homology groups of moduli spaces of stable sheaves.

For a collection of exceptional sheaves \(S=\{E_1,\dots, E_s\}\), i. e., \(\mathrm{Ext}^1(E_i, E_i)=0\), satisfying some technical conditions, the author defines certain exact triangles in the bounded derived category of coherent sheaves on \(X\), called universal extensions and universal divisions.

This constitutes the main idea of the paper as it allows to construct operators \(e_i\), \(h_i\), \(f_i\), \(i=1,\dots, s\), on \(\bigoplus_v H_*(M_H(v), \mathbb C)\) that satisfy the commutation relations for Chevalley generators. To be precise, this allows to show \([e_i, f_j]=0\), \(i\neq j\); other arguments are included in Nakajima’s papers mentioned above.

The paper consists of an introduction and 7 sections. Some important facts about moduli of stable sheaves of minimal degree as well as the definitions of a universal extension and a universal division are collected in Section 1. An action of a Lie algebra on \(\bigoplus_v H_*(M_H(v), \mathbb C)\), the main result of the paper, is constructed in Section 2. In Section 3, examples of such actions are given in the cases of \(K3\) and Enriques surfaces. Section 4 deals with actions associated to purely \(1\)-dimensional exceptional sheaves. In particular, the author constructs an action of the affine Lie algebra associated to a singular fibre of an elliptic surface. In Section 5, a remark for the moduli of equivariant sheaves is given. Examples of actions of affine Lie algebras on the moduli of stable perverse sheaves on a resolution of a rational double point are presented in Section 6. The last section of the paper is an appendix, where some facts about moduli of coherent systems and the existence of semistable sheaves on \(K3\) surfaces and rational elliptic surfaces are presented.

For the entire collection see [Zbl 1193.14002].

While studying in [J. Reine Angew. Math. 515, 97–123 (1999; Zbl 0940.14026)] a special type of Fourier-Mukai transforms, the author introduced the Brill-Noether locus of the moduli space. Similar results were obtained by E. Markman in [J. Algebr. Geom. 10, No. 4, 623–694 (2001; Zbl 1074.14525)]. The Brill-Noether locus turns out to be a Grassmannian bundle over a smooth manifold. A similar Grassmannian structure appears in the paper by H. Nakajima [Duke Math. J. 91, No. 3, 515–560 (1998; Zbl 0970.17017)].

Based on the desciption of the Brill-Noether locus mentioned above, N. Nakajima constructed in [Contemp. Math. 322, 75–87 (2003; Zbl 1064.14043)] an \(\mathfrak{sl}_2\)-action on \(\bigoplus_v H_*(M_H(v), \mathbb C)\), where \(v\) runs through a suitable set of Mukai vectors satisfying some minimality conditions, and \(H_*\) denotes the Borel-Moore homology group.

The paper under review aims to generalize Nakajima’s result. The author constructs a Lie algebra action on the homology groups of moduli spaces of stable sheaves.

For a collection of exceptional sheaves \(S=\{E_1,\dots, E_s\}\), i. e., \(\mathrm{Ext}^1(E_i, E_i)=0\), satisfying some technical conditions, the author defines certain exact triangles in the bounded derived category of coherent sheaves on \(X\), called universal extensions and universal divisions.

This constitutes the main idea of the paper as it allows to construct operators \(e_i\), \(h_i\), \(f_i\), \(i=1,\dots, s\), on \(\bigoplus_v H_*(M_H(v), \mathbb C)\) that satisfy the commutation relations for Chevalley generators. To be precise, this allows to show \([e_i, f_j]=0\), \(i\neq j\); other arguments are included in Nakajima’s papers mentioned above.

The paper consists of an introduction and 7 sections. Some important facts about moduli of stable sheaves of minimal degree as well as the definitions of a universal extension and a universal division are collected in Section 1. An action of a Lie algebra on \(\bigoplus_v H_*(M_H(v), \mathbb C)\), the main result of the paper, is constructed in Section 2. In Section 3, examples of such actions are given in the cases of \(K3\) and Enriques surfaces. Section 4 deals with actions associated to purely \(1\)-dimensional exceptional sheaves. In particular, the author constructs an action of the affine Lie algebra associated to a singular fibre of an elliptic surface. In Section 5, a remark for the moduli of equivariant sheaves is given. Examples of actions of affine Lie algebras on the moduli of stable perverse sheaves on a resolution of a rational double point are presented in Section 6. The last section of the paper is an appendix, where some facts about moduli of coherent systems and the existence of semistable sheaves on \(K3\) surfaces and rational elliptic surfaces are presented.

For the entire collection see [Zbl 1193.14002].

Reviewer: Oleksandr Iena (Trieste)

### MSC:

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

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |

14J28 | \(K3\) surfaces and Enriques surfaces |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |