Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
type: Item
instance of: scholarly article
title: Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
zbMATH Open document ID: 1331.14023
author: Martijn Kool
published in: Geometriae Dedicata
publication date: 2015-06-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.3393
review text: The author computes Euler characteristics of moduli spaces \(\mathcal{M}\) of torsion free \(\mu-\)stable sheaves over a toric surface \(S\). The action of the torus \(T\) on \(S\) lifts to \(\mathcal{M}\) and the fixed point locus \(\mathcal{M}^T\), by the author's previous result [\textit{M. Kool}, Adv. Math. 227, No. 4, 1700--1755 (2011; Zbl 1227.14018)], consists of \(T-\)equivariant \(\mu-\)stable sheaves. Hence Klyachko's description of \(T-\)equivariant torsion free sheaves applies. The author follows Göttsche's strategy to reduce the computation of \(e(\mathcal{M}_S^H(r,c_1,c_2))\) to \(e(\mathcal{N}_S^H(r,c_1,c_2))\), where \(\mathcal{N}_S^H(r,c_1,c_2)\) is the moduli space of \(\mu-\)stable locally free sheaves with rank \(r\) and \(i\)-th Chern class \(c_i\). By \textit{A. A. Klyachko}'s result [Math. USSR, Izv. 35, No. 2, 337--375 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001--1039 (1989; Zbl 0706.14010)], the fixed point locus \(\mathcal{N}_S^H(r,c_1,c_2)^T\) can be stratified into finite strata, each of which can be realized as a good geometric \(\mathrm{SL}(r,\mathbb{C})-\)quotient of a locally closed subset \(\mathcal{D}_{v,\delta}^s\) of some flag variety, and has a very explicit combinatorial description. The main theorem of the paper expresses the generating function \(\displaystyle{\sum_{c_2}}e(\mathcal{M}_S^H(r,c_1,c_2))q^{c_2}\) in terms of \(e(\mathcal{D}_{v,\delta}^s/\mathrm{SL}(r,\mathbb{C}))\). Finally, the author applies the main theorem to the following examples: rank 1 case, rank 2 and 3 case on the projective plan, and rank 2 case on a Hirzebruch surface, also he explains the concordance between his results and the results in the literature.
reviewed by: Yao Yuan
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14N20
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 6446914
zbMATH Keywords: moduli spaces of sheaves
zbMATH Keywords: toric surfaces
zbMATH Keywords: generating functions
zbMATH Keywords: configurations of linear subspaces
MaRDI profile type: Publication
OpenAlex ID: W2057712419
arXiv ID: 0906.3393
cites work: Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces
cites work: Torus fixed points of moduli spaces of stable bundles of rank three
cites work: Torus action on the moduli spaces of torsion plane sheaves of multiplicity four
cites work: Q4780265
cites work: On the homology of the Hilbert scheme of points in the plane
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Algebraic stacks
cites work: The Betti numbers of the Hilbert scheme of points on a smooth projective surface
cites work: Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces
cites work: Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces
cites work: Stable reflexive sheaves
cites work: Q3128810
cites work: MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'
cites work: Configurations in abelian categories. IV: Invariants and changing stability conditions
cites work: On Equivariant Vector Bundles on an Almost Homogeneous Variety
cites work: Toric bundles and problems of linear algebra
cites work: EQUIVARIANT BUNDLES ON TORAL VARIETIES
cites work: Stable bundles, representation theory and Hermitian operators
cites work: Fixed point loci of moduli spaces of sheaves on toric varieties
cites work: The Betti numbers of the moduli space of stable sheaves of rank 3 on \({\mathbb{P}^2}\)
cites work: BPS invariants of \(\mathcal N=4\) gauge theory on Hirzebruch surfaces
cites work: BPS invariants of semi-stable sheaves on rational surfaces
cites work: Poincaré polynomials of moduli spaces of stable bundles over curves
cites work: Graded rings and equivariant sheaves on toric varieties
cites work: Equivalence classes of polarizations and moduli spaces of sheaves
cites work: Murphy's law in algebraic geometry: Badly-behaved deformation spaces
cites work: The Betti numbers of the moduli space of stable sheaves of rank2 on P2.
DOI: 10.1007/S10711-014-9966-2
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description: scientific article