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Mosc. Math. J., 2007, Volume 7, Number 4, Pages 673–697 (Mi mmj306)  

This article is cited in 13 scientific papers (total in 13 papers)

Quiver varieties and Hilbert schemes

A. G. Kuznetsovab

a Steklov Mathematical Institute, Russian Academy of Sciences
b Laboratoire J.-V. Poncelet, Independent University of Moscow

Abstract: In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, if $X=\mathbb C^2$, $\Gamma\subsetSL(\mathbb C^2)$ is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$, we show that $X^{\Gamma[n]}$ (the $\Gamma$-equivariant Hilbert scheme of $X$), and $X_{\Gamma}^{[n]}$ (the Hilbert scheme of $X_{\Gamma}$) are quiver varieties for the affine Dynkin graph corresponding to $\Gamma$ via the McKay correspondence with the same dimension vectors but different parameters $\zeta$ (for earlier results in this direction see works by M. Haiman, M. Varagnolo and E. Vasserot, and W. Wang). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_{\Gamma}^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\mathbb Z/d\mathbb Z$) via the fixed points of a $(\mathbb C^*\times\mathbb C^*)$-action we deduce the following combinatorial identity: the number $UCY(n,d)$ of Young diagrams consisting of $nd$ boxes and uniformly colored in $d$ colors equals the number $CY(n,d)$ of collections of $d$ Young diagrams with the total number of boxes equal to $n$.

Key words and phrases: Quiver variety, Hilbert scheme, McKay correspondence, moduli space.


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MSC: Primary 14D21; Secondary 53C26, 16G20
Received: January 4, 2007

Citation: A. G. Kuznetsov, “Quiver varieties and Hilbert schemes”, Mosc. Math. J., 7:4 (2007), 673–697

Citation in format AMSBIB
\by A.~G.~Kuznetsov
\paper Quiver varieties and Hilbert schemes
\jour Mosc. Math.~J.
\yr 2007
\vol 7
\issue 4
\pages 673--697

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    This publication is cited in the following articles:
    1. S. Gusein-Zade, I. Luengo, A. Melle-Hernández, “On generating series of classes of equivariant Hilbert schemes of fat points”, Mosc. Math. J., 10:3 (2010), 593–602  mathnet  crossref  mathscinet
    2. Szabo R.J., “Instantons, Topological Strings, and Enumerative Geometry”, Adv. Math. Phys., 2010, 107857  crossref  mathscinet  zmath  isi  elib  scopus
    3. Belavin A., Belavin V., Bershtein M., “Instantons and 2d superconformal field theory”, J. High Energy Phys., 2011, no. 9, 117  crossref  zmath  isi  elib  scopus
    4. Nagao K., “Derived Categories of Small Toric Calabi-Yau 3-Folds and Curve Counting Invariants”, Q. J. Math., 63:4 (2012), 965–1007  crossref  mathscinet  zmath  isi  scopus
    5. Bruzzo U., Sala F., Szabo R.J., “N=2 Quiver Gauge Theories on a-Type Ale Spaces”, Lett. Math. Phys., 105:3 (2015), 401–445  crossref  mathscinet  zmath  isi  scopus
    6. Pedrini M., Sala F., Szabo R.J., “AGT relations for abelian quiver gauge theories on ALE spaces”, J. Geom. Phys., 103 (2016), 43–89  crossref  mathscinet  zmath  isi  elib  scopus
    7. Bruzzo U., Pedrini M., Sala F., Szabo R.J., “Framed Sheaves on Root Stacks and Supersymmetric Gauge Theories on Ale Spaces”, Adv. Math., 288 (2016), 1175–1308  crossref  mathscinet  zmath  isi  elib  scopus
    8. Szabo R.J., “N=2 Gauge Theories, Instanton Moduli Spaces and Geometric Representation Theory”, J. Geom. Phys., 109:SI (2016), 83–121  crossref  mathscinet  zmath  isi  scopus
    9. Bartocci C., Bruzzo U., Lanza V., Rava C.L.S., “Hilbert Schemes of Points of Phi(P1) (-N) as Quiver Varieties”, J. Pure Appl. Algebr., 221:8 (2017), 2132–2155  crossref  mathscinet  isi  scopus
    10. Shigeyuki Fujii, Satoshi Minabe, “A Combinatorial Study on Quiver Varieties”, SIGMA, 13 (2017), 052, 28 pp.  mathnet  crossref
    11. Bartocci C., Lanza V., Rava C.L.S., “Moduli Spaces of Framed Sheaves and Quiver Varieties”, J. Geom. Phys., 118:SI (2017), 20–39  crossref  zmath  isi  scopus
    12. Zhou Z., “Donaldson-Thomas Theory of [C-2/Z(N+1)] X P-1”, Sel. Math.-New Ser., 24:4 (2018), 3663–3722  crossref  mathscinet  zmath  isi  scopus
    13. Claudio Bartocci, Ugo Bruzzo, Valeriano Lanza, Claudio L. S. Rava, “On the Irreducibility of Some Quiver Varieties”, SIGMA, 16 (2020), 069, 13 pp.  mathnet  crossref
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