Sheaves on ALE spaces and quiver varieties <script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=default"> </script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\(","\)"]] }, TeX: { extensions: ["AMSmath.js", "AMSsymbols.js"], Macros: { CC: "{\\mathbb C}", C: "{\\mathbb C}", R: "{\\mathbb R}", Q: "{\\mathbb Q}", Z: "{\\mathbb Z}", cO: "{\\mathcal O}", diag: "\\mathop{\\mathrm{diag}}\\nolimits", np: "[{#1}#2{#1}, 2]", operatorname: ["\\mathrm{#1}",1], } } }); </script> <script type="text/javascript" src="MyConfig.js"> </script> <meta http-equiv="X-UA-Compatible" CONTENT="IE=EmulateIE7" /> </head> <body bgcolor= "#90C0C0"> <center> <h1>Sheaves on ALE spaces and quiver varieties </center> <center> Hiraku Nakajima (Kyoto) </center> Abstract <p> The space of the stability parameters, used to define a quiver variety, has a chamber structure. For an affine type, there is a distinguished chamber $C_0$ where the corresponding quiver variety is the framed moduli space of $\Gamma$-equivariant torsion free sheaves on ${\mathbb C}^2$. Here $\Gamma$ is the finite subgroup of $SL_2({\mathbb C})$ corresponding to the affine quiver via the McKay correspondence. There is also another chamber $C_\infty$, most far way from $C_0$, where the corresponding quiver variety is a framed moduli space of torsion free sheaves on the ALE space, the minimal resolution of ${\mathbb C}^2/\Gamma$. This result is an analog of a similar identification for a framed moduli space of anti-self-dual connections on an ALE space, given by Kronheimer and the author in 1989. <hr> <address> <a href="mailto:nakajima@math.kyoto-u.ac.jp">nakajima@math.kyoto-u.ac.jp</a> </address>