<p>In this article, we study the decomposition into irreducible components of the fixed point locus under the action of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> a finite subgroup of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{SL}_2(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the smooth Nakajima quiver variety of the Jordan quiver. The quiver variety associated with the Jordan quiver is either isomorphic to the punctual Hilbert scheme of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {C}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> or to the Calogero-Moser space. We describe the irreducible components using quiver varieties over the McKay’s quiver associated with the finite subgroup <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. We moreover give a general combinatorial model of the indexing set of these irreducible components in terms of certain elements of the root lattice of the affine Lie algebra associated with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. Finally, we prove that for every projective, symplectic resolution of a wreath product singularity, there exists an irreducible component of the fixed point locus of the punctual Hilbert scheme of the plane that is isomorphic to the resolution.</p>

The Fixed Point Locus of the Smooth Jordan Quiver Variety Under the Action of the Finite Subgroups of $\textrm{SL}_2(\mathbb {C})$

期刊论文
发表日期：

2025年9月11日

查看全文

Raphaël Paegelow

摘要

In this article, we study the decomposition into irreducible components of the fixed point locus under the action of $\Gamma $ $Γ$ a finite subgroup of $\textrm{SL}_2(\mathbb {C})$ ${SL}_{2} (C)$ of the smooth Nakajima quiver variety of the Jordan quiver. The quiver variety associated with the Jordan quiver is either isomorphic to the punctual Hilbert scheme of $\mathbb {C}^2$ $C^{2}$ or to the Calogero-Moser space. We describe the irreducible components using quiver varieties over the McKay’s quiver associated with the finite subgroup $\Gamma $ $Γ$ . We moreover give a general combinatorial model of the indexing set of these irreducible components in terms of certain elements of the root lattice of the affine Lie algebra associated with $\Gamma $ $Γ$ . Finally, we prove that for every projective, symplectic resolution of a wreath product singularity, there exists an irreducible component of the fixed point locus of the punctual Hilbert scheme of the plane that is isomorphic to the resolution.

The Fixed Point Locus of the Smooth Jordan Quiver Variety Under the Action of the Finite Subgroups of \(\textrm{SL}_2(\mathbb {C})\)

摘要