Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(X, \omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a conical symplectic variety of dimension 2n which has a projective symplectic resolution. Assume that X admits an effective Hamiltonian action of an n-dimensional algebraic torus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, compatible with the conical <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{C}^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-action. A typical example of X is a toric hyperkähler variety Y(A, 0). In this article, we prove that this property characterizes Y(A, 0) with A unimodular. More precisely, if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(X, \omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is such a conical symplectic variety, then there is a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>-equivariant (complex analytic) isomorphism <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varphi : (X, \omega ) \rightarrow (Y(A,0), \omega _{Y(A,0)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>ω</mi> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> under which both moment maps are identified. Moreover, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> sends the center <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$0_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>0</mn> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> of X to the center <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$0_{Y(A,0)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>0</mn> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> of Y(A, 0).

Towards a characterization of toric hyperkähler varieties among symplectic singularities

期刊论文
发表日期：

2025年12月16日

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Yoshinori Namikawa

摘要

Let $(X, \omega )$ $(X, ω)$ be a conical symplectic variety of dimension 2n which has a projective symplectic resolution. Assume that X admits an effective Hamiltonian action of an n-dimensional algebraic torus $T^n$ $T^{n}$ , compatible with the conical $\textbf{C}^*$ $C^{*}$ -action. A typical example of X is a toric hyperkähler variety Y(A, 0). In this article, we prove that this property characterizes Y(A, 0) with A unimodular. More precisely, if $(X, \omega )$ $(X, ω)$ is such a conical symplectic variety, then there is a $T^n$ $T^{n}$ -equivariant (complex analytic) isomorphism $\varphi : (X, \omega ) \rightarrow (Y(A,0), \omega _{Y(A,0)})$ $φ : (X, ω) \to (Y (A, 0), ω_{Y (A, 0)})$ under which both moment maps are identified. Moreover, $\varphi $ $φ$ sends the center $0_X$ $0_{X}$ of X to the center $0_{Y(A,0)}$ $0_{Y (A, 0)}$ of Y(A, 0).