A characterization and solvability of quasihomogeneous singularities

Let $(V, 0)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ be an isolated hypersurface singularity defined by the holomorphic function $f : (C^{n + 1}, 0) \to (C, 0)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi> <mo class="MathClass-punc">:</mo> <mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mn>1</mn></mrow></msup><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo> <mo class="MathClass-rel">→</mo> <mo class="MathClass-open" stretchy="false">(</mo><mi>ℂ</mi><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ . A local $k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi></math>$ -th ( $0 \leq k \leq n + 1$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn> <mo class="MathClass-rel">≤</mo> <mi>k</mi> <mo class="MathClass-rel">≤</mo> <mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>1</mn></math>$ ) Hessian algebra $H_{k} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ of isolated hypersurface singularity $(V, 0)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ is a finite-dimensional $C$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℂ</mi></math>$ -algebra and it depends only on the isomorphism class of the germ $(V, 0)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ . It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasihomogeneous in terms of its local $k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi></math>$ -th Hessian algebra $H_{k} (f)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>f</mi><mo class="MathClass-close" stretchy="false">)</mo></math>$ . Xu and Yau proved that $(V, 0)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ admits a quasihomogeneous structure if and only if $H_{0} (f)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>f</mi><mo class="MathClass-close" stretchy="false">)</mo></math>$ is isomorphic to a finite-dimensional nonnegatively graded algebra in the early 1980s. In this paper, on the one hand, we generalize Xu and Yau’s result to $H_{n + 1} (f)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mn>1</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>f</mi><mo class="MathClass-close" stretchy="false">)</mo></math>$ . On the other hand, a new series of finite-dimensional Lie algebras $L_{k} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ (resp. $L^{k} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ ) was defined to be the Lie algebra of derivations of the $k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi></math>$ -th ( $0 \leq k \leq n + 1$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn> <mo class="MathClass-rel">≤</mo> <mi>k</mi> <mo class="MathClass-rel">≤</mo> <mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>1</mn></math>$ ) Hessian algebra $H_{k} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ (resp. $A^{k} (V) : = O_{n + 1} / (f, m^{k} J_{f})$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo> <mo class="MathClass-punc">:</mo><mo class="MathClass-rel">=</mo> <msub><mrow><mi mathvariant="bold-script">𝒪</mi></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mn>1</mn></mrow></msub><mo class="MathClass-bin">∕</mo><mo class="MathClass-open" stretchy="false">(</mo><mi>f</mi><mo class="MathClass-punc">,</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msup><msub><mrow><mi>J</mi></mrow><mrow><mi>f</mi></mrow></msub><mo class="MathClass-close" stretchy="false">)</mo></math>$ ) and is finite-dimensional. We prove that $(V, 0)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-punc">,</mo><mn>0</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ is quasihomogeneous singularity if $L_{n + 1} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mn>1</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ (resp. $L^{k} (V) : = Der (A^{k} (V))$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo> <mo class="MathClass-punc">:</mo><mo class="MathClass-rel">=</mo> <mtext> Der</mtext><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">)</mo></math>$ ) satisfies certain conditions. Moreover, we investigate whether the Lie algebras $L_{k} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ (resp. $L^{k} (V)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi> <mo class="MathClass-close" stretchy="false">)</mo></math>$ ) are solvable.

Keywords

solvability of derivation Lie algebra, isolated quasihomogeneous singularities.

Mathematical Subject Classification

Primary: 14B05, 32S05

Milestones

Received: 9 August 2023

Revised: 15 January 2024

Accepted: 14 April 2024

Published: 12 June 2024

Authors

Guorui Ma
Yau Mathematical Sciences Center Tsinghua University Beijing China

Stephen S.-T. Yau
Beijing Institute of Mathematical Sciences and Applications (Bimsa) Beijing China	Department of Mathematical Sciences Tsinghua University Beijing China

Qiwei Zhu
Department of Mathematical Sciences Tsinghua University Beijing China

Huaiqing Zuo
Department of Mathematical Sciences Tsinghua University Beijing China

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Vol. 329, No. 1, 2024

Guorui Ma, Stephen S.-T. Yau, Qiwei Zhu and Huaiqing Zuo

Abstract

Keywords

Mathematical Subject Classification

Milestones

Authors