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Non-existence of negative weight derivations on Jacobian algebras

  • Bingyi Chen
  • Zhiwen Liu
  • Stephen S. -T. Yau
  • Huaiqing Zuo
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Abstract

Let S=𝕂[x1,β‹―,xn] be a polynomial ring in n variables with positive integer weights over a field 𝕂 of characteristic zero. Let f∈ S be a weighted homogeneous polynomial such that the Jacobian algebra A=.S middle/ ((βˆ‚ f)/(βˆ‚ x1),β‹―,(βˆ‚ f)/(βˆ‚ xn)). is a finite-dimensional 𝕂-vector space. We prove that A has no nonzero negative weight derivation, which confirms a conjecture of the third author. An immediate consequence of this conjecture is the micro-local characterization of quasi-homogeneous singularities.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026nonexistence,
  title = {{Non-existence of negative weight derivations on Jacobian algebras}},
  author = {Bingyi Chen and Zhiwen Liu and Stephen S. -T. Yau and Huaiqing Zuo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25615},
  eprint = {2610.25615},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25615},
}
RIS
TY  - UNPB
TI  - Non-existence of negative weight derivations on Jacobian algebras
AU  - Bingyi Chen
AU  - Zhiwen Liu
AU  - Stephen S. -T. Yau
AU  - Huaiqing Zuo
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.25615
UR  - https://arxiv.org/abs/2610.25615
AB  - Let S=𝕂[x_1,β‹―,x_n] be a polynomial ring in n variables with positive integer weights over a field 𝕂 of characteristic zero. Let f∈ S be a weighted homogeneous polynomial such that the Jacobian algebra A=.S middle/ ((βˆ‚ f)/(βˆ‚ x_1),β‹―,(βˆ‚ f)/(βˆ‚ x_n)). is a finite-dimensional 𝕂-vector space. We prove that A has no nonzero negative weight derivation, which confirms a conjecture of the third author. An immediate consequence of this conjecture is the micro-local characterization of quasi-homogeneous singularities.
ER  -