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Transactions of the American Mathematical Society

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Non-existence of negative weight derivations on positively graded Artinian algebras


Authors: Hao Chen, Stephen S.-T. Yau and Huaiqing Zuo
Journal: Trans. Amer. Math. Soc. 372 (2019), 2493-2535
MSC (2010): Primary 14B05, 32S05
DOI: https://doi.org/10.1090/tran/7628
Published electronically: March 20, 2019
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Abstract: Let $ R= {\Bbb C}[x_1,x_2,\ldots , x_n]/(f_1,\ldots , f_m)$ be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on $ R$. Alexsandrov conjectured that there are no negative weight derivations when $ R$ is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on $ R$ when $ R$ is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on $ R$ when the degrees of $ f_1,\ldots ,f_m$ are bounded below by a constant $ C$ depending only on the weights of $ x_1,\ldots ,x_n$. Moreover this bound $ C$ is improved in several special cases.


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Additional Information

Hao Chen
Affiliation: The College of Information Science and Technology, Jinan University, Guangzhou 510632, Guangdong, China
Email: haochen@jnu.edu.cn

Stephen S.-T. Yau
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
Email: yau@uic.edu

Huaiqing Zuo
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
Email: hqzuo@mail.tsinghua.edu.cn

DOI: https://doi.org/10.1090/tran/7628
Keywords: Derivations, weighted homogeneous singularities, Aleksandrov Conjecture, Halperin Conjecture, Yau Conjecture
Received by editor(s): March 7, 2018
Received by editor(s) in revised form: May 29, 2018
Published electronically: March 20, 2019
Additional Notes: The first author was supported by NSFC Grants 11371138 and 11531002.
Both the second and third authors were supported by NSFC Grant 11531007 and the start-up fund from Tsinghua University.
The third author was supported by NSFC Grants 11771231, 11401335, and Tsinghua University Initiative Scientific Research Program.
Dedicated: Dedicated to Professor Shigefumi Mori on the occasion of his 65th birthday
Article copyright: © Copyright 2019 American Mathematical Society