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

HAO CHEN Jinan University Stephen S.-T. Yau Tsinghua University Huaiqing Zuo Tsinghua University

mathscidoc:1906.01003

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 372, (4), 2493–2535, 2019.8
Let R = C[x1, x2, . . . , xn]/(f1, . . . , fm) 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 f1, . . . ,fm are bounded below by a constant C depending only on the weights of x1, . . . , xn. Moreover this bound C is improved in several special cases.
Derivations, weighted homogeneous singularities, Aleksandrov Conjecture, Halperin Conjecture, Yau Conjecture.
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@inproceedings{hao2019non-existence,
  title={NON-EXISTENCE OF NEGATIVE WEIGHT DERIVATIONS ON POSITIVELY GRADED ARTINIAN ALGEBRAS},
  author={HAO CHEN, Stephen S.-T. Yau, and Huaiqing Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190626023742046285361},
  booktitle={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY},
  volume={372},
  number={4},
  pages={2493–2535},
  year={2019},
}
HAO CHEN, Stephen S.-T. Yau, and Huaiqing Zuo. NON-EXISTENCE OF NEGATIVE WEIGHT DERIVATIONS ON POSITIVELY GRADED ARTINIAN ALGEBRAS. 2019. Vol. 372. In TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. pp.2493–2535. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190626023742046285361.
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