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

Chen, Hao; Yau, Stephen; Zuo, Huaiqing

doi:10.1090/tran/7628

Non-existence of negative weight derivations on positively graded Artinian algebras
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by Hao Chen, Stephen S.-T. Yau and Huaiqing Zuo PDF

Trans. Amer. Math. Soc. 372 (2019), 2493-2535 Request permission

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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