The non-existence of negative weight derivations on positive dimensional isolated singularities: generalized wahl conjecture

Stephen Yau tsinghua University Huaiqing Zuo tsinghua University Hao Chen Bingyi Chen

mathscidoc:1803.01004

to appear J. Differential geometry
Let R = C[x1; x2;    ; xn]=(f) where f is a weighted homogeneous polynomial de ning an isolated singularity at the origin. Then R and Der(R;R) are graded. It is well-known that Der(R;R) does not have a negatively graded component. Wahl conjectured that this is still true for R = C[x1; x2;    ; xn]=(f1; f2;    ; fm) which de nes an isolated, normal and complete intersection singularity and f1; f2;    ; fm weighted homogeneous polynomials with the same weight type (w1;w2;    ;wn). Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for R when the degree of fi; 1  i  m are bounded below by a constant C depending only on the weights w1;w2;    ;wn. Moreover this bound C is improved when any two of w1;w2;    ;wn are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when fi are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.
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@inproceedings{stephenthe,
  title={THE NON-EXISTENCE OF NEGATIVE WEIGHT DERIVATIONS ON POSITIVE DIMENSIONAL ISOLATED SINGULARITIES: GENERALIZED WAHL CONJECTURE},
  author={Stephen Yau, Huaiqing Zuo, Hao Chen, and Bingyi Chen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180307145911794532973},
  booktitle={to appear J. Differential geometry},
}
Stephen Yau, Huaiqing Zuo, Hao Chen, and Bingyi Chen. THE NON-EXISTENCE OF NEGATIVE WEIGHT DERIVATIONS ON POSITIVE DIMENSIONAL ISOLATED SINGULARITIES: GENERALIZED WAHL CONJECTURE. In to appear J. Differential geometry. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180307145911794532973.
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