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arXiv math/9805127math.DGmath.AG
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Dirac算子核的消灭定理

Vanishing theorems for the kernel of a Dirac operator

Maxim Braverman

备注：
A mistake in Theorem 3.13 is corrected. Some othe misprints are removed
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摘要：

得到了Dirac算子在Clifford模上的核的消灭定理，该Clifford模被一个曲率在基流形上任意点都是非退化的线丛的足够大的幂扭曲.特别地，当基流形几乎是复流形时，我们证明了被曲率为混合符号的线丛扭曲的$\spin^c$ Dirac算子的核的消失定理。在这种情况下，我们也放松了曲率非退化的假设。这些结果推广了Borthwick和Uribe的一个消灭定理。作为应用，我们得到了经典Andreotti-Grauert消灭定理的一个新的证明，证明了紧致复流形的上同调性，该流形的值在全纯向量丛的全纯截面的层中，该向量丛被曲率为混合符号的全纯线丛的大幂扭曲.作为另一个应用，我们计算了被线丛的大幂扭转的签名算子的指数的符号。

英文摘要：

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a $\spin^c$ Dirac operator twisted by a line bundle with curvature of a mixed sign. In this case we also relax the assumption of non-degeneracy of the curvature. These results are generalization of a vanishing theorem of Borthwick and Uribe. As an application we obtain a new proof of the classical Andreotti-Grauert vanishing theorem for the cohomology of a compact complex manifold with values in the sheaf of holomorphic sections of a holomorphic vector bundle, twisted by a large power of a holomorphic line bundle with curvature of a mixed sign. As another application we calculate the sign of the index of a signature operator twisted by a large power of a line bundle.

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