硬Lefschetz定理与半小映射的拓扑
The Hard Lefschetz Theorem and the topology of semismall maps
摘要：
在复射影流形上引入左线丛的概念。证明了左线丛满足硬Lefschetz定理、Lefschetz分解和Hodge-Riemann双线性关系。研究了复流形上的真全纯半小映射，证明了对于常系数，分解定理等价于某些交形式的非退化性。通过证明所讨论的形式是非退化的，给出了定义域与目标是射影的情况下常数层的复直象的分解定理的一个证明。我们的证明揭示了一个新的特点，即形式是确定的。
英文摘要：
We introduce the notion of lef line bundles on a complex projective manifold. We prove that lef line bundles satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We study proper holomorphic semismall maps from complex manifolds and prove that, for constant coefficients, the Decomposition Theorem is equivalent to the non-degeneracy of certain intersection forms. We give a proof of the Decomposition Theorem for the complex direct image of the constant sheaf when the domain and the target are projective by proving that the forms in question are non-degenerate. A new feature uncovered by our proof is that the forms are definite.
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