摘要翻译：
类似于Hard Lefschetz定理(HLT)和Hodge-Riemann双线性关系(HRR)的陈述在各种上下文中成立：它们对光滑紧K\\Ahler流形的上同调代数或射影toric簇的交上同调施加限制；它们限制了Hodge结构极化变化的局部单极性；它们对凸多边形的可能的$F$-向量施加条件。虽然这些定理的陈述依赖于k“Ahler类或其类似类的选择，但通常存在一个可能的k”Ahler类的锥。因此，很自然地要问HLT和HRR在混合环境中是否仍然是真的。本文给出了一种证明混合HLT和HRR的统一方法，推广了已有的结果，并在新的情况下证明了它，如非有理多边形的交上同调。
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英文标题：
《Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations》
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作者：
Eduardo Cattani
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible $f$-vectors of convex polytopes. While the statements of these theorems depend on the choice of a K\"ahler class, or its analog, there is usually a cone of possible K\"ahler classes. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note we present a unified approach to proving the mixed HLT and HRR, generalizing the previously known results, and proving it in new cases such as the intersection cohomology of non-rational polytopes.
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PDF链接：
https://arxiv.org/pdf/0707.1352