摘要翻译：
本文继续了作者通过对数几何和曲面退化、仿射流形与奇点、对数Calabi-Yau空间和Calabi-Yaus的曲面退化等研究镜像对称性的程序。本文的主要工作是计算给定的具有奇点的仿射流形B上的Calabi-Yau簇的上同调。我们证明了B上的Calabi-Yau簇的Dolbeault上同调群是用B上束的上同调群来描述的。首先通过计算与B相关的对数Calabi-Yau空间上的对数de Rham上同调群和对数Dolbeault上同调群来证明这一点，然后在对数条件下证明上同调的一个变基定理。作为应用，这表明了我们通过仿射流形的Legendre对偶构造的镜像对称，导致了镜像对称中通常所期望的Hodge数的交换，并给出了光滑的单性的显式描述。
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英文标题：
《Mirror Symmetry via Logarithmic Degeneration Data II》
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作者：
Mark Gross, Bernd Siebert
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最新提交年份：
2009
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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：
  This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting.   As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.
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PDF链接：
https://arxiv.org/pdf/0709.2290