摘要翻译：
本文致力于研究对数对变形的各个方面，特别是与奇点不变性和对数复数有关的问题。特别地，利用最小模型程序的最新结果，我们在Siu和Kawamata以及Hacon和McKernan的最新工作的精神下，得到了伴随因子的一个扩张定理。然而，我们的主要动机来自于对Fano变种变形的研究。我们的第一个应用是关于Fano变体族中Mori chamber分解的性质：我们证明了在弱奇点的情况下，当维数较小时，这种分解在变形下是刚性的。然后我们分析了toric Fano变体的变形性质，证明了每一个最多具有末端奇点的单纯toric Fano变体在变形下都是刚性的（如果是奇异的，尤其是不光滑的）。
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英文标题：
《Deformations of canonical pairs and Fano varieties》
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作者：
Tommaso de Fernex, Christopher D. Hacon
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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 is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case of mild singularities, such decomposition is rigid under deformation when the dimension is small. We then turn to analyze deformation properties of toric Fano varieties, and prove that every simplicial toric Fano variety with at most terminal singularities is rigid under deformations (and in particular is not smoothable, if singular).
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PDF链接：
https://arxiv.org/pdf/0901.0389