Noncommutative enhancements of contractions
Noncommutative enhancements of contractions
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DOI：
10.1016/j.aim.2018.11.019
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发表时间：
2016-12
影响因子：
1.7
通讯作者：
W. Donovan;M. Wemyss
中科院分区：
文献类型：
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作者：
W. Donovan;M. Wemyss
给定一个簇X到基Y的收缩，我们在（1）允许一个有平凡和项的倾斜丛的可裂部分分解和（2）纤维维数至多为1的所有收缩的情况下，在温和的假设下，增强了Y中的轨迹，在该轨迹上的收缩与非交换环D的某层不是同构的.在所有情况下，这都会产生一个全局不变式。在crepant设置，我们然后应用此研究导出的自等价的X。我们一般的工作，放弃了许多通常的限制，因此既扩展又统一了现有的方法。在充分的一般性，我们构造一个新的endofunctor的衍生范畴的X通过扭转在D上，然后，在适当的限制下，奇异性，给出条件时，它是一个自等价。我们表明，这些条件自动保持时，在Y的非同构轨迹有余维3或更多，其中包括行列式触发器，我们也控制的条件时，非同构轨迹有余维2，其中包括3倍因子曲线收缩。
Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.