Noncommutative deformations and flops
Noncommutative deformations and flops
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DOI：
10.1215/00127094-3449887
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发表时间：
2013-09
影响因子：
2.5
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
我们证明了在三维空间中每一条翻转或翻动的不可约有理曲线的非交换形变函子是可表示的，因此，我们给每一条这样的曲线关联一个非交换形变代数\(A_{con}\)。这个新的不变量扩展并统一了三维空间中翻动曲线的已知不变量，比如里德宽度和法丛的双次数。它也适用于翻转和奇异概型的情形。我们表明非交换形变代数\(A_{con}\)是有限维的，并给出了一种获得曲线交换形变的新方法，这使我们能够对某些\((-3,1)\)曲线的这些形变进行明确计算。然后我们展示我们的新不变量\(A_{con}\)如何也控制翻动的同调代数。对于任何在只有戈伦斯坦终端奇点的射影三维空间中的翻动曲线，我们通过围绕非交换形变代数\(A_{con}\)上的一个泛族扭转来构造三维空间导出范畴的一个自等价，并证明这个自等价是布里奇兰德的翻动 - 翻动函子的逆。这表明为了理解三维空间的导出自等价以及布里奇兰德稳定性流形，严格有必要考虑曲线的非交换形变。
We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33-fold is representable, and hence, we associate to every such curve a noncommutative deformation algebra AconAcon. This new invariant extends and unifies known invariants for flopping curves in 33-folds, such as the width of Reid and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra AconAcon is finite-dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us to make explicit calculations of these deformations for certain (−3,1)(−3,1)-curves. We then show how our new invariant AconAcon also controls the homological algebra of flops. For any flopping curve in a projective 33-fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the 33-fold by twisting around a universal family over the noncommutative deformation algebra AconAcon, and prove that this autoequivalence is an inverse of Bridgeland’s flop-flop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33-fold and, thus, the Bridgeland stability manifold.