Spherical functors and the flop-flop autoequivalence
Federico Barbacovi
备注：
87 pages; comments are welcome; v2 37 pages, the paper was split in two, the second part is arXiv:2103.02555, minor changes, comments are welcome; v3: 25 pages, examples merged back into this version, main result stated for enhanced triangulated categories, exposition improved, proofs simplified, comments are welcome
Flops are birational transformations which, conjecturally, induce derived equivalences. In many cases an equivalence can be produced as pull-push via a resolution of the birational transformation; when this happens, we have a non-trivial autoequivalence of either sides of the flop known as the \emph{flop-flop autoequivalence}. We prove that such autoequivalence can be realised as the inverse of a spherical twist around a conservative, spherical functor in a natural way. More precisely, we prove that a natural, conservative spherical functor exists in a more general framework and that the flop-flop autoequivalence fits into this picture. We also give an explicit description of the source category of the spherical functor for standard flops (local model and family case) and Mukai flops. We conclude with some speculation about Grassmannian flops and the Abuaf flop.
对于每个光滑的拟射影曲面X，我们利用导出的McKay对应在n>max{1，1}的Hilbert点方案的导出范畴之间构造一系列P^{n-1}-函子H_{l,n}：D（X x X^[1]）--> D（X^[n+1]）.它们可以被认为是Nakajima算子的类似物。函子也限于广义Kummer簇上的P^{n-1}-函子。我们还研究了诱导的自等价，并获得，例如，一个普遍的辫子关系的导出自等价的希尔伯特平方的K3曲面和库默尔四倍群。
