对邦达尔 - 波利索丘克猜想的一个光滑射影反例
A smooth projective counterexample to Bondal-Polishchuk's conjecture
浏览论文内容
中文总结 AI 辅助
针对邦达尔 - 波利索丘克猜想，以特定光滑射影三维簇\(X\)为例，通过证明辫子群在\(D^b(X)\)中全例外集合集上作用不可迁，给出反例，此前该形式的反例未知。
中文摘要 AI 辅助
对于一个特定的光滑射影（弱法诺）三维簇\(X\)，我们证明了辫子群在\(D^b(X)\)中全例外集合集上的作用不是可迁的。这为邦达尔和波利索丘克1993年的一个猜想提供了反例。该猜想首先被张、海登和施罗尔反驳，他们构造了一族部分包裹的福卡亚范畴，其可迁性不成立。然而，之前尚不知道对于光滑射影簇\(X\)形如\(D^b(X)\)的反例。
英文摘要
For a particular smooth projective (weak Fano) threefold $X$ we show that the braid group action on the set of full exceptional collections in $D^b(X)$ is not transitive. This provides a counterexample to a conjecture of Bondal and Polishchuk from 1993. The conjecture was first disproved by Chang, Haiden, and Schroll, who constructed a family of partially wrapped Fukaya categories for which the transitivity fails. However, no counterexample of the form $D^b(X)$ for $X$ a smooth projective variety was previously known. In addition, we show that the space of Bridgeland stability conditions $Stab(X)$ on $X$ has infinitely many connected components. This is the first known example of a smooth projective variety whose space of Bridgeland stability conditions is disconnected. Finally, we apply a similar method to establish that $Stab(Y)$ for a symmetric quintic threefold $Y$ is also disconnected.
