Tilting exercises. (English) Zbl 1075.14015
Given an algebraic variety \(X\) over an algebraically closed field, and a closed embedding \(i : Y \hookrightarrow X\), a \(Y\)-tilting perverse sheaf \(M\) on \(X\) is defined to a be a perverse sheaf of \(X\) such that \(i^{!}\,M\) and \(i^*\,M\) are perverse sheaves on \(Y\). The authors first discuss this notion in terms of tilting extensions of a perverse sheaf defined on the open complement \(X \backslash Y\). Then, in the case of a stratified variety with smooth connected strata, they exhibit a bijection between the set of strata and the set of isomorphism classes of indecomposable tilting perverse sheaves.
The theory is applied to the case of the Schubert stratification of a flag variety \(G/B\), where \(G\) is a semisimple algebraic group. Among other results, W. Soergel’s Struktursatz [J. Am. Math. Soc. 3, No. 2, 421–445 (1990; Zbl 0747.17008)] is obtained as a corollary. A conjecture by Kapranov is also proved, namely that the Serre functor on the category of perverse sheaves on \(X\) is the square of the Radon transform.
The theory is applied to the case of the Schubert stratification of a flag variety \(G/B\), where \(G\) is a semisimple algebraic group. Among other results, W. Soergel’s Struktursatz [J. Am. Math. Soc. 3, No. 2, 421–445 (1990; Zbl 0747.17008)] is obtained as a corollary. A conjecture by Kapranov is also proved, namely that the Serre functor on the category of perverse sheaves on \(X\) is the square of the Radon transform.
Reviewer: Daniele Faenzi (Firenze)
MSC:
14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
18E30 | Derived categories, triangulated categories (MSC2010) |