Residual categories for (co)adjoint Grassmannians in classical types. (20th May 2021)

• Record Type: Journal Article
• Authors: Kuznetsov, Alexander; Smirnov, Maxim
• Is Part Of: Compositio mathematica. Volume 157:Number 6(2021)
• Page Start: 1172
• Page End: 1206
• Publication Date: 2021-05-20
• Subjects: Lefschetz collections, -- residual categories, -- quantum cohomology, -- adjoint and coadjoint homogeneous varieties; 14M17, -- 14N35, -- 18G80; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007090
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$, that is, flag varieties $\operatorname {Fl}(1, n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2, 2n)$ ; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2, 2n)$ this is the first exceptional collection proved to be full.