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Residual categories for (co)adjoint Grassmannians in classical types

Published online by Cambridge University Press:  20 May 2021

Alexander Kuznetsov
Affiliation:
Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin str., Moscow119991, Russiaakuznet@mi-ras.ru
Maxim Smirnov
Affiliation:
Universität Augsburg, Institut für Mathematik, Universitätsstr. 14, 86159Augsburg, Germanymaxim.smirnov@math.uni-augsburg.de

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.

Type
Research Article
Copyright
© The Author(s) 2021

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Footnotes

This work is supported by the Russian Science Foundation under grant 19-11-00164.

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