# Residual categories for (co)adjoint Grassmannians in classical types

@article{Kuznetsov2021ResidualCF,
title={Residual categories for (co)adjoint Grassmannians in classical types},
author={Alexander Kuznetsov and Maxim Smirnov},
journal={Compositio Mathematica},
year={2021},
volume={157},
pages={1172 - 1206}
}
• Published 13 January 2020
• Mathematics
• Compositio Mathematica
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…
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## References

SHOWING 1-10 OF 41 REFERENCES
On residual categories for Grassmannians
• Mathematics
Proceedings of the London Mathematical Society
• 2019
We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the
Quantum cohomology of a product (with Appendix by R. Kaufmann2)
• Mathematics
• 1996
The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of
On the quantum cohomology of adjoint varieties
• Mathematics
• 2011
We study the quantum cohomology of adjoint and coadjoint homogeneous spaces. The product of any two Schubert classes does not involve powers of the quantum parameter greater than 2. With the help of
Quantum Pieri rules for isotropic Grassmannians
• Mathematics
• 2009
We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or
Exceptional collections for Grassmannians of isotropic lines
We construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the Grassmannian of isotropic two‐dimensional subspaces in a symplectic vector space of
On the big quantum cohomology of coadjoint varieties
• Mathematics
• 2021
. This paper is devoted to the study of the quantum cohomology of coadjoint varieties of simple algebraic groups across all Dynkin types. We determine the non-semisimple factors of the small quantum
Lefschetz exceptional collections in S k-equivariant categories of ( P n ) k
We consider the bounded derived category of Sk-equivariant coherent sheaves on (P). The goal of this paper is to construct in this category a rectangular Lefschetz exceptional collection when this is
Derived categories of the Cayley plane and the coadjoint Grassmannian of type F
• Mathematics
• 2020
For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type $\mathrm{E}_6$, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A
Hochschild cohomology of generalised Grassmannians
• Mathematics
• 2019
We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e. partial flag varieties associated to maximal parabolic subgroups in a simple