On semisimplicity of quantum cohomology of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si5.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">P</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>-orbifolds

Ke, Hua-Zhong

doi:10.1016/j.geomphys.2019.05.007

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2021

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“…Thus, from Dubrovin's conjecture we conclude that generic semisimplicity of should imply the existence of a full exceptional collection in . On the other hand, it was observed that the opposite implication is not true; see [CKMPS19, GMS15, Ke19, Per14].…”

Section: Introductionmentioning

confidence: 91%

Residual categories for (co)adjoint Grassmannians in classical types

Kuznetsov¹,

Smirnov²

2021

Compositio Math.

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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.

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