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The homological projective dual of $\operatorname{Sym}^{2}\mathbb{P}(V)$

Part of: Symplectic geometry, contact geometry (Co)homology theory Homological methods

Published online by Cambridge University Press:  17 January 2020

Jørgen Vold Rennemo
Jørgen Vold Rennemo*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK email jvrennemo@gmail.com
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Abstract

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

Keywords

derived category of coherent sheaveshomological projective dualitymatrix factorisations

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions 16E35: Derived categories 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 3 , March 2020 , pp. 476 - 525
Copyright
© The Author 2020

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Footnotes

1

Current address: Departmet of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway

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