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arXiv · 1907.07624

Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

Cheuk Yu Mak·Ivan Smith
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Abstract

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.

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BibTeXRIS

Cheuk Yu Mak, Ivan Smith. 2019-07-16. Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$. https://arxiv.org/abs/1907.07624

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