JournalsjemsVol. 24, No. 9pp. 3215–3332

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

  • Cheuk Yu Mak

    University of Cambridge, UK
  • Ivan Smith

    University of Cambridge, UK
Fukaya–Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$ cover
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Abstract

We realise Stroppel’s extended arc algebra [13, 51] in the Fukaya–Seidel category of a natural Lefschetz fibration on the generic fibre of the adjoint quotient map on a type AA nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein–Gel’fand–Gel’fand category O\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.

Cite this article

Cheuk Yu Mak, Ivan Smith, Fukaya–Seidel categories of Hilbert schemes and parabolic category O\mathcal{O}. J. Eur. Math. Soc. 24 (2022), no. 9, pp. 3215–3332

DOI 10.4171/JEMS/1159