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Homological mirror symmetry of elementary birational cobordisms

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Abstract

The derived category of coherent sheaves \({\mathcal {T}}_B\) associated to a birational cobordism which is either a weighted projective space, a stacky Atiyah flip, or a stacky blow-up of a point has a conjectural mirror Fukaya–Seidel category \({\mathcal {T}}_A\). The potential W defining \({\mathcal {T}}_{A}\) has base \({\mathbb {C}}^*\) and exhibits a great deal of symmetry. This paper investigates the structure of the Fukaya–Seidel category for the mirror potentials. A proof of homological mirror symmetry \({\mathcal {T}}_A \cong {\mathcal {T}}_B\) for these birational cobordisms is then given.

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Acknowledgements

The author thanks M. Ballard, C. Diemer, D. Favero, L. Katzarkov, P. Seidel and Y. Soibelman for helpful discussions. The author also thanks the referee for several insightful comments.

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  1. Department of Mathematics, Kansas State University, Manhattan, KS, 66502, USA

    Gabriel Kerr

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  1. Gabriel Kerr
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Kerr, G. Homological mirror symmetry of elementary birational cobordisms. Sel. Math. New Ser. 23, 2801–2847 (2017). https://doi.org/10.1007/s00029-017-0325-7

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