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Mirror Symmetry for Perverse Schobers from Birational Geometry

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Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent–constructible correspondence and a recent result of Ganatra et al. (Microlocal morse theory of wrapped fukaya categories. arXiv:1809.08807) to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal et al. (Selecta Math 24(1):85–143, 2018) on our mirror schobers.

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Both authors are grateful for conversations with A. Bondal and M. Kapranov at Kavli IPMU. The first author is grateful for discussions with T. Logvinenko, and thanks D. Auroux, T. Coates, and D. Nadler for helpful conversations at an early stage of the Project. The authors thank M. Romo for comments on the manuscript, and the organizers of the conference ‘Categorical and Analytic Invariants in Algebraic Geometry V’ at Osaka in 2018, where this collaboration started. Finally, the authors are grateful to an anonymous referee for many helpful comments. The authors acknowledge the support of WPI Initiative, MEXT, Japan, and of JSPS KAKENHI Grants JP16K17561 and JP18K13405 respectively which were held at Kavli IPMU, University of Tokyo. The first author is supported by the Yau MSC, Tsinghua University, and the Thousand Talents Plan.

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Correspondence to W. Donovan.

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Communicated by H. T. Yau

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Donovan, W., Kuwagaki, T. Mirror Symmetry for Perverse Schobers from Birational Geometry. Commun. Math. Phys. 381, 453–490 (2021).

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