[Paper Review] Mirror symmetry for perverse schobers from birational geometry

Will Donovan, Tatsuki Kuwagaki|arXiv (Cornell University)|Mar 27, 2019

Algebraic structures and combinatorial models40 references4 citations

TL;DR

This paper constructs mirror partners to perverse schobers arising from birational geometry in dimensions 2 and 3, using Fukaya categories with stops and the coherent-constructible correspondence. It establishes homological mirror symmetry equivalences by showing that the 2nd compact cohomology of the mirror schober recovers the derived category of coherent sheaves on singular varieties, providing a new proof for such mirror symmetries.

ABSTRACT

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-Pardon-Shende 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-Kapranov-Schechtman on our mirror schobers.

Motivation & Objective

• To construct mirror partners to perverse schobers from birational geometry in 2D and 3D Calabi–Yau varieties.
• To establish homological mirror symmetry equivalences between derived categories of coherent sheaves and Fukaya categories with stops.
• To interpret schobers as supported on loci in mirror moduli spaces and relate them to constructible sheaves via the coherent-constructible correspondence.
• To provide a new proof of homological mirror symmetry for singular varieties by evaluating categorified cohomology operators on the mirror schobers.

Proposed method

• The authors use the coherent-constructible correspondence to relate derived categories of coherent sheaves to categories of constructible sheaves.
• They apply a recent result of Ganatra–Pardon–Shende to relate schobers to constructible sheaves via microlocal theory.
• The construction involves defining spherical pairs and flobers from flops in 3-folds, using derived categories of the fiber product and quotient constructions.
• Fukaya categories with stops are used to model the mirror side, with stops encoded by Lagrangian skeletons.
• The method employs Verdier–Drinfeld quotients and homotopy push-outs in the Morita model to relate categories of constructible sheaves.
• The 2nd compact cohomology of the mirror schober is computed via push-out diagrams, leading to the equivalence with the derived category of the singular variety.

Experimental results

Research questions

• RQ1 Can a locally constant family of Fukaya categories on the complex structure moduli space be extended to a perverse schober?
• RQ2 Does this mirror schober satisfy a homological mirror symmetry equivalence with the derived category of coherent sheaves on the singular variety?
• RQ3 How can the categorified cohomology operators of Bondal–Kapranov–Schechtman be evaluated on the mirror schober to recover the derived category of the singular variety?
• RQ4 Is the mirror schober construction compatible with the coherent-constructible correspondence for singular toric varieties?
• RQ5 Can the flobers and spherical pairs arising from flops yield equivalent 2nd compact cohomology in the mirror setting?

Key findings

• The mirror schober is constructed using Fukaya categories with stops, providing a categorical mirror partner to perverse schobers from birational geometry.
• Homological mirror symmetry is established via an equivalence between the derived category of coherent sheaves on the singular variety and the 2nd compact cohomology of the mirror schober.
• The 2nd compact cohomology of the mirror schober recovers the derived category $ D(X_0) $, confirming the mirror symmetry equivalence.
• The construction uses a homotopy push-out diagram in the Morita model, showing that the mirror schober arises as a pushout of categories of constructible sheaves.
• The method provides a new proof of homological mirror symmetry for singular varieties by evaluating categorified cohomology operators on the mirror schober.
• The results suggest a generalization of the coherent-constructible correspondence to singular toric varieties via limits over smooth refinements.