Simons Collaboration on Homological Mirror Symmetry 2017 Annual Meeting

This was the second annual meeting of the Simons Collaboration on Homological Mirror Symmetry.

Collaboration website: https://schms.math.berkeley.edu/

Yau gave an overview of the results obtained by the various groups in the collaboration aimed at proving and categorizing the SYZ mirror symmetry, which constructs mirror Calabi-Yau varieties by taking dual torus fibrations. He discussed recent breakthroughs in this direction obtained jointly with Collins and Jacob, emphasizing analytic advances on the existence of solutions to the deformed Yang-Mills equation necessary for constructing the SYZ mirrors of stable objects. He also discussed the connection of these special solutions to the rigid analytic constructions proposed last year by Abouzaid and the former collaboration member Tony Yu. In the second part of his talk, Yau discussed new work for constructing mirrors of three-dimensional Calabi-Yau canonical singularities.

Auroux presented a new definition of admissibility for Lagrangian submanifolds in Landau-Ginzburg models. The introduction of this concept completes one of the major goals of the Collaboration program, as it bridges the gap between the setup in Abouzaid’s thesis and the subsequent work of Fang-Liu-Treumann-Zaslow. Auroux demonstrated how revisiting homological mirror symmetry for toric varieties from this perspective provides a more functorial approach on the subject. The novel approach is being developed in detail in the Ph.D. thesis of the Collaboration Ph.D. student Andrew Hanlon (University of California, Berkeley) to be defended in May 2018. The new notion of admissibility also plays a key technical role in the ongoing work of Abouzaid and Auroux to prove homological mirror symmetry for complete intersections in toric varieties.

Sheridan reported on a remarkable extension of Seidel’s proof of homological mirror symmetry for K3 surfaces to the noncommutative realm. The extension reveals an interesting connection with the rationality problem for cubic fourfolds. There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett) or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a noncommutative K3 surface to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is ‘trigonometric,’ i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to Batyrev-Borisov). Sheridan and Smith show that in contrast to the algebraic side of the story, the mirror is always ‘geometric,’ i.e., it is the Shioda-Inose K3 surface equipped with an appropriate Kähler form. This leads to their main theorem: homological mirror symmetry holds in this context.

Fukaya, Seidel and Pardon presented important technical advances in Floer theory that are essential for the algebraic construction of the Fukaya category, which is the second major goal of our Collaboration. Fukaya explained a new technique for dealing with the analytic difficulties appearing in compactifying the moduli space of pseudo-holomorphic curves in the complement of a divisor. In contrast to the stable map compactication in this situation, one has to deal with strata, which, while smooth by themselves, can be singular in the compactication. Fukaya explained how to apply the implicit function theorem in such a situation. Seidel presented a new construction of an important algebraic structure on endo functors on Fukaya categories. Bottman-Ma’u-Wehrheim-Woodward, and later, from a different viewpoint, Lekili-Lipyanskiy-Fukaya, have considered the tensor category structure on the Fukaya category of M x M. By specializing to a formal neighborhood of the diagonal, one obtains a formal group. Seidel explained how to construct and compute this formal group and why it is potentially particularly interesting for Fano manifolds. John Pardon reported on major progress in the development of Floer theory for noncompact manifolds. Ganatra, Shende and Pardon discovered a class of Liouville manifolds with boundary, called ‘Liouville sectors,’ on which wrapped Floer theory is well behaved. Pardon explained why wrapped Floer theory on Liouville sectors is covariantly functorial under inclusions of Liouville sectors. The formalism of Liouville sectors and their Fukaya categories allows one to formalize a conjectural “cosheaf property” for wrapped Fukaya categories, with potential applications to the study of Fukaya categories of Weinstein manifolds. He also explained a “stop removal” that may potentially lead to a proof of the cosheaf property for nice covers.

Finally, Abouzaid and Pantev presented important results from last year addressing the constructions of mirrors and the general proof of homological mirror symmetry. After outlining the recent proof of Homological Mirror Symmetry for smooth Lagrangian torus brations via family Floer homology, Abouzaid explained a program for extending the approach to more general SYZ brations consisting of two parts: (i) the construction of immersed Lagrangians controlling the singularities of such brations, and (ii) the study of a new Floer-theoretic notion of a ‘loop space’ for immersed Lagrangians. This is complemented by a new nc localization formalism of PIs Katzarkov, Kontsevich and Pantev, which explains how the rigid analytic spaces arising from the symplectic geometry constructions can be glued. The formalism allows one to carry out descent constructions in general algebraic and analytic frameworks without resorting to generators. Pantev described the requisite descent technology and showed how it can be used to glue local mirror symmetry functors to ultimately give a local to global proof of Homological Mirror Symmetry. Pantev also discussed various applications, such as the connection to classical Zariski descent.

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