Simons Collaboration on Homological Mirror Symmetry Research activities update for 2019/20
During the past year, the collaboration advanced and expanded the homological mirror symmetry program in unexpected ways. The team produced robust mirror proofs in various settings and leveraged the new methodology to discover ubiquitous manifestations of the mirror phenomenon in surprising new settings. In parallel with the continuing development of the non-Archimedean descent techniques and the foundations of virtual fundamental chains in Floer theory, the collaboration PIs and postdocs obtained key new results that go far beyond the state-of-the-art in core mirror symmetry constructions.
We made substantial progress on the foundational steps of the program.
- Abouzaid, Groman and Varolgunes advanced their big project rebuilding Lagrangian Floer theory geared toward concrete local-to-global constructions of mirrors for symplectic manifolds that are presented as the total space of a Lagrangian fibration.
- Sheridan established the technical foundations for computing Gromov-Witten invariants via homological mirror symmetry by defining the relative Fukaya category of a semipositive variety and gave a general framework for computing Floer-theoretic operations from chains in Deligne-Mumford moduli spaces.
- Fukaya, Oh, Ohta and Ono completed the foundational book on algebraic and homotopical models of spaces with Kuranishi structures, which appeared in Springer Monographs in Mathemat ics.
- Lee, Lian and Yau described the symmetries of the mirror correspondence by analyzing the stability manifold of orbifolds. Several team members and postdocs combined their analysis with noncommutative geometry to give a detailed description of wall crossing for the motivic generating series for Gieseker stable coherent sheaves with compact support on local Calabi-Yau spaces which lead to a construction of a large class of new topological mirror pairs of Calabi–Yau orbifolds.
- Katzarkov, Kontsevich and Pantev developed a new approach to the quantum blow-up formula and derived from it a novel structure theory for Frobenius manifolds and a gluing theorem for noncommutative Hodge structures. The framework has immediate applications to classical geometry, including the construction of new, strong birational invariant over non-algebraically closed base fields.
Another major theme in our work this year was the construction and comparison of mirrors and their spaces of quantum branes in the presence of singularities. The following is a sampling of our results in this direction.
- Auroux, Efimov and Katzarkov gave a new construction of Fukaya categories of the trivalent configurations of rational curves which arise as mirrors of higher genus Riemann surfaces. • Arinkin, Pantev and Toën gave a universal construction of symplectic structures along the fibers of a perverse sheaf of categories and produced canonical quantizations of fibered mirrors. Specific constructions of mirrors of such perverse sheaves were worked out by Lau-Lee-Lin in the context of SYZ mirror symmetry for del Pezzo surfaces and by Cheung-Fan-Lin in the context of noncommutative Calabi- Yau spaces associated with quivers.
- Lian, Yau and collaborators extended the recent novel mirror construction of singular Calabi Yau mirrors utilizing gauge fixing and a fractional version of Batyrev-Borisov’s duality. Lian and Kim developed a parallel theory for CY cyclic covers over a homogeneous space and used toric de generations to give a complete proof of Hodge theoretic mirror symmetry for covers of homogeneous spaces of type A and C.
- Abouzaid and Sylvan proved homological mirror symmetry for the local singularities of the toric degenerations introduced by Gross and Siebert while Lau-Lee-Lin proved that the mirror given by Floer-theoretical gluing method using immersed Lagrangians agrees with the mirror constructed by Carl–Pomperla–Siebert.
- Balasubramanian, Distler and Donagi described a flat degeneration of the Hitchin system to a nodal base curve and classified the limiting behavior in terms of Lie theoretic data. They also proved a surprising flatness result for family the Hitchin bases over the compactified moduli space. Beck,Donagi and Wendland investigated the folding by graph automorphisms for Hitchin integrable systems. They showed that the fixed point loci of such automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers and are also isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by Donagi and Pantev.
- Borissov, Sheshmani and Yau constructed a globally defined Lagrangian distribution on the stable locus of the derived Quot-stack of coherent sheaves on a Calabi-Yau four fold. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined (−1)-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form.
- Yau and collaborators made a breakthrough in understanding how mirror symmetry interacts with non-conifold extremal transitions. They compared the A-model orbifold Gromov-Witten in variants of the transition to the B-model period integrals for the new class of orbifolds constructed by Hosono-Lee-Lian-Yau and developed a new method for proving mirror theorems that utilizes wall-crossing formula for epsilon-stable quasimap invariants with a GIT quotient as the target.