- Organized by
Tony Pantev, Ph.D.University of Pennsylvania
The fourth annual meeting of the Simons Collaboration on Homological Mirror Symmetry focused on recent work extracting deep geometric and arithmetic statements from the homological mirror interactions between symplectic topology, Hodge theory, non-commutative geometry and differential geometry. Talks highlighted progress on the motivic and cycle theoretic consequences of recently developed arithmetic enhancements of Floer theory and explored constructive and birational geometry aspects of tropical and quantum invariants over non-algebraically closed fields. Additionally talks focused on differential geometry showcased novel existence results for special Lagrangian tori and their implications for the homological mirror correspondence.
Collaboration website: https://schms.math.berkeley.edu/
Many of the new discoveries and results obtained by the collaboration members in the past year focus on the deeper algebraic and arithmetic aspects of the mirror correspondence.
Varolgunes reported on joint works with Abouzaid, Groman and Tonkong, which introduced Floer theoretic invariants for arbitrary compact subsets of symplectic manifolds and proved a descent statement. An application is a proof of the equivalence for local models of the degenerations studied by Gross and Siebert which works over nn-algebraically closed fields. Other applications give applications to arithmetic quantum invariants and local-to-global construction of mirror symmetry for symplectic manifolds that are presented as the total space of a Lagrangian fibration or the union of a standard neighborhood of a symplectic divisor and a Liouville domain.
In his public lecture, Kontsevich described a sweeping program for constructing new birational invariants of algebraic varieties. In one direction, he described a surprising new discovery obtained in joint works with Tschinkel and Pestun giving constructive and computable birational invariants of algebraic varieties endowed with an action of a finite group. Another direction, currently under investigation in a joint work with Katzarkov and Pantev, uses quantum cohomology data for a generic polarization to produce novel birational invariants over non-algebraically closed fields. The new invariants arise from a new structure theory of Frobenius manifolds and a gluing construction for quantum D-modules. An essential technical ingredient here is a deep comparison theorem of Mark Mclean showing that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. During the meeting, Mclean explained the key idea of the proof identifying both small quantum products as identical deformations of symplectic cohomology of some common open affine subspace.
Seidel explained how chain-level genus zero Gromov-Witten theory can be used to associate to any closed monotone symplectic manifold a formal group whose Lie algebra is the odd degree cohomology of the manifold. When taken with coefficients in Fp for some prime p, the p-th power map of the formal group captures the quantum Steenrod operations and allows one to compute derived Picard groups of Fukaya categories and to verify arithmetic refinements of homological mirror symmetry.
Sheridan described a new joint work with Ivan Smith that builds a the symplectic technology needed to understand the motivic aspects and consequences of the homological mirror correspondence. The work builds a bridge between the algebraic K-theory of a variety and the Lagrangian cobordism group of the mirror and uses it to probe interesting questions on algebraic cycles going back to the deep conjectures of Beauville-Voisin, Beilinson and Bloch.
Collins and Yau described exciting recent progress in the construction of physical quantum branes in Ricci flat backgrounds. Their work from the past year focused on developing new technology in geometric analysis that lead Collins, Jacob and Yau to a new nonlinear stability condition that guarantees the existence of solutions to the deformed Hermitian-Yang-Mills equations and allowed them to obtain precise mirrors of special Lagrangian branes in dimensions two and three. Surprising theorems guaranteeing existence and completeness of moduli were obtained in a variety of settings including the case of log Calabi-Yau varieties, the case of singular Calabi-Yau spaces arising as fractional complete intersections, and the case of Landau-Ginzburg models on toric varieties. The analysis is based on new infinite-dimensional symplectic reduction picture in complex geometry that is a mirror of Jake Solomon’s space of infinite Lagrangians. Collins also described a striking recent joint work with Jacob and Lin that applies these ideas to Tian-Yau metrics on the complement of anticanoical divisors in Fano varieties and in particular proves the existence of special Lagrangian torus fibrations on Tian-Yau surfaces and their mirrors.
Auroux described a remarkable solution to an old puzzle in homological mirror symmetry — understanding the symplectic mirrors of algebraic cycles which are Noether-Lefschetz special. The problem is very delicate and highlights a tantalizing discrepancy in our current understanding of the mirror correspondence. On the complex side, the Noether-Lefschetz special cycles are the algebraic cycles that are most readily accessible by constructions in projective geometry. On the symplectic side, however, the corresponding objects are only visible through an unexplicit construction that formally adds images of projectors. Almost twenty years ago, Kapustin and Orlov understood that the geometric content of the symplectic mirrors of special algebraic cycles should be coisotropic submanifolds equipped with a special rank one connection, which induces a symplectic holomorphic structure on the transversal of the null foliation. The problem is that this geometry does not fit in the formalism of Fukaya categories, and all previous attempts to try and define interactions between coisotropic and Lagrangian branes have failed. The breakthrough comes from a new idea of Auroux implemented in the work of his student Yingdi Qin. In the case of tori, Auroux and Qin use a doubling construction that allows them to incorporate coisotropic branes in the Fukaya category of a bigger torus and to understand the mirror process explicitly in homological and SYZ terms. In his forthcoming thesis, Qin uses this construction to define homomorphisms between coisotropic and Lagrangian branes to understand their mirrors in terms of real, complex or quaternionic multiplication and to describe higher rank coisotropic branes. Qin’s work opens a new path for in-depth understanding of the mirror correspondence, which we plan to explore. A deformed semi-flat version of the construction matches perfectly with the new solutions of the deformed Hermitian-Yang-Mills equation obtained by Collins and Yau, which will give new insight into the existence problem for physical branes. At the same time, the natural extension of the doubling construction beyond the case of tori is naturally captured by deformation quantization and the mirror functor is manifestly realized by the Kontsevich-Soibelman nonlinear Riemann-Hilbert correspondence.
Rietsch presented a striking structural result for toric Landau-Ginzburg models. She described new joint work with Jamie Judd showing that any complete Laurent polynomial with coefficients in the positive part of the field of Puiseux series has a unique positive critical point. She also explained a tropical interpretation of the statement, which gives a finite recursive procedure for computing such critical points. This structural result has several deep applications that are currently investigated. In particular, Judd and Rietsch prove a comparison with mutations and deduce a streamlined version of mirror symmetry for cluster varieties. They also give applications to mirror symmetry for toric varieties providing, in particular, a construction of canonical nondisplaceable Lagrangian tori for toric symplectic manifolds.
Thursday, November 14
9:30 AM Nick Sheridan | Lagrangian Cobordism and Chow groups 11:00 AM Umut Varolgunes | Local-to-Global Principles in Floer Theory 1:00 PM Denis Auroux | Coisotropic Branes and HMS for Tori 2:30 PM Konstanze Rietsch | The Tropical Critical Point and Mirror Symmetry 4:00 PM Tristan Collins | Special Lagrangians in Tian-Yau Manifolds and SYZ Mirror Symmetry
Friday, November 15
9:30 AM Paul Seidel | Automorphisms of Fukaya Categories and Quantum Steenrod Operations 11:00 AM Mark Mclean | Birational Calabi-Yau Manifolds have the Same Small Quantum Products 1:00 PM Shing-Tung Yau | The HMS/SYZ Program: A Progress Report
From Equivariant Blow-Ups to Modular Symbols
Maxim Kontsevich, IHES
Part of the beauty of mathematics is the interplay between its different branches. In this lecture, Maxim Kontsevich will talk about a recent discovery of an unexpected relation between questions in birational algebraic geometry and the theory of automorphic forms. Computer experiments with an easy-looking system of equations played an essential role in the discovery.
Kontsevich’s recent work on this topic has been with Yuri Tschinkel and Vasily Pestun. At the origin of this work was a question of constructing birational invariants of algebraic varieties endowed with an action of a finite group (e.g., a cyclic group generated by an automorphism of finite order). Based on a simple ansatz that ignores almost all non-trivial geometric information except spectra of group action at fixed points, they arrived at an overdetermined system of linear equations. To their surprise, the system has sporadic non-trivial solutions giving, for instance, an invariant of 3-dimensional manifolds with a birational automorphism of order 43. The existence of such a solution is explained by the existence of a certain arithmetic object (a motive, or an automorphic form).
About the Speaker
Kontsevich was born in 1964 in the USSR. He studied mathematics at Moscow State University where he was a student of Israel Gelfand. In 1992, he received his Ph.D. from the University of Bonn in Germany. Don Zagier served as his thesis advisor. For one year, he was a professor at the University of California, Berkeley. Since 1995, he has been a permanent professor at the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, France. Kontsevich works in many areas of modern mathematics and mathematical physics and has received numerous prizes, including the Fields Medal in 1998.
More information is available here.
University of California, Berkeley
Coisotropic Branes and HMS for Tori
Auroux will describe an approach, developed in the Ph.D. thesis of his student Yingdi Qin, to the problem of incorporating coisotropic branes into the Fukaya category of a torus, as well as the motivation for this construction from the perspective of SYZ and homological mirror symmetry. Qin’s work also gives insight into the equivalence between the Fukaya categories of dual symplectic tori (e.g., elliptic curves with inverse areas), which Auroux will explain if time permits.
Massachusetts Institute of Technology
Special Lagrangians in Tian-Yau Manifolds and SYZ Mirror Symmetry
Collins will discuss the existence of special Lagrangian submanifolds in noncompact Calabi-Yau manifolds constructed by Tian-Yau and connections with mirror symmetry. Collins will discuss the existence of special Lagrangian torus fibrations in complex dimension 2 on Tian-Yau surfaces and a conjectural dual special Lagrangian fibration on the mirror surface. This is joint work with A. Jacob and Y.-S. Lin.
Stony Brook University
Birational Calabi-Yau Manifolds Have the Same Small Quantum Products
We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of symplectic cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.
King’s College London
The Tropical Critical Point and Mirror Symmetry
Call a (generalized) Puiseaux series positive if the leading term is a positive real number. Suppose we are given a Laurent polynomial f(x_1,…, x_n) over the field of generalized Puiseaux series, and that f has positive coefficients. Rietsch shows that, under a mild hypothesis on the Newton polytope, such a Laurent polynomial has a unique positive critical point. Rietsch will give some applications of this result related to mirror symmetry. This is joint work with Jamie Judd.
Massachusetts Institute of Technology
Automorphisms of Fukaya Categories and Quantum Steenrod Operations
Seidel will consider the formal germ of the autoequivalence group of a Fukaya category. This contains arithmetic information, partly encoded in quantum Steenrod operations.
University of Edinburgh
Lagrangian Cobordism and Chow Groups
Homological mirror symmetry implies an isomorphism between the Grothendieck group of the derived category and that of the Fukaya category. The former is related to the Chow group, via the Chern character, whereas the latter is related to the Lagrangian cobordism group through the work of Biran-Cornea. One can try to compare these two groups directly. Sheridan will describe joint work with Ivan Smith in which they take some preliminary steps in this direction and find some interesting analogies.
Local-to-Global Principles in Floer Theory
Varolgunes will start by reviewing his thesis work, which introduced Floer theoretic invariants for arbitrary compact subsets of symplectic manifolds and proved a descent statement under restrictive assumptions. He will then report on ongoing work regarding applications to symplectic manifolds that are presented as the total space of a Lagrangian fibration or the union of a standard neighborhood of a symplectic SC divisor and a Liouville domain. These are joint works with D. Tonkonog, M. Abouzaid and Y. Groman.
The HMS/SYZ Program: A Progress Report
Yau will survey recent progress, with particular focus on the dHYM approach to stability, mirror functors by gluing construction and construction of a new class of singular CY mirror pairs by cyclic covers, as well as their applications.