1/4 The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
By Tony Yue Yu
Appears in collection : Tony Yue Yu - The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
Notes: https://nextcloud.ihes.fr/index.php/s/GwJbsQ8xMW2ifb8 1/4 - Motivation and ideas from mirror symmetry, main results.
We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is a plan for each session of the mini-course: 1) Motivation and ideas from mirror symmetry, main results. 2) Skeletal curves: a key notion in the theory. 3) Naive counts, tail conditions and deformation invariance. 4) Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs.