Séminaire de Mathématique

The Frobenius Structure Conjecture for Log Calabi-Yau Varieties (2/4)

by Tony Yue Yu (LMO, Université Paris-Sud)

Europe/Paris
Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane

IHES

Description

Mini-Cours

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. 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 log Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is the 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 and deformation invariance.

4. Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras.

From the same series
1 3 4
Organized by

Maxim Kontsevich

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