Séminaire de Mathématique

CANCELLED and POSTPONED : 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

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

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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