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SUMMARY:The Frobenius Structure Conjecture for Log Calabi-Yau Varieties (4
/4)
DTSTART;VALUE=DATE-TIME:20200730T083000Z
DTEND;VALUE=DATE-TIME:20200730T103000Z
DTSTAMP;VALUE=DATE-TIME:20220128T164120Z
UID:indico-event-5874@indico.math.cnrs.fr
DESCRIPTION:Mini-Cours\n\nWe show that the naive counts of rational curves
in an affine log Calabi-Yau variety U\, containing an open algebraic toru
s\, determine in a surprisingly simple way\, a family of log Calabi-Yau va
rieties\, as the spectrum of a commutative associative algebra equipped wi
th a multilinear form. This is directly inspired by a very similar conject
ure of Gross-Hacking-Keel in mirror symmetry\, known as the Frobenius stru
cture conjecture. Although the statement involves only elementary algebrai
c geometry\, our proof employs Berkovich non-archimedean analytic methods.
We construct the structure constants of the algebra via counting non-arch
imedean analytic disks in the analytification of U. We establish various p
roperties of the counting\, notably deformation invariance\, symmetry\, gl
uing 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 c
anonical scattering diagram defined by counting infinitesimal non-archimed
ean analytic cylinders\, without using the Kontsevich-Soibelman algorithm.
Several combinatorial conjectures of GHKK follow readily from the geometr
ic description. This is joint work with S. Keel\; the reference is arXiv:1
908.09861. If time permits\, I will mention another application of our the
ory 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 sess
ion of the mini-course:\n\n1) Motivation and ideas from mirror symmetry\,
main results.\n2) Skeletal curves: a key notion in the theory.\n3) Naive c
ounts\, tail conditions and deformation invariance.\n4) Scattering diagram
\, comparison with Gross-Hacking-Keel-Kontsevich\, applications to cluster
algebras\, applications to moduli spaces of Calabi-Yau pairs.\n\nRegistra
tion is compulsory. Please click on the link below to receive the zoom lin
k and password to join the mini-course online:\n\nhttps://us02web.zoom.us/
meeting/register/tZIvcOCorD8iG9ES5hqURXELfgJhQbXND8N1\n\n \n\n \n\nhttps
://indico.math.cnrs.fr/event/5874/
LOCATION:IHES Webinar
URL:https://indico.math.cnrs.fr/event/5874/
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