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SUMMARY:CANCELLED and POSTPONED : The Frobenius Structure Conjecture for L
og Calabi-Yau Varieties
DTSTART;VALUE=DATE-TIME:20200327T100000Z
DTEND;VALUE=DATE-TIME:20200327T113000Z
DTSTAMP;VALUE=DATE-TIME:20200528T152104Z
UID:indico-event-5686@indico.math.cnrs.fr
DESCRIPTION:——– IMPORTANT INFORMATION ——–\n\nDue to the he
alth situation related to the Coronavirus epidemic\, the course has been c
ancelled and postponed at a later date to be confirmed.\n\nMini-Cours\n\nW
e show that the naive counts of rational curves in an affine log Calabi-Ya
u variety U\, containing an open algebraic torus\, determine in a surprisi
ngly simple way\, a family of log Calabi-Yau varieties\, as the spectrum o
f a commutative associative algebra equipped with a multilinear form. This
is directly inspired by a very similar conjecture of Gross-Hacking-Keel i
n mirror symmetry\, known as the Frobenius structure conjecture. Although
the statement involves only elementary algebraic geometry\, our proof empl
oys Berkovich non-archimedean analytic methods. We construct the structure
constants of the algebra via counting non-archimedean analytic disks in t
he 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 v
ariety\, we prove that our algebra generalizes\, and in particular gives a
direct geometric construction of\, the mirror algebra of Gross-Hacking-Ke
el-Kontsevich. Several combinatorial conjectures of GHKK follow readily fr
om the geometric description. This is joint work with S. Keel\; the refere
nce is arXiv:1908.09861. If time permits\, I will mention another applicat
ion 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 th
e plan for each session of the mini-course:\n\n1. Motivation and ideas fro
m mirror symmetry\, main results.\n\n2. Skeletal curves: a key notion in t
he theory.\n\n3. Naive counts and deformation invariance.\n\n4. Scattering
diagram\, comparison with Gross-Hacking-Keel-Kontsevich\, applications to
cluster algebras.\n\n \n\nhttps://indico.math.cnrs.fr/event/5686/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/5686/
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