Smoothing toric Fano varieties via mirror symmetry
A pair (X,E) consisting of a smooth Fano variety together with a choice of smooth anticanonical divisor corresponds under mirror symmetry to a family W : U -> A1 of Calabi-Yau varieties over the affine line such that the monodromy at infinity is maximally unipotent. Moreover, a degeneration of (X,E) to a singular toric Fano variety together with its toric boundary corresponds to an open embedding of an algebraic torus in U. Using this heuristic, we construct smoothings of Gorenstein toric Fano 3-folds determined by combinatorial data encoding the construction of the mirror as a blowup of a toric variety. The smoothing is described using the scattering diagram of Kontsevich-Soibelman and Gross-Siebert which encodes counts of holomorphic discs on the mirror via tropical geometry.
This is a part of a program initiated by Coates, Corti, Galkin, Golyshev, and Kasprzyk to classify smooth Fano varieties using mirror symmetry. Period and Gromov--Witten calculations by these authors suggest that all deformation types of Fano 3-folds are obtained by our construction.
This is a report on work in progress with Alessio Corti, Mark Gross, and Andrea Petracci.
|10:30 - 11:00||Coffee and Breakfast||Room: AKW 100|
|11:00 - 12:00||A primer on toric degenerations
Dan Corey (TBA)
|Room: AKW 100|
|12:00 - 02:30||Lunch||Hockey Rink|
|02:30 - 03:30||K-stability of log Fano hyperplane arrangements
Kento Fujita (RIMS Kyoto)
|Room: ML 104|
|03:30 - 04:30||Tea||Math Dept Lounge|
|04:30 - 05:30||Smoothing toric Fano varieties via mirror symmetry
Paul Hacking (UMass Amherst)
|Room: LOM 215|