Paul Hacking (UMass Amherst) |
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 |
Brown Organizers:
Dan Abramovich (Brown),
Kenny Ascher (MIT),
Dori Bejleri (Brown),
Melody Chan (Brown),
Nathan Pflueger (Amherst),
and
Dhruv Ranganathan (MIT).
Yale Organizers:
Asher Auel (Yale),
Max Kutler (Yale),
Kalina Mincheva (Yale),
Sam Payne (Yale),
Yuchen Liu (Yale).