Bi-annual Algebraic and Tropical Meetings of
Brown and YaLE

Fall 2017 @ Yale
November 30, 2017

The BATMOBYLE is a vehicle for bringing together the algebraic and tropical geometry groups of Brown and Yale for a biannual day of talks alternating between the two universities.

Paul Hacking

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.

10:00 - 11:00 Coffee and BreakfastTBA
11:00 - 12:00 A primer on toric degenerations
Dan Corey (TBA)
Room: TBA
12:00 - 02:00 Lunch
02:00 - 03:00 K-stability of log Fano hyperplane arrangements
Kento Fujita (RIMS Kyoto)
Room: TBA
03:00 - 04:00 Tea TBA
04:00 - 05:00 Smoothing toric Fano varieties via mirror symmetry
Paul Hacking (UMass Amherst)
Room: TBA

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).

Previous BATMOBYLEs: Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017.