Wednesday, December 5th, 2018
11:00 am - 12:00 pm, in McCormack 3-407

Hulya Arguz

Imperial College London (UK)

Mirrors to Fano varieties in dimension three

Abstract: An anti-canonical pair (Y,D) is a smooth projective variety Y of dimension n, together with a normal crossing divisor D in |-K_Y|. Mirrors to such varieties in dimension two were constructed by Gross--Hacking--Keel from a canonical scattering diagram. We outline steps to generalize this construction to higher dimensions. We interpret the GHK canonical scattering diagram in terms of a toric scattering construction, studied by Gross--Pandharipande--Siebert, applied to a toric model of Y. We then express this GPS scattering construction in terms of Gross--Siebert scattering on an appropriate affine manifold with singularities: the canonical scattering diagram of GHK now appears as an "asymptotic limit". We apply these techniques to higher dimensional examples, thereby finding mirrors to some 3-dimensional Fano manifolds, including non-toric blow-ups of three-dimensional projective space. This is work in progress with Tom Coates, Mark Gross and Bernd Siebert.