Helge Ruddat, Univ. of Stavanger: Smoothing Toroidal Crossing Fano Varieties Using Log Structures

Helge Ruddat, University of Stavanger: Smoothing and resolving toroidal crossing Fano varieties using log structures obtained from zero-mutable Laurent polynomials
In a series of joint works with Alessio Corti, we construct generically log smooth Fano varieties from reflexive polytopes. The log structures are obtained locally from a zero mutable Laurent polynomial and we conjecture that all log structures of this type are smoothable. A related conjecture for toric singularities was recently stated by Corti-Filip-Petracci. We also conjecture the existence and uniqueness of log resolutions alongside some interesting new mutation structure of the resolutions. A proof of the conjecture for admissible log singularities, as well as for the singularities known as Tom and Jerry, is current joint work with Tim Gräfnitz. The prospective outcome of this program is a unified construction of compact Fano manifolds, possibly in all dimensions but certainly for Fano 3-folds. The method is also believed to work for Q-Gorenstein Fano varieties.