Toric Fano Varieties and Beyond
Online Workshop, December 4th, 2020
Cinzia Casagrande (Università degli Studi di Torino)
Anne-Sophie Kaloghiros (Brunel University)
Andrea Petracci (Freie Universität Berlin)
Michele Rossi (Università degli Studi di Torino)
Organising and Scientific Committee
Gilberto Bini (Università degli Studi di Palermo)
Donatella Iacono (Università degli Studi di Bari)
Schedule(Times in Central European Time Zone, GTM +1, UTC +1)
10:00-10:50 Cinzia Casagrande
11:10-12:00 Michele Rossi
3:00-3:50 (pm) Andrea Petracci
4:10-5:00 (pm) Anne-Sophie Kaloghiros
Title, Abstract and Slides
Cinzia Casagrande: "ON FANO 4-FOLDS WITH LEFSCHETZ DEFECT 3" (SLIDES)
Abstract. We will talk about a classification result for some (smooth, complex) Fano 4-folds. We recall that if X is a Fano 4-fold, the Lefschetz defect delta(X) is an invariant of X defined as follows. Consider a prime divisor D in X and the restriction r:H^2(X,R)->H^2(D,R). Then delta(X) is the maximal dimension of ker(r), where D varies among all prime divisors in X. In a previous work, we showed that if X is not a product of surfaces, then delta(X) is at most 3, and if moreover delta(X)=3, then X has Picard number 5 or 6. We will explain that in the case where X has Picard number 5, there are 6 possible families for X, among which 4 are toric. This is a joint work with Eleonora Romano.
Anne-Sophie Kaloghiros: "TORIC METHODS AND K-STABILITY OF SMOOTH FANO 3-FOLDS" (SLIDES)
Abstract. In this talk, I will discuss explicit results on K-stability of smooth Fano 3-folds. More precisely, criteria for K-polystability of toric Fano 3-folds were proved by differential geometers. I will consider applications of these criteria to the study of K-stability of smooth non-toric Fano 3-folds.
Andrea Petracci: "ON DEFORMATIONS AND SMOOTHINGS OF TORIC FANO VARIETIES" (SLIDES)
Abstract. In this talk I will review what is known about the defomation theory of toric Fanos. In particular I will state some rigidity criteria and some smoothing criteria, which are expressed in terms of the combinatorics of the associated polytopes.
Michele Rossi: "POLAR DUALITY OF TORIC VARIETIES BEYOND THE FANO CONSTRAINT AND CONSEQUENCES IN MIRROR SYMMETRY" (SLIDES)
Abstract. Polar duality between reflexive polytopes gives the well known Batyrev duality between Fano toric varieties, inducing a mirror symmetry between generic sections of their anti-canonical divisors. Borisov extended this result to complete intersections by means of a nef partition of the anti-canonical divisor of a Fano toric variety. In this talk I would like to outline a recipe to overcoming the Fano constraint in the Batyrev-Borisov construction. The key idea is thinking of Batyrev duality as a duality between toric varieties “framed” by their anti-canonical divisor and then allowing a more general “framing”, in principle just given by an effective divisor.
In particular, a generic projective hypersurface can be thought of a framing of the ambient projective space, so admitting (at least) one mirror dual. This process, when restricted to Calabi-Yau projective hypersurfaces, reduces precisely to Batyrev’s duality and when considered for negative Kodaira dimension hypersurfaces produces Landau-Ginzburg mirror models, analogously to what proposed by Givental.