**Toric Fano Varieties and Beyond**

**Toric Fano Varieties and Beyond**

*Online Workshop, December 4th, 2020*

## Invited Speakers

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**

**Organising and Scientific Committee**Gilberto Bini (Università degli Studi di Palermo)

Donatella Iacono (Università degli Studi di Bari)

**Schedule**

**Schedule****(Times in Central European Time Zone, GTM +1, UTC +1)**

9:55-10:00 Welcome

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**: "*O**N 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.