Victor Przyjalkowski

Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models

Abstract We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.

Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models

We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Double solids, categories and non-rationality

This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic cycles, in preparation) we enhance constructions from A. Kuznetsov (arXiv:0904.4330) by introducing Noether--Lefschetz spectra --- an interplay between Orlov spectra (C. Oliva, Algebraic cycles and Hodge theory on generalized Reye congruences, Compos. Math. 92, No. 1 (1994) 1--22) and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where above techniques might apply. We start by constructing a sextic double solid

Double quadrics with large automorphism groups

X

Generalized Homological Mirror Symmetry and Cubics

with 35 nodes and torsion in

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

H^3(X,\mathbb Z)

On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

. This is a novelty --- after the classical example of Artin and Mumford (1972), this is the second example of a Fano threefold with a torsion in the 3-rd integer homology group. In particular

Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

X

On Landau–Ginzburg models for Fano varieties

is non-rational. We consider other examples as well ---

Weak Landau-Ginzburg models for smooth Fano threefolds

V_{10}