https://pure.mpg.de/pubman/item/item_3121492_1/component/file_3121493/Ilten_toric_oa_2013.pdf

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

The paper is joined with arXiv:0911.5428 and improved. 
We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactifications. We state most of known methods of finding Landau-Ginzburg models in terms of Laurent polynomials. We discuss Laurent polynomial representation of Landau-Ginzburg models of Fano varieties and state some problems related to it.

Weak Landau-Ginzburg models for smooth Fano threefolds

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

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

We give a method to estimate Seshadri constants on toric varieties at any point. By using the estimations and toric degenerations, we can obtain some new computations or estimations of Seshadri constants on non-toric varieties. In particular, we investigate Seshadri constants on hypersurfaces in projective spaces and Fano 3-folds with Picard number one in detail.

/pdf/seshadri-constants-via-toric-degenerations-dk4rbmz8l5.pdf

Seshadri constants via toric degenerations

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2014.0704

Four-dimensional Fano toric complete intersections.

The geometric objects of study in this paper are K3 surfaces which admit a polarization by the unique even unimodular lattice of signature (1,17). A standard Hodge-theoretic observation about this special class of K3 surfaces is that their polarized Hodge structures are identical with the polarized Hodge structures of abelian surfaces that are cartesian products of elliptic curves. Earlier work of the first two authors gives an explicit normal form and construction of the moduli space for these surfaces. In the present work, this normal form is used to derive Picard-Fuchs differential equations satisfied by periods of these surfaces. We also investigate the subloci of the moduli space on which the polarization is enhanced. In these cases, we derive information about the Picard-Fuchs differential equations satisfied by periods of these subfamilies, and we relate this information to the theory of genus zero quotients of the upper half-plane by Moonshine groups. For comparison, we also examine the analogous theory for elliptic curves in Weierstrass form.

http://www.cs.umsl.edu/~clingher/cdlw_mckay.pdf

Normal Forms, K3 Surface Moduli, and Modular Parametrizations

We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these families is motivated by the Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds with $h^{2,1} = 1$ over the thrice-punctured sphere. We explore their torically induced fibrations by $M$-polarized K3 surfaces and use these fibrations to construct an explicit geometric transition between an anticanonical hypersurface and a nef complete intersection through a singular subfamily of hypersurfaces. Moreover, we show that another singular subfamily provides a geometric realization of the missing "14th case" variation of Hodge structure from the Doran-Morgan list.

/pdf/the-14th-case-vhs-via-k3-fibrations-2xp6tddz2y.pdf

The 14th case VHS via K3 fibrations

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S 4. There are three pairs of three- dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that S 4 acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard–Fuchs equation for the third Picard rank 19 family by extending the Griffiths–Dwork technique for computing Picard–Fuchs equations to the case of semi-ample hypersurfaces in toric varieties. The holomorphic solutions to our Picard–Fuchs equation exhibit modularity properties known as “Mirror Moonshine”; we relate these properties to the geometric structure of our family.

On a Family of K3 Surfaces with $$\mathcal{S}_{4}$$ Symmetry

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S4. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that S4 acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard-Fuchs equation for the third Picard rank 19 family by extending the Griffiths-Dwork technique for computing Picard-Fuchs equations to the case of semiample hypersurfaces in toric varieties. The holomorphic solutions to our Picard-Fuchs equation exhibit modularity properties known as “Mirror Moonshine”; we relate these properties to the geometric structure of our family.

Jacob Lewis

Papers

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

Normal Forms, K3 Surface Moduli, and Modular Parametrizations

The 14th case VHS via K3 fibrations

On a Family of K3 Surfaces with $$\mathcal{S}_{4}$$ Symmetry

On a Family of K3 Surfaces with S₄ Symmetry