ID: 1901.05503

Complete intersection Calabi--Yau threefolds in Hibi toric varieties and their smoothing

January 16, 2019

View on ArXiv
Makoto Miura
Mathematics
Algebraic Geometry
Combinatorics

In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular Calabi-Yau threefolds. We focus on such non-singular Calabi-Yau threefolds of Picard number one, and illustrate the calculation of topological invariants, using new motivating examples.

Similar papers 1

Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

February 22, 2008

90% Match
Victor Batyrev, Maximilian Kreuzer
Algebraic Geometry

We construct a surprisingly large class of new Calabi-Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^*$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi-Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflex...

Find SimilarView on arXiv

Minuscule Schubert Varieties and Mirror Symmetry

January 31, 2013

89% Match
Makoto Miura
Algebraic Geometry
Combinatorics

We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi-Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety ${\boldsymbol{\Sigma}}$ of the Cayley plane ${\mathbb{OP}}^2...

Find SimilarView on arXiv

Strings on Calabi--Yau spaces and Toric Geometry

March 29, 2001

89% Match
Maximilian Kreuzer
High Energy Physics - Theory

After a brief introduction into the use of Calabi--Yau varieties in string dualities, and the role of toric geometry in that context, we review the classification of toric Calabi-Yau hypersurfaces and present some results on complete intersections. While no proof of the existence of a finite bound on the Hodge numbers is known, all new data stay inside the familiar range $h_{11}+h_{12}\le 502$.

Find SimilarView on arXiv

On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

September 13, 2018

88% Match
Yu-Chien Huang, Washington Taylor
Algebraic Geometry

We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with $h^{1, 1} \geq 140$ or $h^{2, 1} \geq 140$ that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number i...

Find SimilarView on arXiv

A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List

November 5, 2014

88% Match
Ross Altman, James Gray, Yang-Hui He, ... , Nelson Brent D.
Algebraic Geometry

Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, i...

Find SimilarView on arXiv

Calabi-Yau construction by smoothing normal crossing varieties

April 27, 2006

88% Match
Nam-Hoon Lee
Algebraic Geometry
Differential Geometry

We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds with Picard number one with some interesting properties.

Find SimilarView on arXiv

On smooth Calabi-Yau threefolds of Picard number two

April 10, 2021

88% Match
Christian Mauz
Algebraic Geometry

We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.

Find SimilarView on arXiv

Toric Geometry and Calabi-Yau Compactifications

December 29, 2006

88% Match
Maximilian Kreuzer
High Energy Physics - Theory

These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues and topological transitions.

Find SimilarView on arXiv

Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties

May 2, 2006

87% Match
Charles F. Doran, John W. Morgan
Algebraic Geometry
Algebraic Topology

We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological.

Find SimilarView on arXiv

Smoothing Calabi-Yau toric hypersurfaces using the Gross-Siebert algorithm

September 4, 2019

87% Match
Thomas Prince
Algebraic Geometry

We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. We expect these to degenerate to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities. In particular, we explain how to `smooth the boundary' of a class of $4$-dimensional reflexive polytopes to obtain a polarised tropical manifolds. We compute topological invariants of a compactified torus fibration over each such tropic...

Find SimilarView on arXiv