### Abstract

Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be understood by the various restrictions of a line bundle with a section that defines the hypersurface. These general toric geometry discussions give rise to a computational scheme for the details of a toric morphism and the induced fibration of toric hypersurfaces therein. We apply this scheme to study the family of complex 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. Some directions for future work are listed in the end.

Original language | English (US) |
---|---|

Pages (from-to) | 457-505 |

Number of pages | 49 |

Journal | Advances in Theoretical and Mathematical Physics |

Volume | 6 |

Issue number | 3 |

State | Published - May 2002 |

### Fingerprint

### Keywords

- Elliptic calabi-yau manifold
- Fibration
- Fibred calabi-yau manifold
- Heterotic-string/F-theory duality
- Primitive cone
- Relative star
- Toric calabi-yau hypersurface
- Toric morphism

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Mathematics(all)

### Cite this

*Advances in Theoretical and Mathematical Physics*,

*6*(3), 457-505.

**Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces.** / Hu, Yi; Liu, Chien Hao; Yau, Shing Tung.

Research output: Contribution to journal › Article

*Advances in Theoretical and Mathematical Physics*, vol. 6, no. 3, pp. 457-505.

}

TY - JOUR

T1 - Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces

AU - Hu, Yi

AU - Liu, Chien Hao

AU - Yau, Shing Tung

PY - 2002/5

Y1 - 2002/5

N2 - Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be understood by the various restrictions of a line bundle with a section that defines the hypersurface. These general toric geometry discussions give rise to a computational scheme for the details of a toric morphism and the induced fibration of toric hypersurfaces therein. We apply this scheme to study the family of complex 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. Some directions for future work are listed in the end.

AB - Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be understood by the various restrictions of a line bundle with a section that defines the hypersurface. These general toric geometry discussions give rise to a computational scheme for the details of a toric morphism and the induced fibration of toric hypersurfaces therein. We apply this scheme to study the family of complex 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. Some directions for future work are listed in the end.

KW - Elliptic calabi-yau manifold

KW - Fibration

KW - Fibred calabi-yau manifold

KW - Heterotic-string/F-theory duality

KW - Primitive cone

KW - Relative star

KW - Toric calabi-yau hypersurface

KW - Toric morphism

UR - http://www.scopus.com/inward/record.url?scp=21444441312&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21444441312&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 457

EP - 505

JO - Advances in Theoretical and Mathematical Physics

JF - Advances in Theoretical and Mathematical Physics

SN - 1095-0761

IS - 3

ER -