Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces

Yi Hu, Chien Hao Liu, Shing Tung Yau

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


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 languageEnglish (US)
Pages (from-to)457-505
Number of pages49
JournalAdvances in Theoretical and Mathematical Physics
Issue number3
StatePublished - May 2002


  • 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

  • General Mathematics
  • General Physics and Astronomy


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