Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces
The geometry of toric varieties, Calabi-Yau manifolds, and their fibrations has become one of the most important mathematical tools in contemporary quantum field theory. Recently, these methods have been heavily used by physicists, in particular by Philip Candelas and his school, to construct dual pairs of manifolds in the study of heterotic-string/F-theory. Motivated by this important physical background, the authors of the paper under review investigate more closely the structure of toric morphisms between toric varieties, focusing on the various computational details needed for applications to string theory. Accordingly, their work is subdivided into three main parts. After providing the essential mathematical background for physicists, mainly via references to the existing literature on toric geometry and Calabi-Yau manifolds, the first part of the paper deals with the general study of toric morphisms. In particular, a complete description of their fibres, images, and flattening stratifications is given, together with two illustrating examples. The second part is devoted to a fine analysis of line bundles on toric varieties and their sections. This general discussion leads to a computational scheme for the description of a toric morphism and the induced morphism of toric hypersurfaces defined by line bundle sections therein. The third part consists in concrete applications to physics, in that the authors work out their computational scheme for toric morphisms by means of a physically relevant example. Namely, they apply it to the detailed study of a family of \(4\)-dimensional elliptic Calabi-Yau toric hypersurfaces. This particular family has been recently discussed, in a physical context, by \textit{V. Braun, P. Candelas, X. de la Ossa} and \textit{A. Grassi} in their article ``Toric Calabi-Yau fourfolds, duality between \(N=1\) theories, and divisors that contribute to the superpotential'' (preprint, hep-th/0001208). At the end of the paper, some problems and directions for further study are briefly listed. All together, this work under review provides a wealth of extremely subtle, detailed and concrete computations in toric geometry, which are highly propelling in both complex geometry and mathematical physics. Although motivated by recent developments in heterotic-string/F-theory, the paper is mainly mathematical in nature, and certainly a major contribution to the general study of toric varieties and their morphisms.
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