Hirzebruch Surfaces, Tyurin Degenerations and Toric Mirrors: Bridging Generalized Calabi-Yau Constructions

TL;DR

This paper explores diverse constructions of Calabi-Yau manifolds, focusing on Hirzebruch hypersurfaces and their toric counterparts, revealing new deformation connections and mirror symmetry generalizations relevant for string compactifications.

Contribution

It introduces explicit deformation families of Hirzebruch hypersurfaces, identifies their toric models, and generalizes mirror constructions through Laurent deformations and Tyurin degenerations.

Findings

Hirzebruch hypersurfaces have non-Fano properties and Tyurin degenerations.

New Laurent deformations of Calabi-Yau hypersurfaces are identified.

A novel deformation connection between toric spaces is established.

Abstract

There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactifications. Showcasing this diversity, we discuss explicit deformation families of discretely distinct Hirzebruch hypersurfaces in $\mathbb{P}^n \times \mathbb{P}^1$ and identify their toric counterparts in detail. This precise isomorphism is then used to investigate some of their special divisors of interest, and in particular the secondary deformation family of their Calabi-Yau subspaces. In particular, most of the above so called Hirzebruch scrolls are non-Fano, and their (regular) Calabi-Yau hypersurfaces are Tyurin-degenerate, but admit novel (Laurent) deformations by special rational sections as well as a sweeping generalization of the transposition construction of mirror models. This bi-projective embedding also reveals a novel deformation connection between distinct toric spaces, and so also the various divisors of interest including their Calabi-Yau subspaces.