Toric Geometry and Calabi-Yau Compactifications

TL;DR

The notes survey toric constructions of Calabi–Yau manifolds as hypersurfaces and complete intersections, with emphasis on data relevant to string dualities. They present the foundational toric toolkit (fans, homogeneous coordinates) and explain how fibrations and torsion in cohomology arise from polytope combinatorics and lattice refinements, including computational aspects. Key results include explicit descriptions of fundamental groups and Brauer groups in toric CYs, Batyrev–Borisov duality for complete intersections, and practical criteria to detect fibrations via polytope geometry, along with caveats about computational cost. The work highlights ongoing progress on torsion, topological transitions, and open classification questions, with references to PALP and mirror-symmetry considerations.

Abstract

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.