2008/02/22 by Victor V. Batyrev, Maximilian Kreuzer, Batyrev, Victor +1
Chemistry · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Molecular spectroscopy and chirality #Synthesis and Properties of Aromatic Compounds
paper · pdf · doi:10.48550/arxiv.0802.3376
openalex publication_date 2008/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a surprisingly large class of new Calabi-Yau 3-folds X with small Picard numbers and propose a construction of their mirrors X^* using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi-Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive 4-polytopes whose 2-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi-Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive 4-polytopes such that the corresponding Calabi-Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive 4-polytopes defining 68 topologically different Calabi--Yau 3-folds with h11=1. We explain the mirror construction and compute several new Picard--Fuchs operators for the respective 1-parameter families of mirror Calabi-Yau 3-folds.