Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

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 four-polytopes whose two-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 four-polytopes such that the corresponding Calabi-Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive four-polytopes defining 68 topologically different Calabi-Yau 3-folds with $h_11 = 1$. We explain the mirror construction and compute several new Picard-Fuchs operators for the respective oneparameter families of mirror Calabi-Yau 3-folds.

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Published 1 January 2010