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June 2010 Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions
Victor Batyrev, Maximilian Kreuzer
Adv. Theor. Math. Phys. 14(3): 879-898 (June 2010).


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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Victor Batyrev. Maximilian Kreuzer. "Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions." Adv. Theor. Math. Phys. 14 (3) 879 - 898, June 2010.


Published: June 2010
First available in Project Euclid: 1 July 2011

zbMATH: 1242.14037
MathSciNet: MR2801412

Rights: Copyright © 2010 International Press of Boston

Vol.14 • No. 3 • June 2010
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