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arXiv · 2502.07031

Smooth Calabi-Yau varieties with large index and Betti numbers

Jas Singh
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Abstract

A normal variety $X$ is called Calabi-Yau if $K_X \sim_{\mathbb Q} 0$. The index of $X$ is the smallest positive integer $m$ so that $m K_X \sim 0$. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general. This work builds off of the singular Calabi-Yau varieties found by Esser, Totaro, and Wang in arXiv:2209.04597.

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Jas Singh. 2025-02-10. Smooth Calabi-Yau varieties with large index and Betti numbers. https://doi.org/10.1007/s00209-026-03984-y

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