Smooth Calabi-Yau varieties with large index and Betti numbers

TL;DR

This work constructs smooth Calabi–Yau $n$-folds with doubly exponential index $m=(s_{n-1}-1)(2s_{n-1}-3)$ and smooth CYs with extremal sums of Betti numbers, advancing the understanding of maximal topological invariants in all dimensions. It builds on the Esser–Totaro–Wang framework by reducing the problem to toric crepant resolutions of suitable weighted projective spaces and their hypersurfaces, using a detailed combinatorial approach based on Sylvester's sequence to achieve regular unimodular triangulations. The authors develop a toric–combinatorial toolkit (triangulations, dualities, and pulling refinements) to resolve the ambient toric variety and then lift these resolutions to the hypersurface, yielding explicit smooth CY models $V^{(n)}$ and $W^{(n)}$ with prescribed invariants, and they connect these constructions to conjectural extremality and index bounds. They also provide explicit models up to toric automorphisms and discuss implications for the index conjecture and extremal Betti-number behavior across dimensions.

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.

Figures (26)

• Figure 6: $\widecheck{P}_2^{(n + 1)}$
• Figure 7: $\widecheck{P}_2^{(n + 1)}$ with all lattice points shown.
• Figure 8: $\widecheck{P}_2^{(n + 1)} = P^{\leq H} \cup P^{\geq H}$ with its regular subdivision $\mathcal{S}$ glued from $\pi^* \mathcal{T}$ and $\mathrm{Cone}(z, (\pi^* \mathcal{T})|_H)$ along $H$.
• Figure 9: The regular unimodular triangulation $\mathcal{T}^{(n + 1)}$ given by pulling the triangulation $\mathcal{S}$ as in \ref{['triangles/figure/pullbackcone']} at all vertices. Here, the only vertex pulled is $(-1, 0)$, and the new edges depicted with dotted lines.
• Figure : (a) $P_1^{(2)}$ with its cross section $P_2^{(1)} \times \{0\} = P_1^{(2)} \cap \{x_1 = 0\}$.

Theorems & Definitions (47)

• Theorem 1
• Conjecture 2: Index Conjecture
• Conjecture 3
• Theorem 4
• Conjecture 5
• Definition 1.1: A000058 on oeis
• Definition 1.2
• Definition 1.3
• Theorem 1.4: Polar duality
• Definition 1.5