Relations between indices of Calabi--Yau varieties and pairs

TL;DR

This work investigates how indices of Calabi--Yau varieties relate to those of Calabi--Yau pairs with Kawamata log terminal singularities. It develops an inductive framework using Beauville--Bogomolov decomposition to connect the indices of CY varieties in dimension $n$ to those of CY klt pairs in dimension $n-1$, proving that every $m$ with $\varphi(m)\le 2n$ arises as an index of a CY klt pair of dimension $n-1$, and that $I_{\mathrm{sm}}(n)\subseteq I_{\mathrm{klt}}(n-1,\Phi_{\mathrm{st}})\subseteq I_{\mathrm{term}}(n)$. For $n\le3$ these inclusions are equalities, giving a complete picture in low dimension, while explicit weighted-projective constructions yield CY pairs with small indices. The results imply boundedness consequences in higher dimension and illuminate how CY indices propagate across dimension via products and quotients, with potential impact on the CY index conjecture for klt pairs.

Abstract

We show that for any smooth Calabi--Yau variety, its index can be realized as the index of a Kawamata log terminal (klt) Calabi--Yau pair of lower dimension with standard coefficients. Our approach is based on an inductive argument on the dimension using the Beauville--Bogomolov decomposition. A key step in the argument is to prove that for $n\ge3$, any positive integer $m$ satisfying $\varphi(m)\le 2n$ can be realized as the index of a klt Calabi--Yau pair of dimension $n-1$.

Theorems & Definitions (45)

• Conjecture 1.1: Index conjecture for Calabi--Yau pairs
• Theorem 1.2: xu
• Conjecture 1.4
• Theorem 1.5: \ref{['klt st subseteq term', 'index set in low dimension', 'index set thm']}
• Corollary 1.6
• Remark 1.7
• Theorem 1.8: \ref{['phi<2n and index']}
• Remark 1.9
• Lemma 2.1
• Lemma 2.2