Index of coregularity zero log Calabi-Yau pairs

TL;DR

This paper establishes a sharp bound on the index of coregularity zero log Calabi--Yau pairs, showing it is at most 2, with implications for various geometric contexts and automorphism actions.

Contribution

It proves that the Weil index of such pairs satisfies 2λ(K_X+B)~0, extending bounds to generalized and singular cases, and applies to key geometric programs.

Findings

Index of coregularity 0 log Calabi--Yau pairs is at most 2.

The bound extends to generalized, semi-log canonical, and isolated singularities.

Applications include bounds for varieties in the Gross--Siebert and Kontsevich--Soibelman programs.

Abstract

In this article, we study the index of log Calabi--Yau pairs $(X,B)$ of coregularity 0. We show that $2\lambda(K_X+B)\sim 0$, where $\lambda$ is the Weil index of $(X,B)$. This is in contrast to the case of klt Calabi--Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi--Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross--Siebert program or in the Kontsevich--Soibelman program is at most $2$. Finally, we discuss applications to Calabi--Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely non-symplectic automorphism.