Index of coregularity zero log Calabi-Yau pairs

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.

arXiv comment30 pages. We included Section 1.7 and 8 about quotients of Calabi--Yau and holomorphic symplectic varieties

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