Complements and coregularity of Fano varieties
/ Authors
/ Abstract
Abstract We study the relation between the coregularity, the index of log Calabi–Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi–Yau pair $(X,B)$ of coregularity $1$ is at most $120\lambda ^2$ , where $\lambda $ is the Weil index of $K_X+B$ . This extends a recent result due to Filipazzi, Mauri and Moraga. We prove that a Fano variety of absolute coregularity $0$ admits either a $1$ -complement or a $2$ -complement. In the case of Fano varieties of absolute coregularity $1$ , we show that they admit an N-complement with N at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity $0$ admits either a $1$ -complement or $2$ -complement. Furthermore, a klt singularity of absolute coregularity $1$ admits an N-complement with N at most 6. This extends the classic classification of $A,D,E$ -type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity $2$ . In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least $3$ , we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.
Journal: Forum of Mathematics, Sigma
DOI: 10.1017/fms.2024.69