Complements and coregularity of Fano varieties

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.