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.

arXiv comment56 pages. Final version, to appear in Forum Math. Sigma

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• Inversion of adjunction on log canonicity
• Finiteness of algebraic fundamental groups
• The dual complex of Calabi--Yau pairs
• ACC for minimal log discrepancies of exceptional singularities
• Torus invariant divisors
• On the connectedness principle and dual complexes for generalized pairs
• Existence and boundedness of $n$-complements
• A geometric characterization of toric singularities
• Moduli of $\mathbb{Q}$-Gorenstein pairs and applications
• Log canonical thresholds and coregularity

Cited by in corpus (1)

• G-coregularity of del Pezzo surfaces