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Complements and coregularity of Fano varieties

Fernando Figueroa, Stefano Filipazzi, Joaquín Moraga, Junyao Peng

TL;DR

This work analyzes how the coregularity of Fano and Calabi–Yau-type varieties constrains their index and complements. By employing generalized pairs and a canonical bundle formula, it establishes sharp index bounds for log Calabi–Yau pairs of coregularity $0$ and $1$ (the latter ensuring $I(K_X+B+\mathbf{M}_X)\sim 0$ with $I=m\lambda$ and $m\le120\lambda$), and proves that Fano varieties of absolute coregularity $0$ or $1$ admit bounded complements with small $N$ (namely $N\in\{1,2\}$ and $N\in\{1,2,3,4,6\}$, respectively). It also extends these results to klt singularities and outlines how higher coregularity bounds reduce to index conjectures and bounded B-representations, using a novel canonical bundle formula for pairs with bounded relative coregularity. Overall, the paper generalizes classical 2D $A$/$D$/$E$-type classifications to arbitrary dimensions and provides a unified framework for lifting and controlling complements via coregularity. The methods rely on inductive arguments across dimension and coregularity, together with reductions to bases and the use of Kollár–Xu-type models.

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 $120λ^2$, where $λ$ 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.

Complements and coregularity of Fano varieties

TL;DR

This work analyzes how the coregularity of Fano and Calabi–Yau-type varieties constrains their index and complements. By employing generalized pairs and a canonical bundle formula, it establishes sharp index bounds for log Calabi–Yau pairs of coregularity and (the latter ensuring with and ), and proves that Fano varieties of absolute coregularity or admit bounded complements with small (namely and , respectively). It also extends these results to klt singularities and outlines how higher coregularity bounds reduce to index conjectures and bounded B-representations, using a novel canonical bundle formula for pairs with bounded relative coregularity. Overall, the paper generalizes classical 2D //-type classifications to arbitrary dimensions and provides a unified framework for lifting and controlling complements via coregularity. The methods rely on inductive arguments across dimension and coregularity, together with reductions to bases and the use of Kollár–Xu-type models.

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 of coregularity is at most , where is the Weil index of . This extends a recent result due to Filipazzi, Mauri, and Moraga. We prove that a Fano variety of absolute coregularity admits either a -complement or a -complement. In the case of Fano varieties of absolute coregularity , we show that they admit an -complement with at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity admits either a -complement or -complement. Furthermore, a klt singularity of absolute coregularity admits an -complement with at most 6. This extends the classic classification of -type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity . 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 , we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.
Paper Structure (3 sections, 3 theorems)

This paper contains 3 sections, 3 theorems.

Table of Contents

  1. Introduction
  2. Log Calabi--Yau pairs
  3. Index and coregularity of log Calabi--Yau pairs

Key Result

Theorem 1

Let $(X,B,\mathbf{M}_{})$ be a projective generalized log Calabi--Yau pairA generalized log Calabi--Yau pair is a generalized lc pair $(X,B,\mathbf{M}_{})$ with $K_X+B+\mathbf{M}_{X}\sim_\mathbb{Q} 0$. of coregularity $0$ and let be $\lambda$ a positive integer. Assume that $\lambda(K_X+B+\mathbf{M}

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3