Complements and coregularity of Fano varieties
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+...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000690/type/journal_article |
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| author | Fernando Figueroa Stefano Filipazzi Joaquín Moraga Junyao Peng |
| author_facet | Fernando Figueroa Stefano Filipazzi Joaquín Moraga Junyao Peng |
| author_sort | Fernando Figueroa |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-52e6f44ce34343e086ae553a4e4fcc6d |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-52e6f44ce34343e086ae553a4e4fcc6d2025-02-07T07:50:23ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.69Complements and coregularity of Fano varietiesFernando Figueroa0https://orcid.org/0009-0008-0982-8708Stefano Filipazzi1https://orcid.org/0000-0002-0380-5694Joaquín Moraga2Junyao Peng3Department of Mathematics, Northwestern University, Evanston, IL 60208, USA; E-mail:EPFL, SB MATH-CAG, MA C3 625 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland; E-mail:UCLA Mathematics Department, Box 951555, Los Angeles, Los Angeles, CA 90095-1555, USAPrinceton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA; E-mail: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.https://www.cambridge.org/core/product/identifier/S2050509424000690/type/journal_article14E3014B05 |
| spellingShingle | Fernando Figueroa Stefano Filipazzi Joaquín Moraga Junyao Peng Complements and coregularity of Fano varieties Forum of Mathematics, Sigma 14E30 14B05 |
| title | Complements and coregularity of Fano varieties |
| title_full | Complements and coregularity of Fano varieties |
| title_fullStr | Complements and coregularity of Fano varieties |
| title_full_unstemmed | Complements and coregularity of Fano varieties |
| title_short | Complements and coregularity of Fano varieties |
| title_sort | complements and coregularity of fano varieties |
| topic | 14E30 14B05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000690/type/journal_article |
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