Note on quasi-polarized canonical Calabi–Yau threefolds

Jie Liu

doi:10.5802/crmath.55

Géométrie algébrique

Note on quasi-polarized canonical Calabi–Yau threefolds

Jie Liu ¹

¹ Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 415-420.

Résumé

Let $(X, L)$ be a quasi-polarized canonical Calabi–Yau threefold. In this note, we show that $| m L |$ is basepoint free for $m \geq 4$ . Moreover, if the morphism $Φ_{| 4 L |}$ is not birational onto its image and $h^{0} (X, L) \geq 2$ , then $L^{3} = 1$ . As an application, if $Y$ is an $n$ -dimensional Fano manifold such that $- K_{Y} = (n - 3) H$ for some ample divisor $H$ , then $| m H |$ is basepoint free for $m \geq 4$ and if the morphism $Φ_{| 4 H |}$ is not birational onto its image, then either $Y$ is a weighted hypersurface of degree $10$ in the weighted projective space $ℙ (1, \dots, 1, 2, 5)$ or $h^{0} (Y, H) = n - 2$ .

Reçu le : 2019-10-15
Révisé le : 2020-04-17
Accepté le : 2020-04-22
Publié le : 2020-07-28

DOI : 10.5802/crmath.55

Classification : 14E05, 14J30, 14J32, 14J45
Mots clés : birationality, Calabi–Yau threefolds, Fano manifolds, freeness

Affiliations des auteurs :

Jie Liu ¹

¹ Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Note on quasi-polarized canonical {Calabi{\textendash}Yau} threefolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--420},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     doi = {10.5802/crmath.55},
     language = {en},
}

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Jie Liu. Note on quasi-polarized canonical Calabi–Yau threefolds. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 415-420. doi : 10.5802/crmath.55. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.55/

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