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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     author = {Jie Liu},
     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/

Bibliographie
Cité par

[1] Arnaud Beauville A Calabi–Yau threefold with non-abelian fundamental group, New trends in algebraic geometry (Warwick, 1996) (London Mathematical Society Lecture Note Series), Volume 264, Cambridge University Press, 1999, pp. 13-17 | DOI | MR | Zbl

[2] Caucher Birkar; Paolo Cascini; Christopher D. Hacon; James McKernan Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR | Zbl

[3] Enrica Floris Fundamental divisors on Fano varieties of index $n - 3$ , Geom. Dedicata, Volume 162 (2013), pp. 1-7 | DOI | MR | Zbl

[4] Takao Fujita Classification of polarized manifolds of sectional genus two, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, 1988, pp. 73-98 | DOI | MR | Zbl

[5] Takao Fujita Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, 155, Cambridge University Press, 1990, xiv+205 pages | DOI | MR | Zbl

[6] Chen Jiang On birational geometry of minimal threefolds with numerically trivial canonical divisors, Math. Ann., Volume 365 (2016) no. 1-2, pp. 49-76 | DOI | MR | Zbl

[7] Yujiro Kawamata Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces., Ann. Math., Volume 127 (1988) no. 1, pp. 93-163 | DOI | MR | Zbl

[8] Yujiro Kawamata On effective non-vanishing and base-point-freeness, Asian J. Math., Volume 4 (2000) no. 1, pp. 173-181 (Kodaira’s issue) | DOI | MR | Zbl

[9] János Kollár; Shigefumi Mori Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998, viii+254 pages | DOI | MR | Zbl

[10] Oliver Küchle Some remarks and problems concerning the geography of Fano $4$ -folds of index and Picard number one, Quaest. Math., Volume 20 (1997) no. 1, pp. 45-60 | DOI | MR | Zbl

[11] Jie Liu Second Chern class of Fano manifolds and anti-canonical geometry, Math. Ann., Volume 375 (2019) no. 1-2, pp. 655-669 | DOI | MR | Zbl

[12] Wenfei Liu; Sönke Rollenske Pluricanonical maps of stable log surfaces., Adv. Math., Volume 258 (2014), pp. 69-126 | DOI | MR | Zbl

[13] Keiji Oguiso On polarized Calabi-Yau $3$ -folds, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 38 (1991) no. 2, pp. 395-429 | MR | Zbl

[14] Keiji Oguiso; Thomas Peternell On polarized canonical Calabi–Yau threefolds, Math. Ann., Volume 301 (1995) no. 2, pp. 237-248 | DOI | MR | Zbl

[15] Wenhao Ou On generic nefness of tangent sheaves (2017) (https://arxiv.org/abs/1703.03175v1)

[16] Igor Reider Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. Math., Volume 127 (1988) no. 2, pp. 309-316 | DOI | MR | Zbl

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