Note on quasi-polarized canonical Calabi–Yau threefolds

Liu, Jie

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

Let $(X,L)$ be a quasi-polarized canonical Calabi–Yau threefold. In this note, we show that $\left|mL\right|$ is basepoint free for $m\ge 4$. Moreover, if the morphism ${\Phi }_{\left|4L\right|}$ is not birational onto its image and $h^0(X,L)\ge 2$, 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 $\left|mH\right|$ is basepoint free for $m\ge 4$ and if the morphism ${\Phi }_{\left|4H\right|}$ is not birational onto its image, then either $Y$ is a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1,\cdots ,1,2,5)$ or $h^0(Y,H)=n-2$.

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

[1] Beauville, Arnaud 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] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James 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] Floris, Enrica Fundamental divisors on Fano varieties of index $n-3$, Geom. Dedicata, Volume 162 (2013), pp. 1-7 | DOI | MR | Zbl

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

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

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

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

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

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

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