Note on quasi-polarized canonical Calabi-Yau threefolds

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Let (X,L) be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that mL is basepoint free for m≥ 4. Moreover, if the morphism Φ 4L is not birational onto its image and h⁰(X,L)≥ 2, then L³=1. As an application, if Y is a n-dimensional Fano manifold such that -KY=(n-3)H for some ample divisor H, then mH is basepoint free for m≥ 4 and if the morphism Φ 4H is not birational onto its image, then Y is either a weighted hypersurface of degree 10 in the weighted projective space P(1,,1,2,5) or h⁰(Y,H)=n-2.