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Pluricanonical maps of varieties of maximal Albanese dimension

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Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of \(\kappa (X)\) in terms of the set \(V^0(X,\omega_{X} ):=\{ P\in{\mbox{\rm Pic}}^0(X)|h^0(X, \omega_X \otimes P) \ne 0\}\). An immediate consequence of this is that the Kodaira dimension \(\kappa (X)\) is invariant under smooth deformations. We then study the pluricanonical maps \(\varphi _m:X \dashrightarrow \mathbb{P} (H^0(X,mK_X))\). We prove that if X is of general type, \(\varphi _m\) is generically finite for \(m\geq 5\) and birational for \(m\geq 5 \mbox{\rm dim} (X) +1\). More generally, we show that for \(m\geq 6\) the image of \(\varphi _m\) is of dimension equal to \(\kappa (X)\) and for \(m\geq 6\kappa (X)+2\), \(\varphi _m\) is the stable canonical map.

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Received July 7, 2000 / Published online April 12, 2001

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Chen, J., Hacon, C. Pluricanonical maps of varieties of maximal Albanese dimension. Math Ann 320, 367–380 (2001). https://doi.org/10.1007/PL00004478

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  • DOI: https://doi.org/10.1007/PL00004478

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