Pluricanonical maps of varieties of maximal Albanese dimension

Abstract

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.