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 $V^0(X,\omega_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, \omega_X \otimes P) \ne 0\}$. An immediate consequence of this is that the Kodaira dimension $\kappa (X)$ is invariant under smooth deformations. We then study the pluricanonical maps $\phi_m:X -> \Bbb{P} (H^0(X,mK_X))$. We prove that if $X$ is of general type, $\phi_m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \text{\rm dim} (X) +1$. More generally, we show that for $m\geq 6$ the image of $\phi_m$ is of dimension equal to $\kappa (X)$ and for $m\geq 6\kappa (X)+2$, $\phi_m$ is the stable canonical map.