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

13 pages