Symplectic invariance of uniruled affine varieties and log Kodaira dimension

Abstract

We introduce some definitions of uniruledness for affine varieties and use these ideas to show symplectic invariance of various algebraic invariants of affine varieties. For instance we show that if $A$ and $B$ are symplectomorphic smooth affine varieties, then any compactification of $A$ by a projective variety is uniruled if and only if any such compactification of $B$ is uniruled. If $A$ is acylic of dimension $2$, then we show that $B$ has the same log Kodaira dimension as $A$. If $A$ has dimension $3$, has log Kodaira dimension $2$, and satisfies some other conditions, then $B$ cannot be of log general type.