# Kodaira dimension and fundamental groups

In Riemannian geometry the non-negativity of the Ricci curvature $R$ of a manifold $X$ has strong implications on the size of the fundamental group $\pi_1(X)$:

1. If $R>0$, then $\pi_1(X)$ is finite.
2. If $R=0$, it is known that $\pi_1(X)$ is almost abelian, i.e., it contains an abelian subgroup of finite index. Also, $\pi_1(X)$ has polynomial growth.

In the case $X$ is a smooth complex projective variety, the positivity of Ricci curvature is related to ampleness properties of of $-K_X$, so it would be interesting to see whether analogous results of the above hold in algebraic geometry, with Ricci curvature replaced by the Kodaira dimension $\kappa(X)=\sup_n\dim\phi_{nK}(X)$. So my question is:

What implications do non-positive Kodaira dimension have for the fundamental group of $X$? In particular, does some versions of the above results hold with Ricci curvature replaced by Kodaira dimension?

For example, if $X$ is a smooth projective variety with $\kappa(X)=0$, is $\pi_1(X)$ almost abelian?

One could also ask for refined versions of the above statements. For example, when $X$ is Fano it is well-known that $\pi_1(X)=0$. Does the same condition hold for all $X$ with big $-K_X$?

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Hi John, regarding your last question: there are ruled, non-rational surfaces with big anticanonical divisor. In fact, the projectivization of a sufficiently unstable rank two vector bundle on a curve is an example of such a surface. –  damiano Mar 7 '11 at 12:05
The following article ams.org/mathscinet/search/… of Campana gives a condition for the fundamental group to be almost abelian, mimicking what is known for the rationally connected picture. –  Frank Mar 7 '11 at 14:11
Maybe I should expand a little on my comment. For every curve $C$, there are ruled surfaces $S \to C$ with big anticanonical divisor. In particular, the fundamental group of a surface with big anticanonical divisor can be the fundamental group of any curve. Thus, for surfaces, the answer to your last question is 'no'. –  damiano Mar 9 '11 at 10:30