Let $M$ be a compact complex manifold, non necessairly projective (in my case of interset, it has no nonconstant meromorphic functions). Let $D$ be an effective divisor.
I couldn't find results on the (non) abelianity of $\pi_1(M\setminus D)$ in this context. I've seen that when $M$ is projective and $D$ is ample, then it is true, but i didn't understood well the proof.
Are there results linked with the normal bundle of $D$, or if we know something about the line bundle $L$ such that $D=(s)_0$ for $s\in H^0(M,L)$ ? Precisely, if $M$ is a compact and simply connected complex manifold, and $D =(s)_0$ where $s\in H^0(M,L)$ for some line bundle $L$ on $M$, can we say something about the abelianity of $\pi_1(M\setminus D,x)$ ?
Thank you very much,