Suppose $X$ is a smooth Fano $3$-fold, and $D$ a smooth divisor, with $|D|=-K_{X}$. By adjuction, $K_{D} \equiv 0$, so $D$ is K3 or abelian surface. According to what I read, D has to be K3.
Question: How to rule out abelian surface case?
In the case $k=\mathbb{C}$ is there a "direct" way to see that $D$ is simply connected in the complex topology?
(I looked at two introductory articles on Fano 3-folds and did not find it)
For a Fano, $-K$ is ample and $H^1(O_X)=0$. So, from the exact sequence $0\to O_X(-D)\to O_X\to O_D\to 0$, you have $0=H^1(O_X)\to H^1(O_D)\to H^2(O_X(-D))=0$, the last by Kodaira vanishing. So, $H^1(O_D)=0$, which says $D$ is not abelian.