Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K\_M$).
Let $k$ be the number of components of $D$.


Some cheap thoughts give:


If $M$ is a Fano manifold of dimension higher than one, $k=1$.


If $M=\mathbb P^1$ or $M= \mathbb P^1 \times X$, where $X$ is a projective manifold with trivial canonical class, then $k=2$.


Is it possible to have $k$ bigger than two? Is $k$ bounded?



Possible number of components of anticanonical sections of projective manifolds
(Let me turn my comment into an answer to give a correct attribution.)


Yes, in the setup you describe, the maximum number of connected components of the divisor $D$ is 2. This is related to the so-called *connectedness principle* in birational geometry.


The statement that you want follows from Proposition 5.1 of [*Log canonical singularities are Du Bois*](https://arxiv.org/pdf/0902.0648.pdf) by Kollár and Kovács. They prove in particular that if $(Y,\Delta)$ is an lc pair such that $K\_Y + \Delta$ is $\mathbf Q$-linearly equivalent to 0, then the non-klt locus $\operatorname{nklt}(Y,\Delta)$ has 1 or 2 components.


In your case, taking $\Delta=D$, then $\operatorname{nklt}(Y,\Delta)$ is just (the support of) $D$, so this set has at most 2 connected components.


