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We provide a lower bound for the number of components of the algebraicmonodromy group in the situation of a $K3$ surface over a number field $k$. Inthe CM case, this bound is conjecturally sharp, which is proved in the cases ofCM by an imaginary quadratic field and, for $k={\mathbb Q}$, of CM by a cyclicfield. As an application, we show that, in the case of CM by $\smash{{\mathbbQ}(\sqrt{-\delta})}$, the jump character is given by either$\smash{(\frac{-\delta}.)}$ or $1$, the case distinction only depending on thegeometric Picard rank. The cyclic case is similar.

On the component group of the algebraic monodromy group of a $K3$ surface