The truncated complex $K$-moment problem

Let $\gamma \equiv \gamma^{\left( 2n\right) }$ denote a sequence of complex numbers $\gamma _{00}, \gamma _{01}, \gamma _{10}, \dots , \gamma _{0,2n}, \dots , \gamma _{2n,0}$ ( $\gamma _{00}>0, \gamma _{ij}=\bar{\gamma}_{ji}$), and let $K$ denote a closed subset of the complex plane $\mathbb{C}$. The Truncated Complex $K$-Moment Problem for $\gamma$ entails determining whether there exists a positive Borel measure $\mu$ on $\mathbb{C}$ such that $\gamma _{ij}=\int \bar{z}^{i}z^{j}\,d\mu$ ( $0\leq i+j\leq 2n$) and $\operatorname{supp}\mu \subseteq K$. For $K\equiv K_{\mathcal{P}}$ a semi-algebraic set determined by a collection of complex polynomials $\mathcal{P} =\left\{ p_{i}\left( z,\bar{z}\right) \right\} _{i=1}^{m}$, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix $M\left( n\right) \left( \gamma \right)$and the localizing matrices $M_{p_{i}}$. We prove that there exists a $\operatorname{rank}M\left( n\right)$-atomic representing measure for $\gamma ^{\left( 2n\right) }$supported in $K_{\mathcal{P}}$if and only if $M\left( n\right) \geq 0$and there is some rank-preserving extension $M\left( n+1\right)$for which $M_{p_{i}}\left( n+k_{i}\right) \geq 0$, where $\deg p_{i}=2k_{i}$ or $2k_{i}-1$ $(1\leq i\leq m)$.