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In this paper we present results obtained from the unification of $SU(2)$coherent states with $\mathbb{C}P^N$ sigma models defined on the Riemann spherehaving finite actions. The set of coherent states generated by a vectorbelonging to a carrier space of an irreducible representation of the groupgives rise to a map from the sphere into the set of rank-1 Hermitian projectorsin that space. The map can be identified with a particular solution of the$\mathbb{C}P^N$ sigma model, where $N+1$ is equal to the dimension of therepresentation space. In particular a choice of the generating vector as thehighest weight vector of the representation gives rise to the map known as aVeronese immersion. Using a description of the matrix elements of theserepresentations in terms of Jacobi polynomials, we obtain an explicitparametrization of the solutions of the $\mathbb{C}P^N$ models, which has notbeen previously found. We relate the analytical properties of the solutions,which are known to belong to separate classes --- holomorphic, anti-holomorphicand various types of mixed ones --- to the weight corresponding to the chosenweight vector. Some examples of the described constructions are elaborated indetail in this paper.

$\mathbb{C}P^N$ sigma models via the $SU(2)$ coherent states approach