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Given a complex balanced manifold $X$ and a compact complex manifold $S$equipped with a positive volume form $dV>0$ and satisfying an extra conditionsuch that $\mbox{dim}\,S\geq\mbox{dim}\,X -1$, we construct a moment map forthe action of the Lie group of biholomorphisms of $S$ that preserve $dV$ ontothe space of holomorphic maps $f:S\longrightarrow X$ that satisfy a certaincondition with respect to the Bott-Chern cohomology class of the balancedmetric of $X$. The purpose is twofold: to study such maps as a possibleaddition to some very recent hyperbolicity notions involving holomorphic mapswith a certain type of growth from some $\C^p$, rather than $S$, to $X$; and tolay the groundwork for a possible future construction of balanced quotients asan analogue of the classical symplectic quotients.

A Moment Map for the Space of Maps to a Balanced Manifold