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

Given a complex balanced manifold $X$ and a compact complex manifold $S$ equipped with a positive volume form $dV>0$ and satisfying an extra condition such that $dimS\text{geqdim}X-1$, we construct a moment map for the action of the Lie group of biholomorphisms of $S$ that preserve $dV$ onto the space of holomorphic maps $f:S⟶X$ that satisfy a certain condition with respect to the Bott-Chern cohomology class of the balanced metric of $X$. The purpose is twofold: to study such maps as a possible addition to some very recent hyperbolicity notions involving holomorphic maps with a certain type of growth from some ${\text{C}}^p$, rather than $S$, to $X$; and to lay the groundwork for a possible future construction of balanced quotients as an analogue of the classical symplectic quotients.