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

Author:

Dan Popovici, Luis Ugarte

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Algebraic Geometry (math.AG), Complex Variables (math.CV)

journal:

date:

2023-10-31 16:00:00

Abstract

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 $\mbox{dim}\,S\geq\mbox{dim}\,X -1$, 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 $f:S\longrightarrow X$ 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 $\C^p$, 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.

PDF: A Moment Map for the Space of Maps to a Balanced Manifold.pdf