The universal structure of moment maps in complex geometry

We introduce a geometric approach to the construction of moment maps in finite and infinite-dimensional complex geometry. We apply this to two settings: Kähler manifolds and holomorphic vector bundles. We first give a new, geometric proof of Donaldson–Fujiki’s moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern characters of the manifold—namely to a central charge—we further introduce a geometric PDE determining a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -critical Kähler metric, and show that these general equations also satisfy moment map properties. For holomorphic vector bundles, we similarly give a geometric proof that the PDE determining a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -critical connection can be viewed as a moment map. Our main assertion is that this is the canonical way of producing moment maps in complex geometry, and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.