The universal structure of moment maps in complex geometry

Abstract

We introduce a geometric approach to the construction of moment maps in finite and infinite-dimensional complex geometry. We apply this to two settings: K\"ahler manifolds and holomorphic vector bundles. Our new approach exploits the existence of universal families and the theory of equivariant differential forms. For K\"ahler manifolds we give a new, geometric proof of Donaldson-Fujiki's moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern classes of the manifold - namely to a central charge - we further introduce a geometric PDE determining a $Z$-critical K\"ahler metric, and show that these general equations also satisfy moment map properties. For holomorphic vector bundles, using a similar strategy we give a geometric proof of Atiyah-Bott's moment map interpretation of the Hermitian Yang-Mills condition. We then go on to give a new, geometric proof that the PDE determining a $Z$-critical connection - again associated to a choice of central charge - can be viewed as a moment map; deformed Hermitian Yang-Mills connections are a special case, in which our work gives a geometric proof of a result of Collins-Yau. Our main assertion is that this is the canonical way of producing moment maps in complex geometry - associated to any geometric problem along with a choice of stability condition - and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.