The universal structure of moment maps in complex geometry

TL;DR

This paper presents a unified geometric framework for constructing moment maps in complex geometry, linking stability conditions with PDEs in Kähler manifolds and holomorphic vector bundles, and providing new proofs and generalizations.

Contribution

It introduces a universal geometric approach to moment maps in complex geometry, applicable to various settings, and connects stability conditions with PDEs through a canonical construction.

Findings

New geometric proof of Donaldson-Fujiki's moment map interpretation of scalar curvature.

Introduction of a geometric PDE for Z-critical Kähler metrics related to stability.

Proof that Z-critical connections can be viewed as moment maps, generalizing Hermitian Yang-Mills.

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.