The universal structure of moment maps in complex geometry

Abstract

We introduce a natural, geometric approach to constructing moment maps in finite and infinite-dimensional complex geometry. This is applied 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. In the setting of K\"ahler manifolds we first 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 then introduce a geometric PDE determining a Z-critical K\"ahler metric, and show that these general equations also satisfy moment map properties. On 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 go on to give a new, geometric proof that the PDE determining a Z-critical connection-associated to a choice of central charge on the category of coherent sheaves-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.Comment: 41 page