The universal structure of moment maps in complex geometry
Ruadhaí Dervan, Michael Hallam
TL;DR
This work develops a canonical, geometric framework to produce moment maps in complex geometry by leveraging equivariant differential forms and universal families. It shows that, for both manifolds and holomorphic vector bundles, central charges $Z$ induce $Z$-critical PDEs that arise as moment maps in finite- and infinite-dimensional settings, unifying the scalar curvature (cscK) and Hermitian Yang–Mills equations as natural special cases and extending to deformed Hermitian Yang–Mills. The approach uses Chern–Weil theory, fibre integration, and the geometry of families to construct equivariant representatives whose fibre integrals define the appropriate moment maps on bases like the space of almost complex structures or the moduli base $B$. This framework provides a general, canonical route to PDEs associated with stability conditions, potentially enabling widespread construction of analytic counterparts to stability 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: Kähler manifolds and holomorphic vector bundles. Our new approach exploits the existence of universal families and the theory of equivariant differential forms. For Kähler 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 Kähler 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.