2023/04/03 by Ruadhaí Dervan, Dervan, Ruadhaí, Michael Hallam +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2304.01149
openalex publication_date 2023/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.