Equivariant Moment Maps and the Ricci Cocycle in Weighted Fujiki--Donaldson Theory

Abstract

Let $(M,ω)$ be a closed connected symplectic manifold of dimension $2n$,. We discuss weighted Fujiki--Donaldson theory with respect to the measure $dν=e^hω^n/n! $ for an arbitrary smooth real-valued function $h$ on $M$. We prove that the group generated by the flows of Hamiltonians that Poisson-commute with $h$ always admits an equivariant moment map for its action on the full space of $ω$-compatible almost complex structures. The moment map takes values in the continuous dual of the Lie algebra. Its density is expressed in terms of the Hermitian scalar curvature, a local weight correction, and an $L^\infty$ function $r_h^0$ constructed from the long-time average of the polar determinant phase. The function $r_h^0$ can be regarded as a Ruelle-type asymptotic density incorporating phase comparison via a connection. The key step in the proof is to represent the Ricci cocycle as the coboundary of a continuous cochain constructed from $r_h^0$. We also consider the action of the exact volume-preserving diffeomorphism group on a connected component of the space of almost complex structures inducing the given orientation on a closed connected oriented manifold of real dimension $2n$. In contrast to the weighted Hamiltonian setting, we prove that, for $n>1$, this action admits an equivariant moment map if and only if the real first Chern class vanishes.