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∞ function rh0 constructed from the long-time average of the polar determinant phase. The function rh0 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 rh0. 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.
Related
- Equivariant Poisson structures on moment-angle manifolds and their quotients
- Moment Polyptychs and the Equivariant Quantisation of Hypertoric Varieties
- Second Chern class and Fujiki constants of hyperkähler manifolds
- Monge-Ampère degenerations and conifold contractions in Fujiki class 𝒞
- Shintani Cocycle Decomposition of Topological Polylogarithms for Totally Real Fields
BibTeX
@misc{goto2026equivariant,
title = {{Equivariant Moment Maps and the Ricci Cocycle in Weighted Fujiki--Donaldson Theory}},
author = {Ryushi Goto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24543},
eprint = {2610.24543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24543},
}
RIS
TY - UNPB
TI - Equivariant Moment Maps and the Ricci Cocycle in Weighted Fujiki--Donaldson Theory
AU - Ryushi Goto
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24543
UR - https://arxiv.org/abs/2610.24543
AB - 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^∞ 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.
ER -