2025/10/29 by Eum, Kiyoon
#32Q15 #58J52 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.25456
In Kähler geometry, the Donaldson-Fujiki moment map picture interprets the scalar curvature of a Kähler metric as a moment map on the space of compatible almost complex structures on a fixed symplectic manifold. In this paper, we generalize this picture using the framework of equivariant determinant line bundles. Given a prequantization P=(L,h,∇) of a compact symplectic manifold (M,ω), let G=Aut(P). We construct for each k∈ℕ a G-equivariant determinant line bundle λ(k)\rightarrowJint on the space of integrable compatible almost complex structures, equipped with the G-invariant Quillen metric. The curvature form of λ(k) admits an asymptotic expansion whose coefficients yield a sequence of G-invariant closed two-forms Ωj on Jint and corresponding moment maps μj:Jint→ C^∞(M). Each μj arises from the asymptotic expansion of the variation of the log of the Quillen metric with respect to Kähler potentials, keeping the complex structure fixed. This provides a natural generalization of the Donaldson-Fujiki moment map interpretation of scalar curvature. Moreover, we show that μj coincide with the Z-critical equations introduced by Dervan-Hallam, and we state a generalization of Fujiki's fiber integral formula.