Yet Another Family of Complex Structures on $S^6$

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 $P=(L,h,\nabla)$ of a compact symplectic manifold $(M,ω)$, let $\mathcal{G}=\mathrm{Aut}(P)$. For each $k\in\mathbb{N}$, we construct a $\mathcal{G}$-equivariant determinant line bundle $λ^{(k)}\rightarrow\mathcal{J}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of $λ^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed $2$-forms $Ω_j$ on $\mathcal{J}_{int}$ and corresponding moment maps $μ_j:\mathcal{J}_{int}\rightarrow C^\infty(M)$. Each $μ_j$ arises from the asymptotic expansion of the variation of the logarithm of the Quillen metric with respect to Kähler potentials, with the complex structure held fixed. This provides a natural generalization of the Donaldson--Fujiki moment map interpretation of scalar curvature. Moreover, we show that $μ_j$ coincide with the $Z$--critical equations introduced by Dervan--Hallam, and we state a generalization of Fujiki's fiber integral formula.

math.DG↗

arXiv2026

A new boundary mass for asymptotically flat half-manifolds

We introduce a boundary analogue of the Gauss--Bonnet--Chern mass for asymptotically flat half-manifolds with non-compact boundary. We prove that this mass is well defined and establish the corresponding positive mass theorems for graphical and conformally flat graphs. Also we provide a Penrose-type inequality for the mass $\mathfrak{m}_{a,B}(g)$.

math.DG↗

arXiv2026

Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler--Ricci shrinker satisfying \[ \operatorname{Ric}+\nabla^2 f=\frac12 g. \] We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in\{0,\ldots,m\}$ such that \[ R\equiv k \] and \[ (M^{2m},g,J)\cong \bigl(N^{2k}\times\mathbb C^{m-k}, g_N+g_{\mathrm{Euc}},J_N\oplus J_0\bigr), \] where $(N^{2k},g_N,J_N)$ is Kähler--Einstein and \[ \operatorname{Ric}_{g_N}=\frac12 g_N. \] As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition $\mathcal{L}_{\nabla f}\operatorname{Ric}=0$ implies radial flatness and hence rigidity.

math.DG↗