Branson's cocycle as an action, polarization, and kernel detector

Definitions and mathematical statement

Problem 25 asks for deeper properties of Branson's cocycle $H[\widehat g,g]$, then begins an incomplete transition to a curvature-prescription problem. The paired aid uses a closed connected conformal manifold of dimension $n$, with a formally self-adjoint conformally covariant operator $P_g$, $P_g1=0$, and \[ e^{nu}Q_{e^{2u}g}=Q_g+P_gu. \] This includes the appropriate critical even-dimensional $Q$-curvature setting. Pairings and integrals below use $dv_g$, and $\kappa=\int Q_g\,dv_g$ is conformally invariant.

The reported normal form and cocycle identity are \[ H_g(u):=H[e^{2u}g,g]=2\int uQ_g\,dv_g+\int uP_gu\,dv_g, \] \[ H_g(u+v)=H_g(u)+H_{e^{2u}g}(v). \] Half this expression is a path-independent conformal action, $F_g=H_g/2$, with \[ dF_g|_u(v)=\int v(Q_g+P_gu)\,dv_g =\int vQ_{e^{2u}g}\,dv_{e^{2u}g}. \] The aid also recovers the defining data from a one-base cocycle: \[ \int uQ_g\,dv_g=\tfrac14(H_g(u)-H_g(-u)), \] \[ B_g(u,v):=\int uP_gv\,dv_g =\tfrac12\left(H_g(u+v)-H_g(u)-H_g(v)\right). \] Self-adjointness matters: polarization recovers the symmetric form and hence this specified $P$.

For $h\in\ker P_g$, the reported exact slope is \[ H_g(u+th)-H_g(u)=2t\int hQ_g\,dv_g. \] The pairing on the right is conformally invariant when $h$ is identified through the common conformal kernel. In particular $H_g(u+c)=H_g(u)+2c\kappa$. Thus the raw cocycle is unbounded along constant conformal factors if $\kappa\ne0$, and descends under independent constant rescalings of its endpoint metrics exactly when $\kappa=0$. This is not the same as simultaneous rescaling of both endpoints.

The aid attributes a volume correction to later Branson--Gover work. With \[ V_u=\int e^{nu}\,dv_g,\qquad \mathcal F_g(u)=F_g(u)-\frac\kappa n\log(V_u/V_0), \] this action is invariant under $u\mapsto u+c$, and its critical metrics have constant $Q$. Let $d\mu_u=e^{nu}dv_g/V_u$. Its reported Hessian is \[ d^2\mathcal F_g|_u(v,w)=B_g(v,w) -n\kappa\left(\int vw\,d\mu_u -\int v\,d\mu_u\int w\,d\mu_u\right). \] If $P$ is nonnegative and $\kappa<0$, this is strictly positive modulo constants. The normalized action is strictly convex modulo scale, giving uniqueness of a critical constant-$Q$ metric in the conformal class if one exists. Neither existence nor an exhaustive account of deeper cocycle properties follows from this conditional convexity statement. The missing prescribed-curvature problem at the final colon is not invented.

Short English statement

The supplied aid interprets Branson's cocycle as twice a conformal action, recovers $Q$ and the symmetric $P$-pairing by polarization, and describes exact kernel slopes and scale dependence. Its volume-normalized action has constant-$Q$ critical points and is strictly convex modulo scale under the stated sign assumptions, yielding conditional uniqueness rather than existence.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[384], Problem 25. Title: "Branson's cocycle as an action, polarization, and kernel detector."
• AIM workshop Conformal structure in geometry, analysis, and physics, supplied reference https://aimath.org/WWN/confstruct/confstruct.pdf. The original ends at the introduction of a curvature-prescription problem; no remarks or literature entry is supplied. This draft does not access a PDF.
• Supplied attributions: Branson for the cocycle, and later Branson--Gover work for the volume correction. Full bibliographic entries are absent. Attempt 1 gives the polarization, kernel-slope and exact Hessian synthesis.
• Registry dated 14 August 2026: "partially_solved," "PARTIAL-PROGRESS," statement status "exact"; attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.