Form Q-operators: recovered status and a two-gate cohomology descent criterion

Definitions and mathematical statement

This extraction combines background on form analogues of $Q$-curvature with a nonspecific "Problems $k$" program. Work on a closed oriented Riemannian manifold $M$ of even dimension $n$, not necessarily connected. Write $\Omega^k$ for smooth real forms, $C^k=\ker d$, and use de Rham cohomology. For a conformal change $\widehat g=e^{2\omega}g$, the scalar $Q$-curvature is best recorded as its top-form density $q_g$. The source gives \[ q_{\widehat g}=q_g+P_n\omega, \] where the critical GJMS operator $P_n:\Omega^0\to\Omega^n$ is formally self-adjoint and has a detour factorization of type $dM_1d$, up to conventions. Consequently multiplication by $q_g$ on locally constant functions induces a conformally invariant map $C^0\to H^n(M;{\mathbb{R}})$, and $\int_M u\,q_gc$ is invariant for $c\in C^0$, $u\in\ker P_n$. Its value on $1$ is $q_g$. One may alter an operator extension off locally constant functions by terms annihilating them, while the source asks that formal self-adjointness be retained.

The general reported construction has $M_k^g:\Omega^k\to\Omega^{n-k}$, $0\leq k\leq n/2-1$, obtained using ambient metrics and tractors. In the form/density identifications intended by the source, its axioms are \[ M_k^{\widehat g}c=M_k^gc+L_k(\omega c)\quad(c\in C^k),\qquad L_k=\beta_k dM_{k+1}^gd, \] with fixed normalization constants, and formal self-adjointness for both $M_k^g$ and $L_k$. The terminal factor at the middle degree requires the construction's endpoint convention, not supplied here. Put \[ \mathcal H_g^k=\{c\in C^k:dM_k^gc=0\},\qquad \Phi_g(c)=[M_k^gc]\in H^{n-k}(M;{\mathbb{R}}). \] The source states conformal invariance of $\mathcal H_g^k$ and $\Phi_g$, nontriviality on some conformal manifolds, and the invariant pairing \[ \int_M u\wedge M_k^gc\quad(c\in C^k,\ u\in\ker L_k), \] with signs/weighted identifications absorbed in the formal-adjoint convention. Also $M_0^g1=q_g$. These are reported established form-$Q$ properties, not hypotheses already verified for every arbitrary operator.

The aid's minimal-axiom conclusion is that the transformation and detour laws alone make $dM_k^g c$ independent of conformal representative on closed inputs, and the change in $M_k^g c$ exact. Formal self-adjointness gives the kernel-pairing invariance. A separate cohomological descent issue remains. Let \[ q:\mathcal H^k\to H^k(M;{\mathbb{R}}),\qquad c\mapsto[c]. \] Then $\Phi$ factors through the represented subspace $q(\mathcal H^k)$ if and only if \[ [M_k^g(d\alpha)]=0 \quad\text{for every }d\alpha\in\mathcal H^k. \] This exact-input condition is necessary and sufficient for independence of the chosen conformal-harmonic representative. Obtaining the resulting map on all of $H^k(M;{\mathbb{R}})$ additionally requires surjectivity of $q$. Neither conformal invariance nor equality of a dimension with a Betti number automatically supplies both gates.

The source's near-middle illustration, for $n\geq4$ and $p=n/2-1$, uses the weighted operator \[ \mathcal M_g=d\delta+2J-4P^{\sharp}:\Omega^p\to\Omega^p[-2], \qquad J=\frac{\operatorname{Scal}}{2(n-1)},\quad P=\frac{\operatorname{Ric}-Jg}{n-2}, \] where $P^{\sharp}$ acts on form slots. The weighted target identifies by conformal Hodge star with $(p+2)$-forms. Exercise $7$ asks to check on closed inputs the law \[ \mathcal M_{\widehat g}c=\mathcal M_gc+\beta\delta d(\omega c),\qquad\beta\ne0, \] under the intended weighted identification. It notes formal self-adjointness of the Maxwell operator $\delta d$, invariant pairings with its kernel, invariance modulo $\operatorname{im}\delta$, and invariance of $\delta\mathcal M_g$ on closed forms. On a closed Riemannian manifold $\ker(\delta d)$ consists of closed forms. The source also reports a conformally invariant $\delta\mathcal M_gd$ on $(n/2-2)$-forms, giving Paneitz in dimension four, and defines the corresponding conformal harmonics by $\delta\mathcal M_gc=0$.

It cautions that their dimension need not always equal the Betti number, suggests a generic equality through elliptic coercivity of $(d,\delta\mathcal M)$ without defining a small exceptional set, and gives $S^p\times S^{p+2}$ with standard product metric as a nontrivial example where conformal harmonics in degree $p$ are ordinary harmonic forms. It further asserts a non-removability-by-invariant-operator-and-divergence property, referring to Maxwell leading-order uniqueness and an unavailable formula $(44)$. These background claims need the precise conventions and references.

The final "Problems $k$" only asks for analogues of unspecified $Q$-curvature conundrums for the $M_k^g$. The initial "Problem $2$ have a role" fragment and the following Chapter C heading with Thomas Branson's "Anti-conformal perturbations" are boundary material, not complete additional problems. No characterization of $Q$, general cohomology-surjectivity theorem, or precise new open-program statement is invented from them.

Short English statement

The form-$Q$ construction gives conformally invariant harmonic-input spaces, output cohomology classes and kernel pairings. The aid isolates two separate requirements for a map on ordinary de Rham cohomology: exact inputs must map to exact outputs, and every cohomology class must have an admissible representative. The source's remaining open prompt is nonspecific.

Sources

• Local UnsolvedMath registry, permanent ID $20002016$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[353]. The source-item field is $2$, but the extraction covers the end of Chapter B, equations $(35),(36)$, Exercise $7$, and "Problems $k$."
• AIM workshop "Conformal Structure in Geometry, Analysis, and Physics," 12--16 August 2003, Palo Alto, California: https://aimath.org/WWN/confstruct/confstruct.pdf. The supplied verification reports comparison of official HTML and PDF. Page numbers $22,23$, the Chapter C heading and Thomas Branson's next-topic attribution are boundary material. No external source or PDF was opened in this session.
• The aid attributes the central established construction to Branson and Gover; the source names $Q$-curvature, GJMS operators, Maxwell, Paneitz, ambient metrics and tractors. No full authoritative bibliographic entry, formula $(44)$, or endpoint/normalization conventions are supplied. Original remarks and literature fields are empty, with no proposer or year.
• Supplied research attempt $1$, partial_result, run status partially_solved, rigor label "proved," novelty confidence "low." Registry classification PARTIAL-PROGRESS, updated 14 August 2026. Its candidate contribution is the two-gate descent criterion, not a claim to invent the form-$Q$ construction.