Intrinsic contact moment maps and a realizability sieve

Definitions and mathematical statement

G. Landweber's Question 5.7 asks for an analogue of abstract moment maps that models local contact geometry. The source does not define the desired axioms or the exact local-model statement. The paired aid works with a compact Lie group $G$ preserving a cooriented contact distribution $\xi$ on $M$.

Let $\xi_+^\circ\subset T^*M$ be its positive annihilator cone, with tautological one-form $\lambda$ and symplectic form $d\lambda$. The reported intrinsic contact moment map is \[ \Psi:\xi_+^\circ\to\mathfrak g^*,\qquad \langle\Psi(\beta_x),\eta\rangle=\beta_x(\eta_M(x)),\qquad \Psi(r\beta_x)=r\Psi(\beta_x),\quad r>0. \] Use the Hamiltonian sign convention $\iota_{\eta}\omega=-d\langle\Psi,\eta\rangle$. A $G$-invariant positive contact form $\alpha$ defining $\xi$ is a section of this cone, and the representative on the base is $\mu_\alpha=\alpha^*\Psi$, equivalently $\langle\mu_\alpha,\eta\rangle=\alpha(\eta_M)$. Changing the form to $f\alpha$, for smooth positive invariant $f$, changes this representative to $f\mu_\alpha$.

The aid records the exact pointwise kernel condition \[ \ker(\mu_\alpha(x))= \{\eta\in\mathfrak g:\eta_M(x)\in\xi_x\}. \] In particular isotropy Lie algebras are annihilated, and $\mu_\alpha(x)|_{\mathfrak h}=0$ on $M^H$. These are stronger contact-specific restrictions than allowing arbitrary constant projected values on fixed sets. Conversely the aid reports descent from homogeneous symplectic cones, with a free proper positive-dilation action and compatible $G$-action, to the corresponding cooriented contact quotient. Its full local-model hypotheses are not supplied.

For a fixed distribution and an invariant positive defining form $\alpha_0$, the reported necessary-and-sufficient realizability test for a smooth equivariant map $\nu$ is \[ \nu\text{ is induced by another invariant positive form defining }\xi \quad\Longleftrightarrow\quad \nu=f\mu_0\text{ for a smooth positive invariant }f, \qquad\mu_0=\mu_{\alpha_0}. \] The fixed-distribution and coorientation qualifications are essential.

The aid gives an explicit test case on standard contact $S^3\subset\mathbb C^2$. For the circle action \[ e^{it}(z_1,z_2)=(e^{it}z_1,e^{-it}z_2), \] choose the standard invariant form normalized by $\mu_0=|z_1|^2-|z_2|^2$, and let $\nu=\mu_0^3$. The aid calls $\nu$ a GGK exact abstract moment map, obtainable from the invariant one-form $\mu_0^2\alpha_0$. It has the same zero torus and the same positive rays away from zero as $\mu_0$, yet any conformal multiplier off that torus is forced to be $\mu_0^2$, which tends to zero there. Hence no everywhere positive smooth multiplier exists, and $\nu$ is not contact-realizable in this fixed positive contact-form class. Abstract exactness alone therefore does not supply the stated contact realizability. The construction is partial progress on the requested general theory, not a complete axiomatization or universal local-normal-form theorem.

Short English statement

The aid formulates contact moment data intrinsically on the positive annihilator symplectic cone. On a fixed cooriented contact distribution, changing its defining form multiplies the representative moment map by a positive invariant function. A cubed moment map on the standard three-sphere is an abstract exact example that fails this fixed-distribution realizability test.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[296], source item 5.7 attributed to G. Landweber. Title: "Intrinsic contact moment maps and a realizability sieve."
• AIM workshop list Moment maps and surjectivity in various geometries, supplied reference https://aimath.org/WWN/momentmaps/momentmaps.pdf. Original remarks and literature fields are empty. The supplied verification repairs only line-break hyphenation "mo-ment" to "moment"; this draft does not access a PDF.
• Supplied research attempt 1 invokes the GGK exact abstract-moment-map framework and gives the cone description, realizability criterion and weight-$(1,-1)$ sphere example. No full GGK bibliographic entry or authoritative separate citation is supplied.
• Registry dated 14 August 2026: "partially_solved," "PARTIAL-PROGRESS," statement status "corrected_verified"; attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.