A boundary fragment and the specialization behind it

Definitions and mathematical statement

The entire canonical mathematical text is the boundary clause: "Conjecture 1.1, but we state it again here in T. Hausel's formulation, for good measure:" It contains neither the earlier conjecture nor the displayed formulation that follows it. Accordingly this record does not supply a stand-alone conjecture with recoverable quantifiers and hypotheses. The aid locates the clause immediately before a separately extracted display in Section 6.1; no permanent identity for that neighboring display is supplied here.

The supplied specialization aid can be described conditionally. Let $G$ be a compact Lie group acting linearly on a complex representation $V$, with the induced cotangent hyperkähler action on $W=T^*V$ and moment map $\boldsymbol\mu:W\to\mathfrak g^*\otimes{\mathbb{R}}^3$. Let $\zeta$ be an invariant level and put $Z=\boldsymbol\mu^{-1}(\zeta)$, $Q=Z/G$. With coefficients in a chosen ring $R$, the equivariant restriction is \[ H_G^*(W;R)\longrightarrow H_G^*(Z;R). \] The linear equivariant contraction of $W$ identifies its source with $H^*(BG;R)$. When the action on $Z$ is free, the usual Borel-to-quotient comparison identifies $H_G^*(Z;R)$ with $H^*(Q;R)$, giving the quotient-form Kirwan map \[ \kappa_\zeta:H^*(BG;R)\longrightarrow H^*(Q;R). \] For merely locally free actions, finite stabilizers introduce a coefficient issue: this comparison is available rationally under the usual quotient-space hypotheses, but cannot be asserted integrally without further conditions. Thus the source and target identifications alone do not imply surjectivity in every coefficient system.

The aid separately requires checking the earlier conjecture's negative-gradient compactness hypothesis for the shifted function \[ F_\zeta(w)=\|\boldsymbol\mu(w)-\zeta\|^2, \] rather than importing a zero-level flow assertion without checking the shift. It reports properness of $\boldsymbol\mu$ as a sufficient uniform condition: each descending trajectory remains in a compact sublevel set of $F_\zeta$. Properness and the full earlier conjecture's hypotheses are not supplied for an arbitrary cotangent representation. This conditional specialization does not reconstruct or prove the omitted Hausel formulation. The record calls its candidate a five-part proposition, but does not reproduce the five parts.

Short English statement

This item is an introductory clause, not a complete conjecture. The aid explains how a linear cotangent action can turn an equivariant restriction map into a quotient-form Kirwan map. It keeps the coefficient restriction for finite stabilizers and the separate shifted-flow compactness check explicit.

Sources

• Local UnsolvedMath registry, permanent ID 20001962, exact source code catalogue entry; canonical aim-geometry-notes.json, notes[299], tag conjecture; AIM Workshop Problem Lists, Geometry. Created and updated 14 August 2026.
• AIM Moment maps and surjectivity in various geometries, source metadata item 1.1; the aid places the boundary clause in Section 6.1 immediately before a separately extracted display attributed to T. Hausel. Source URL https://aimath.org/WWN/momentmaps/momentmaps.pdf. The earlier Conjecture 1.1 and following display are not supplied; the PDF was not accessed.
• Supplied attempt 1: context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; classification OPEN-TRIAGE, suggested difficulty L2, registry status open. Candidate: conditional specialization lemma. No separate authoritative theorem citation is supplied for its five-part package.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.