A sharp circle criterion for intersecting Kirwan kernels

Definitions and mathematical statement

N. Proudfoot asks, "in this setting," whether \[ \ker f=\bigcap_{\alpha\text{ central regular}}\ker\kappa_\alpha. \] The setting and the maps $f,\kappa_\alpha$ are not defined in this record. Comment 7.2 says the abelian case is true by a proof using hyperkähler geometry and asks for a direct proof. The aid instead gives an abelian counterexample under a particular locally-free Borel-restriction interpretation. This is a substantive formulation conflict: the following computations are conditional on that interpretation and cannot certify refutation of the omitted intended setting.

For the aid's explicit reading, let \[ V=\bigoplus_{i=1}^{p+q}{\mathbb{C}}_{w_i}\oplus{\mathbb{C}}^k, \quad w_i\in{\mathbb{Z}}\setminus\{0\},\quad p+q\geq1, \] be a linear $S^1$-representation with $p$ positive weights, $q$ negative weights and $k$ trivial complex summands. Its moment map is \[ \mu(z)=\tfrac12\sum_iw_i|z_i|^2. \] Its locally free locus is $V_{\mathrm{lf}}=({\mathbb{C}}^{p+q}\setminus\{0\})\times{\mathbb{C}}^k$. Use rational Borel equivariant cohomology, with $H^*_{S^1}(V;{\mathbb{Q}})={\mathbb{Q}}[u]$, $\deg u=2$, and define \[ f:H^*_{S^1}(V;{\mathbb{Q}})\to H^*_{S^1}(V_{\mathrm{lf}};{\mathbb{Q}}), \qquad \kappa_\alpha:H^*_{S^1}(V;{\mathbb{Q}}) \to H^*_{S^1}(\mu^{-1}(\alpha);{\mathbb{Q}}). \] All real values are central; the nonzero values are regular, with empty levels allowed and assigned zero target cohomology. For nonempty positive and negative chambers respectively, the aid's formulas are \[ \ker f=(u^{p+q}),\qquad \ker\kappa_{\alpha>0}=(u^p),\qquad \ker\kappa_{\alpha<0}=(u^q). \] If $p,q>0$, both signs occur and \[ \bigcap_{\alpha\text{ regular}}\ker\kappa_\alpha =(u^{\max(p,q)})\supsetneq(u^{p+q}). \] If only one sign occurs, the empty opposite levels have full kernel and do not affect the intersection; the two sides agree. Trivial summands are contractible factors and do not change these computations. Thus, for this rational representation model, equality holds exactly when all nonzero weights have one sign.

For weights $+1,-1$ on ${\mathbb{C}}^2$, the same counterexample is integral as well: \[ \ker f=(u^2)\subsetneq(u) =\bigcap_{\alpha\ne0}\ker\kappa_\alpha \quad\text{in }{\mathbb{Z}}[u]. \] Here the locally free sphere quotient is a Hopf-type $S^2$ quotient, while each nonzero-level quotient is contractible. The general-weight ideal formulas above are asserted over ${\mathbb{Q}}$, where weight products are units. Over ${\mathbb{Z}}$, nonunit weights produce Euler coefficients and finite-isotropy torsion; one cannot simply reuse the same monomial ideals. For example, a single weight $a\ne0$ has restriction kernel $(au)$, not necessarily $(u)$.

These examples use mixed-sign moment maps that are not proper. If properness, a different regular locus, an alternative map $f$, a different cohomology theory or other conditions were part of the missing source setting, the computation must be reassessed. In particular, the original comment asserting an abelian theorem and seeking a direct proof remains unreconciled by the supplied data.

Short English statement

Under the aid's locally-free Borel interpretation, mixed positive and negative circle weights give unequal kernels; the $(+1,-1)$ example works integrally and rationally. The original comment says the abelian statement is true, so missing definitions or hypotheses are essential and the intended conjecture cannot be declared refuted from this record alone.

Sources

• Permanent UnsolvedMath ID $20001970$, exact public code catalogue entry; local registry aim-geometry-notes.json, notes[307], canonical tag question.
• AIM workshop "Moment maps and surjectivity in various geometries," Question 7.1 and Comment 7.2, both attributed to N. Proudfoot: https://aimath.org/WWN/momentmaps/momentmaps.pdf. The intervening 14 is an extracted page number. No PDF was accessed.
• Research attempt 1, "A sharp circle criterion for intersecting Kirwan kernels," labelled counterexample, refuted at the run, proved and low novelty confidence; registry solved and SOLVED-IN-LITERATURE. Its claimed surrounding Borel interpretation is not quoted in the supplied verification field. No full theorem citation, original literature entry or year is supplied.