equivariance-forced-question
Status and infobox
- status:
—→ open - kind:
—→ question - year:
—→ 2026 - statement:
—→ Is every non-automorphic Keller map of $\mathbb C^n$, after a polynomial change of coordinates, equivariant for a nontrivial $\mathbb G_m$-action? - short:
—→ O5 — must every Keller counterexample carry a torus symmetry?
Relations
- + implies jc-plane: If O5 holds in dimension two, there is no plane counterexample, so a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed).
Source +16 −0
Revision 250 as rendered
Alpöge's map, Gallagher's family, the factory maps and Harish's map are all graded by a torus with weights of mixed sign. Is that forced?
The evidence is local. At Alpöge's map and at Gallagher's , the Fong–Fable ledger reports the first-order Keller deformations of degree at most 7. Every one is either a composition move or an equivariant deformation. At a single residual direction is obstructed at second order, and none remains at . The ledger calls this local, capped and two points deep, not a classification.[1]
Why it matters: in the plane, Shaska's theorem makes every torus-symmetric Keller map an automorphism. If O5 holds in dimension two, there is no plane counterexample, so a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed). A negative answer in dimension three would be a counterexample by a genuinely new mechanism.
- ↑ D. Fong and Claude Fable 5, theorems register (theorem 8) and claims ledger, jacobianconjectures.com.