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Math-2464 · mathematics · Geometry

The Tamed-to-Compatible Symplectic Conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
If an almost-complex structure $J$ on a compact $4$-manifold is tamed by a symplectic form, does there exist a symplectic form compatible with $J$?

Formulation references: [S1], [S2].

Short English statement

If an almost-complex structure $J$ on a compact $4$-manifold is tamed by a symplectic form, does there exist a symplectic form compatible with $J$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2464-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2464

Taming versus compatibility: Keeping the tamed almost complex structure fixed, the theorem supplies a smooth compatible symplectic form on every closed connected four-manifold. The compact boundaryless convention is the closed-manifold convention of this conjecture; no prescribed class or integrability is required.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

The Tamed-to-Compatible Symplectic Conjecture

OpenAI's claim

Taming versus compatibility: The manuscript claims that an almost-complex structure on a closed smooth four-manifold admits a compatible symplectic form whenever it admits a taming one. It keeps the almost-complex structure fixed.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a smooth almost complex structure $J$ on a smooth real four-manifold $X$, assumed connected and closed. If some closed real two-form $\omega$ satisfies $\omega(v,Jv)>0$ for all nonzero tangent $v$, the claim produces a closed real two-form $\eta$ with \[\eta(v,Jv)>0\quad(v\ne0),\qquad\eta(Ju,Jv)=\eta(u,v).\] Thus a symplectic form taming $J$ can be replaced by one compatible with $J$. No equality of their cohomology classes or prescribed-volume assertion is included.

Outline of the argument

Taming versus compatibility: Failure of compatibility yields a positive current annihilating closed invariant forms. An elliptic correction makes it annihilate all closed two-forms, with resolvent estimates controlling its anti-invariant part and nearby complex-line directions. Splitting off the positive-density region and refining the pairing removes the negative error, contradicting the existence of the original tamer.

Formal verification

Family 342 describes Lean coverage for a compatible form for a fixed almost-complex structure, assuming a taming form and a closed four-dimensional manifold.

Taming versus compatibility: The manuscript links TamingCompatibility.lean (declaration OAI.TamingCompatibility.taming_implies_compatibility).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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