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Math-1476 · mathematics · Topology

Kirby Problem 4.93

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $X$ be the standard smooth K3 surface, represented for example by any nonsingular quartic complex hypersurface in $\mathbb{CP}^3$, with its underlying smooth $4$-manifold. A symplectic form on $X$ is a smooth real $2$-form $\omega$ with $d\omega=0$ and with $\omega_x$ nondegenerate on every tangent space. A symplectomorphism from $(X,\omega)$ to $(X,\eta)$ is a diffeomorphism $f$ with $f^*\eta=\omega$.

A Kähler form means a symplectic form $\eta$ compatible with an integrable complex structure $J$ on $X$: it satisfies $\eta(Ju,Jv)=\eta(u,v)$ and $\eta(u,Ju)>0$ for all nonzero $u$. Integrability means $J$ arises from complex coordinate charts with holomorphic transition maps.

Is every symplectic form $\omega$ on the standard K3 smooth manifold symplectomorphic to a Kähler form? Thus both $J$ and its Kähler form $\eta$ may be chosen in response to $\omega$, as may the diffeomorphism $f$.

No fixed complex structure is prescribed in the conclusion. On the other hand, equality of cohomology classes, deformation through symplectic forms with varying classes, or agreement of characteristic classes alone is weaker than the requested symplectomorphism. The underlying differentiable manifold remains the standard K3 throughout; replacing it by an exotic manifold homeomorphic to K3 would be a different question.

Short English statement

Is every symplectic form on the standard K3 smooth manifold carried by a diffeomorphism to a Kähler form for some complex structure?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1476-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1476

Hypersymplectic deformation: The individual-Kahler corollary treats symplectic forms that occur in a positive triple of closed two-forms. It gives the desired Kahler realization for that restricted K3 class, whereas our question concerns every symplectic form on standard K3.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Kirby Problem 4.93

OpenAI's claim

Hypersymplectic deformation: On a closed oriented four-manifold, a positive triple satisfying the paper’s normalization and consisting of closed two-forms is claimed to deform smoothly to a hyperkähler triple while keeping each cohomology class fixed. The starting triple need not already define a hyperkähler metric.

Mathematical statement of the claimed result

Mathematical claim 1.
On an oriented four-manifold, a hypersymplectic structure consists of three real forms $\omega_i$ of degree two, satisfying $d\omega_i=0$, with a pointwise span of dimension three that is positive definite for the wedge pairing specified by the orientation. For a closed connected smooth $X$ and such a triple normalized by $\int_X\omega_i\wedge\omega_j=\delta_{ij}$, the claim gives a smooth hypersymplectic deformation $\omega(t)$ with \[\omega(0)=\omega,\qquad[\omega_i(t)]=[\omega_i],\qquad\omega_i(1)\wedge\omega_j(1)=2\delta_{ij}\mu_1.\] The endpoint forms are parallel self-dual forms for a hyperkähler metric. A further corollary states that each nonzero real combination $\alpha_c=\sum c_i\omega_i$ of the original triple is individually a Kähler form for some hyperkähler metric and parallel complex structure, chosen separately for that combination.

Outline of the argument

Hypersymplectic deformation: Two forms determine an almost-complex structure, and the third form must keep a taming representative throughout the pair deformation. Positive-current separation and quadratic mass estimates establish a criterion for retaining that representative. An auxiliary fibration and an equal-square pair supply the deformation framework, after which classical Kähler arguments reach the hyperkähler endpoint.

Formal verification

Hypersymplectic deformation: The pinned catalogue lists no Lean formalization for this result.

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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