CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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A compatible almost-complex structure that is not integrable

Counterexample

On $R^{4}$ with coordinates $(x_{1}, y_{1}, x_{2}, y_{2})$ and $ω = d x_{1} \land d y_{1} + d x_{2} \land d y_{2}$ , there is an explicit compatible almost-complex structure with nonzero Nijenhuis tensor.

Facts & Assumptions

Given: Let $J_{0} \partial_{x_{i}} = \partial_{y_{i}}$ and $J_{0} \partial_{y_{i}} = - \partial_{x_{i}}$ . Put $A = diag (e^{x_{2}}, e^{- x_{2}}, 1, 1)$ in the displayed coordinate frame and $J = A J_{0} A^{- 1}$ .

[F1]

Symplectic conjugation preserves compatibility. Compatible complex structure on a symplectic vector space.

[F2]

An integrable almost-complex structure has vanishing Nijenhuis tensor $N_{J} (X, Y) = [J X, J Y] - J [J X, Y] - J [X, J Y] - [X, Y]$ ; this necessary implication is the easy direction of the Newlander--Nirenberg criterion recorded in the cited source. Compatible almost-complex structures and Kähler geometry.

Verification

technique · direct

1.1F1givenalgebra

Pointwise $A$ preserves $ω$ because its reciprocal scalings on $(x_{1}, y_{1})$ preserve $d x_{1} \land d y_{1}$ and it fixes the other pair. Thus [F1] makes $J$ compatible. Explicitly, with $a = e^{- 2 x_{2}}$ , $J \partial_{x_{1}} = a \partial_{y_{1}}$ , $J \partial_{y_{1}} = - a^{- 1} \partial_{x_{1}}$ , and $J$ is standard on the second pair.

2.1F2step 1.1algebra∎

For $X = \partial_{x_{1}}$ and $Y = \partial_{x_{2}}$ , all coordinate brackets vanish, while $[a \partial_{y_{1}}, \partial_{x_{2}}] = - a^{'} \partial_{y_{1}}$ . Hence $N_{J} (X, Y) = J (a^{'} \partial_{y_{1}}) = - (a^{'} / a) \partial_{x_{1}} = 2 \partial_{x_{1}} \neq 0$ . By [F2], $J$ is not integrable.

Depends on

Used by

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Dependency tree · two levels

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Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)