Three anti-invariant classes on a nonintegrable four-manifold

Abstract

We construct smooth nonintegrable almost complex structures on the product of a genus-two curve and an elliptic curve with at least three independent real anti-invariant de Rham classes. The structures converge smoothly to the product complex structure and equal it outside one coordinate ball. The construction gives a negative answer to the unrestricted integrability conjecture of Draghici–Li–Zhang. The separate generic-vanishing statement is a prior theorem of Tan–Wang–Zhang–Zhu.

The idea in plain English

An almost complex structure tells a real even-dimensional space how multiplication by the imaginary unit should act on tangent directions. Integrability is the extra condition that makes this local rule come from actual complex coordinates.

An influential conjecture proposed that, on a compact four-dimensional manifold, three independent anti-invariant cohomology classes would force integrability. These classes are global geometric data represented by closed differential forms with a particular symmetry under the almost complex structure.

The paper constructs a small deformation of a product of two complex curves that preserves three such global classes but becomes nonintegrable. The deformation is confined to one coordinate ball and can be made arbitrarily small in the smooth topology.

The key distinction is between global classes and pointwise forms. The three classes are independent globally, while their local representatives fit into a two-dimensional plane through a relation with varying coefficients. This leaves room to preserve all three classes while breaking integrability.

The precise result

Let Σ be the smooth compact completion of y² = z⁶ − 1, let E = C/(Z + iZ), and let J₀ be the product complex structure on M = Σ × E. Theorem 1 constructs ε₀ > 0 and a smooth family Jε for |ε| < ε₀.

Every Jε agrees with J₀ outside one fixed coordinate ball and Jε → J₀ in C∞. Three fixed, linearly independent classes in H²_dR(M; R) have closed Jε-anti-invariant representatives.

Every nonzero sufficiently small ε gives a nonintegrable structure. Consequently h⁻_Jε ≥ 3; the paper gives this lower bound, not an exact dimension.

This is a counterexample to the unrestricted compact-four-manifold integrability conjecture of Draghici–Li–Zhang. The separate theorem that h⁻_J = 0 is open and dense among almost complex structures is prior work of Tan–Wang–Zhang–Zhu and is compatible with this exceptional family.

Why the result holds

Two holomorphic one-forms on the genus-two curve, paired with the elliptic-curve differential, supply three closed real forms. Positivity of the square of a real holomorphic two-form and Stokes’ theorem establish independence of their de Rham classes.

In one coordinate ball the forms satisfy γ = uα − vβ. A compactly supported perturbation changes β and γ by exact terms while preserving the corresponding identity γε = uα − vβε, so all three global classes stay fixed.

For a sufficiently small parameter, α and βε span a positive two-plane. Normalizing this plane gives a decomposable complex two-form whose contraction kernel defines the new almost complex structure. The construction agrees smoothly with J₀ near the ball’s boundary.

The local relation puts the third representative in the same anti-invariant plane. On the region where the cutoff is one, an explicit coframe has a nonzero integrability obstruction for every nonzero parameter, proving that the new structure is nonintegrable.

Scope and limitations

The construction is on the specified genus-two-curve times elliptic-curve product. It does not assert that every compact four-manifold admits such a deformation.

Three independent de Rham classes do not mean three pointwise independent anti-invariant forms. The local anti-invariant plane has real dimension two, and the third representative is a variable-coefficient combination of the other two.

The theorem proves h⁻_J ≥ 3, without computing the exact value. It concerns smooth almost complex structures and does not impose compatibility with a fixed symplectic form.

The deformation is not metric related to the original complex structure: there is no common positive compatible metric. Thus it lies outside the hypotheses of the earlier Draghici–Li–Zhang metric-related obstruction.

Although the starting real pair is closed, variable normalization produces a complex two-form that is not closed. The established closed normalized-pair integrability criterion therefore does not apply.

Earlier work includes compact nonintegrable examples with h⁻ = 2, higher-dimensional and noncompact examples, and nonintegrable structures on products of curves with many pluricanonical sections. Those sections are not automatically closed real anti-invariant de Rham classes. The bounded comparison does not certify exhaustive novelty or exclude further deductions from older examples.

Status and AI use

AI tools were used extensively in the construction, drafting, literature checking and internal adversarial review. The note is unrefereed, and no independent human peer review or formal verification is claimed.

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Collaborate on this result

I welcome substantive collaboration on independent verification of the global class construction and local integrability obstruction, comparison with earlier almost complex examples, and preparation for journal review. Expertise in four-manifold geometry, differential forms or almost complex cohomology would be particularly useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Three anti-invariant classes on a nonintegrable four-manifold (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23422055

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Background and prior work

• Draghici, Li and Zhang, Symplectic forms and cohomology decomposition of almost complex four-manifolds: Question 3.27 asks for a compact nonintegrable four-dimensional example with h⁻_J ≥ 3.
• Draghici, Li and Zhang, On the J-anti-invariant cohomology of almost complex 4-manifolds: Conjecture 2.5 is the unrestricted integrability claim. Theorem 1.1 concerns the narrower class sharing a compatible metric with an integrable structure.
• Tan, Wang, Zhang and Zhu, On cohomology of almost complex 4-manifolds: Theorem 1.1 proves that h⁻_J = 0 is open and dense in the C∞ topology. This generic result is prior work, not contradicted by the exceptional family.
• Hind and Tomassini, On the anti-invariant cohomology of almost complex manifolds: Proposition 4.2 gives compact four-dimensional examples with h⁻ = 2. Their higher-dimensional and noncompact constructions have different scope.
• Donaldson, Two-forms on four-manifolds and elliptic equations: The positive oriented two-plane description is established background for the local realization of an almost complex structure.
• Bande and Kotschick, The geometry of recursion operators: Theorem 6 gives the closed-pair integrability criterion. The present normalized pair fails its closedness requirement.
• Chen and Zhang, Kodaira dimensions of almost complex manifolds I: Section 6.3 supplies earlier nonintegrable structures on T² × S with many pluricanonical sections. That is a different invariant from closed real anti-invariant cohomology.
• Chen and Zhang, Kodaira dimensions of almost complex manifolds II: Section 5.1 explicitly distinguishes the two invariants and asks for the anti-invariant cohomology of those earlier examples.