On a compact almost complex manifold, every known curvature-theoretic test for integrability is only sufficient: Wan's Bochner-technique identities show that the vanishing of certain global integrals built from the covariant derivative of JJ and the Riemannian curvature tensor forces the Nijenhuis tensor to vanish, but integrable structures can fail to satisfy them. The open problem, isolated explicitly by Wan in his study of harmonic complex structures, is to find a global curvature functional C(J)C(J), analogous to these integral identities, such that N(J)=0N(J)=0 if and only if C(J)=0C(J)=0 — a genuine necessary-and-sufficient curvature criterion for integrability — and, as its purpose, to derive geometric conditions on a Riemannian manifold forcing C(J)≠0C(J)\neq 0 for every almost complex structure, which would prove nonexistence of complex structures, the motivating test case now being the general case of real dimension at least 66 (Yau's problem 52) rather than the six-sphere S6S^6, which in 2026 was shown to carry complex structures. Later work has produced only one-directional curvature obstructions and critical-point-type variational characterizations, so the equivalence itself remains open.

Background

An almost complex structure on a smooth manifold MM is a bundle endomorphism JJ of the tangent bundle with J2=−idJ^2=-\mathrm{id}. It is called integrable when it arises from an actual complex manifold structure; by the Newlander–Nirenberg theorem this holds exactly when the Nijenhuis tensor N(J)(X,Y)=J[JX,Y]+J[X,JY]+[X,Y]−[JX,JY]N(J)(X,Y)=J[JX,Y]+J[X,JY]+[X,Y]-[JX,JY] vanishes identically. Determining which almost complex manifolds admit complex structures is a fundamental problem: in real dimension 4 there are compact almost complex manifolds with no complex structure, but no example is known in dimension 6 or higher, and Yau's problem 52 even asks whether every compact almost complex manifold of dimension at least 6 admits a complex structure. The touchstone is S6S^6, which carries the standard octonionic almost complex structure (the one induced on each tangent space by multiplication with a fixed imaginary unit octonion; it is non-integrable), and it is now known to carry complex structures, the classical Hopf problem of whether S6S^6 admits one having been resolved in the affirmative in 2026 (Alpöge–Claude, with further families by Liu–Rollenske and Viaclovsky; see References); the known negative results are metric-restricted, namely LeBrun's theorem that the round S6S^6 admits no orthogonal (metric-compatible) complex structure, and the theorem of Liu–Yang, recorded in the source paper, that no complex structure on S6S^6 is compatible with a metric of sectional curvature in (1/4,1](1/4,1]. Wan's approach connects integrability to curvature through the Hodge–Laplace operator Δ\Delta acting on JJ, viewed as a tangent-bundle-valued 1-form: on a compact manifold ΔJ=0\Delta J=0 (a harmonic complex structure) forces dJ=0dJ=0, and a direct identity gives dJ(X,Y)−dJ(JX,JY)=−J N(J)(X,Y)dJ(X,Y)-dJ(JX,JY)=-J\,N(J)(X,Y), hence integrability. The Weitzenböck formula for ΔJ\Delta J — the standard identity expressing the Hodge–Laplace operator on bundle-valued forms through covariant derivatives plus a curvature term — then yields Bochner-type integral identities, e.g. JJ is harmonic as soon as ∫M(∣∇J∣2+⟨JR(ei,ej)ei,Jej⟩−⟨R(ei,ej)Jei,Jej⟩) dv=0\int_M(|\nabla J|^2+\langle JR(e_i,e_j)e_i,Je_j\rangle-\langle R(e_i,e_j)Je_i,Je_j\rangle)\,dv=0, and in the almost-Hermitian (i.e. JJ compatible with the metric) case the condition dJ=0dJ=0 is equivalent to the vanishing of an analogous global integral involving the scalar curvature SS and the curvature pairing ⟨R(ei,ej)Jei,Jej⟩\langle R(e_i,e_j)Je_i,Je_j\rangle. Here ∇\nabla is the Levi-Civita connection, RR the curvature tensor, and {ei}\{e_i\} a local orthonormal frame. Wan proves these are sufficient conditions for integrability, giving the nested classes Kähler ⊂\subset harmonic ⊂\subset complex (a Kähler structure being a metric-compatible JJ with ∇J=0\nabla J=0), and uses them to show the round S6S^6 admits no harmonic complex structure; but integrable structures need not satisfy them. The obstruction program he formulates asks for the converse direction: a curvature expression whose vanishing is not merely sufficient but equivalent to integrability.

Problem Statement

Find a global curvature functional C(J)C(J) on a compact almost complex manifold MM — a quantity given by an integral over MM of an expression built from JJ, the curvature tensor RR of a Riemannian metric gg on MM, and covariant derivatives of JJ, in the source's words a global curvature expression related to JJ which is "similar to formula 4.1 or 5.1", i.e. in the spirit of the Bochner-type integral identities ∫M(∣∇J∣2+⟨JR(ei,ej)ei,Jej⟩−⟨R(ei,ej)Jei,Jej⟩) dv=0\int_M(|\nabla J|^2+\langle JR(e_i,e_j)e_i,Je_j\rangle-\langle R(e_i,e_j)Je_i,Je_j\rangle)\,dv=0 and, in the almost-Hermitian case, ∫M(∣∇J∣2+S−⟨R(ei,ej)Jei,Jej⟩) dv=0\int_M(|\nabla J|^2+S-\langle R(e_i,e_j)Je_i,Je_j\rangle)\,dv=0 — such that, provably and without further hypotheses on JJ, the Nijenhuis tensor N(J)N(J) (whose pointwise vanishing is equivalent to integrability) vanishes identically if and only if C(J)=0C(J)=0. This is the requested necessary-and-sufficient curvature characterization of integrability; the sufficient conditions currently available (harmonicity of JJ, or the vanishing of the above integrals) must become equivalences.

Two clarifications of the admissible class, both following the source's setup. (i) The metric gg entering the integrand (through RR, ∇\nabla, the inner products and the volume form) is an arbitrary but fixed Riemannian metric on MM; as in the source, JJ need not be compatible with gg, and the equivalence must hold for every almost complex structure JJ without any compatibility hypothesis. (ii) C(J)C(J) must be a genuine curvature expression of the model type — a scalar integrand assembled from JJ, RR, ∇J\nabla J and metric contractions as in the displayed identities — and must not contain the Nijenhuis tensor itself: since N(J)N(J) is expressible through JJ and ∇J\nabla J, a construction such as ∫M∣N(J)∣2 dv\int_M|N(J)|^2\,dv (or any integral of a pointwise nonnegative expression engineered to vanish exactly where N(J)=0N(J)=0) satisfies the displayed equivalence trivially, but it repackages the integrability condition instead of deriving it from curvature data and therefore does not belong to the class of curvature expressions the source asks for. The content of the problem is precisely that a global Bochner-type curvature quantity controls pointwise integrability.

The stated purpose of the characterization, expressed conditionally in the source as an application rather than as part of the required criterion, is to serve as an obstruction theory: once CC is in hand, identify geometric conditions on (M,g)(M,g) under which C(J)>0C(J)>0 (or <0<0) for every almost complex structure JJ on MM, thereby proving that MM admits no complex structure — with the still-open general question of whether every compact almost complex manifold of real dimension at least 66 admits a complex structure (Yau's problem 52) as the motivating test case, where such a sign-forcing result would produce the first compact almost-complex-but-not-complex manifolds in dimension at least 66.

References

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  2. Gabriella Clemente, A curvature obstruction to integrability, Math. Commun. 28 (2023) 29–48, arXiv:2108.03376 (2021, v6 2023), https://arxiv.org/abs/2108.03376

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  13. Jeff Viaclovsky, A two-parameter family of complex structures on S6S^6, arXiv:2609.33785 (2026), https://arxiv.org/abs/2609.33785

  14. Jeff Viaclovsky, Combs and complex structures on the six-sphere, arXiv:2610.10451 (2026), https://arxiv.org/abs/2610.10451