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In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry.

The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s.[1]

Formal definition

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Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field J of degree (1, 1) such that when regarded as a vector bundle isomorphism on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold.

If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose M is n-dimensional, and let J : TM → TM be an almost complex structure. If J2 = −1 then (det J)2 = (−1)n. But if M is a real manifold, then det J is a real number – thus n must be even if M has an almost complex structure. One can show that it must be orientable as well.

An easy exercise in linear algebra shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a (1, 1)-rank tensor pointwise (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold M is equivalent to a reduction of the structure group of the tangent bundle from GL(2n, R) to GL(n, C). The existence question is then a purely algebraic topological one and is fairly well understood.

Examples

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For every integer n, the flat space R2n admits an almost complex structure. An example for such an almost complex structure is (1 ≤ j, k ≤ 2n): for odd j, for even j.

The only spheres which admit almost complex structures are S2 and S6 (Borel & Serre (1953)). In particular, S4 cannot be given an almost complex structure (Ehresmann and Hopf). In the case of S2, the almost complex structure comes from an honest complex structure on the Riemann sphere. The 6-sphere, S6, when considered as the set of unit norm imaginary octonions, inherits an almost complex structure from the octonion multiplication; the question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.[2]

Differential topology of almost complex manifolds

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Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V− (the eigenspaces of J corresponding to +i and −i, respectively), so an almost complex structure on M allows a decomposition of the complexified tangent bundle TMC (which is the vector bundle of complexified tangent spaces at each point) into TM+ and TM−. A section of TM+ is called a vector field of type (1, 0), while a section of TM− is a vector field of type (0, 1). Thus J corresponds to multiplication by i on the (1, 0)-vector fields of the complexified tangent bundle, and multiplication by −i on the (0, 1)-vector fields.

Just as we build differential forms out of exterior powers of the cotangent bundle, we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle). The almost complex structure induces the decomposition of each space of r-forms

In other words, each Ωr(M)C admits a decomposition into a sum of Ω(p, q)(M), with r = p + q.

As with any direct sum, there is a canonical projection πp,q from Ωr(M)C to Ω(p,q). We also have the exterior derivative d which maps Ωr(M)C to Ωr+1(M)C. Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type

so that is a map which increases the holomorphic part of the type by one (takes forms of type (p, q) to forms of type (p+1, q)), and is a map which increases the antiholomorphic part of the type by one. These operators are called the Dolbeault operators.

Since the sum of all the projections must be the identity map, we note that the exterior derivative can be written

Integrable almost complex structures

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Every complex manifold is itself an almost complex manifold. In local holomorphic coordinates one can define the maps

(just like a counterclockwise rotation of π/2) or

One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.

The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point p. In general, however, it is not possible to find coordinates so that J takes the canonical form on an entire neighborhood of p. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If M admits local holomorphic coordinates for J around every point then these patch together to form a holomorphic atlas for M giving it a complex structure, which moreover induces J. J is then said to be 'integrable'. If J is induced by a complex structure, then it is induced by a unique complex structure.

Given any linear map A on each tangent space of M; i.e., A is a tensor field of rank (1, 1), then the Nijenhuis tensor is a tensor field of rank (1,2) given by

or, for the usual case of an almost complex structure A=J such that ,

The individual expressions on the right depend on the choice of the smooth vector fields X and Y, but the left side actually depends only on the pointwise values of X and Y, which is why NA is a tensor. This is also clear from the component formula

In terms of the Frölicher–Nijenhuis bracket, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor NA is just one-half of [A, A].

The Newlander–Nirenberg theorem states that an almost complex structure J is integrable if and only if NJ = 0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.

There are several other criteria which are equivalent to the vanishing of the Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature):

  • The Lie bracket of any two (1, 0)-vector fields is again of type (1, 0)

Any of these conditions implies the existence of a unique compatible complex structure.

The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether S6 admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For real-analytic J, the Newlander–Nirenberg theorem follows from the Frobenius theorem; for C∞ (and less smooth) J, analysis is required (with more difficult techniques as the regularity hypothesis weakens).

Compatible triples

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Suppose M is equipped with a symplectic form ω, a Riemannian metric g, and an almost complex structure J. Since ω and g are nondegenerate, each induces a bundle isomorphism TM → T*M, where the first map, denoted φω, is given by the interior product φω(u) = iuω = ω(u, •) and the other, denoted φg, is given by the analogous operation for g. With this understood, the three structures (g, ω, J) form a compatible triple when each structure can be specified by the two others as follows:

  • g(u, v) = ω(u, Jv)
  • ω(u, v) = g(Ju, v)
  • J(u) = (φg)−1(φω(u)).

In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified. For example, ω and J are compatible if and only if ω(•, J•) is a Riemannian metric. The bundle on M whose sections are the almost complex structures compatible to ω has contractible fibres: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.

Using elementary properties of the symplectic form ω, one can show that a compatible almost complex structure J is an almost Kähler structure for the Riemannian metric ω(u, Jv). Also, if J is integrable, then (M, ω, J) is a Kähler manifold.

These triples are related to the 2 out of 3 property of the unitary group.

Generalized almost complex structure

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Nigel Hitchin introduced the notion of a generalized almost complex structure on the manifold M, which was elaborated in the doctoral dissertations of his students Marco Gualtieri and Gil Cavalcanti. An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM. A generalized almost complex structure is a choice of a half-dimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles. In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original bundle.

An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the Lie bracket. A generalized almost complex structure integrates to a generalized complex structure if the subspace is closed under the Courant bracket. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing pure spinor then M is a generalized Calabi–Yau manifold.

See also

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References

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Almost complex manifold

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An almost complex manifold is a smooth real manifold MM of even dimension 2n2n equipped with a smooth tensor field JJ of type (1,1) on its tangent bundle TMTM such that J2=−IdJ^2 = -\mathrm{Id}. This structure endows each tangent space TpMT_pM with the properties of a complex vector space of dimension nn, allowing the complexification TM⊗CTM \otimes \mathbb{C} to decompose into eigenspaces T1,0MT^{1,0}M and T0,1MT^{0,1}M corresponding to eigenvalues ii and −i-i, respectively. Such manifolds generalize complex manifolds, as every complex manifold admits a canonical almost complex structure induced by its holomorphic atlas, but the converse requires additional conditions. The key distinction between almost complex and complex manifolds lies in the integrability of the almost complex structure JJ. Integrability is equivalent to the vanishing of the Nijenhuis tensor NJN_J, defined by NJ(X,Y)=[JX,JY]−J[JX,Y]−J[X,JY]+[X,Y]N_J(X,Y) = [JX, JY] - J[JX, Y] - J[X, JY] + [X, Y] for vector fields X,YX, Y, or alternatively, to the complex subbundle T0,1MT^{0,1}M being closed under Lie brackets. When NJ=0N_J = 0, local holomorphic coordinates exist, making (M,J)(M, J) a . This integrability criterion is established by the Newlander-Nirenberg theorem, which asserts that an almost complex structure on a manifold is integrable it arises from a compatible with a holomorphic atlas. Almost complex manifolds play a central role in differential geometry and topology, enabling the study of complex vector bundles and characteristic classes like Chern classes on real manifolds. Notable examples include the standard complex structure on Cn\mathbb{C}^n and Kähler manifolds, while S2S^2 and S6S^6 admit almost complex structures but not complex ones. They also arise in and , where generalized almost complex structures extend the framework to include additional geometric data.

Basic Concepts

Definition

An almost complex structure on a smooth manifold MM is defined as a smooth section JJ of the bundle End(TM)\mathrm{End}(TM) satisfying J2=−IdTMJ^2 = -\mathrm{Id}_{TM}, where TMTM denotes the of MM. Such a structure exists only if dim⁡M=2n\dim M = 2n for some integer n≥1n \geq 1, as the condition J2=−IdJ^2 = -\mathrm{Id} requires the real to be even. Moreover, the of JJ implies that MM is orientable, since JJ provides a consistent choice of orientation via the of tangent spaces into JJ-invariant planes. The almost complex structure JJ extends C\mathbb{C}-linearly to the complexification TCM=TM⊗RCT_{\mathbb{C}}M = TM \otimes_{\mathbb{R}} \mathbb{C}, yielding a direct sum decomposition TCM=T1,0M⊕T0,1M,T_{\mathbb{C}}M = T^{1,0}M \oplus T^{0,1}M, where T1,0MT^{1,0}M is the eigenspace of JJ corresponding to the eigenvalue +i+i, and T0,1MT^{0,1}M is the eigenspace for −i-i. This decomposition is unique and splits the complexified tangent bundle into holomorphic and anti-holomorphic components, each of complex dimension nn. A dual decomposition arises on the complexified cotangent bundle TC∗MT^*_{\mathbb{C}}M, with JJ acting on one-forms via the transpose, yielding (T∗)1,0M(T^*)^{1,0}M and (T∗)0,1M(T^*)^{0,1}M as the respective eigenspaces. The notion of an almost complex structure was introduced in the late 1940s by Charles Ehresmann and as a means to reduce the structure group of the frame bundle of TMTM from GL(2n,R)\mathrm{GL}(2n, \mathbb{R}) to GL(n,C)\mathrm{GL}(n, \mathbb{C}). Ehresmann formalized this in his work on fiber bundles, showing that such a reduction endows the manifold with a GL(n,C)\mathrm{GL}(n, \mathbb{C})-structure compatible with the complex linear group. Independently, Hopf explored topological implications, including the existence of almost complex structures on certain spheres. This reduction captures the essential geometric property of JJ, allowing the tangent spaces to be identified with Cn\mathbb{C}^n in a smooth, varying manner across MM.

Examples

The standard example of an almost complex manifold is the Euclidean space R2n\mathbb{R}^{2n} equipped with the canonical almost complex structure J0J_0, obtained by identifying R2n\mathbb{R}^{2n} with Cn\mathbb{C}^n and defining J0J_0 as multiplication by ii on each tangent space. This structure is integrable, making R2n\mathbb{R}^{2n} biholomorphic to Cn\mathbb{C}^n. Complex Lie groups provide further examples, as they are smooth manifolds equipped with a compatible almost complex structure derived from their complex multiplication. For instance, the special linear group SL(n,C)\mathrm{SL}(n,\mathbb{C}), viewed as a real manifold of dimension 2(n2−1)2(n^2-1), admits a left-invariant almost complex structure that is integrable by construction. Similarly, certain real semisimple Lie groups, such as SL(3,R)\mathrm{SL}(3,\mathbb{R}), admit left-invariant complex structures, hence integrable almost complex structures. Among spheres, only S2S^2 and S6S^6 admit almost complex structures, as established by topological obstructions for higher even-dimensional spheres. The 2-sphere S2S^2 carries a standard almost complex structure via its identification with the CP1\mathbb{CP}^1, induced by the S1→S3→S2S^1 \to S^3 \to S^2. For S6S^6, an almost complex structure arises from its embedding in the space of imaginary octonions, leveraging the octonionic Hopf fibration and related G2G_2-structures. Integrable almost complex manifolds include complex tori and Calabi–Yau manifolds. Complex tori, formed as quotients Cn/Λ\mathbb{C}^n / \Lambda by a lattice Λ\Lambda, inherit an integrable almost complex structure from Cn\mathbb{C}^n. Calabi–Yau manifolds, which are compact Kähler manifolds of complex dimension nn with trivial and vanishing first , possess integrable almost complex structures compatible with their Ricci-flat Kähler metrics. Non-integrable examples abound in homogeneous spaces, particularly nearly Kähler manifolds. The 6-sphere S6S^6 with its standard almost complex structure from the is non-integrable, as shown by the failure of the Newlander–Nirenberg integrability condition; this relates to the longstanding Hopf problem, where the existence of an integrable structure remains open despite the almost complex case being resolved affirmatively. Other homogeneous examples include the nearly Kähler structure on S3×S3S^3 \times S^3, which is non-integrable and arises from its SU(2)×SU(2)SU(2) \times SU(2) symmetry.

Properties and Topology

Differential topology

An almost complex structure JJ on a smooth manifold MM of real dimension 2n2n endows the complexified tangent bundle TM⊗CTM \otimes \mathbb{C} with a C\mathbb{C}-linear decomposition into eigenspaces T1,0M⊕T0,1MT^{1,0}M \oplus T^{0,1}M, where T1,0M={v−iJv∣v∈TM}T^{1,0}M = \{ v - i Jv \mid v \in TM \} is the +i+i-eigenspace and T0,1M={v+iJv∣v∈TM}T^{0,1}M = \{ v + i Jv \mid v \in TM \} is the −i-i-eigenspace, each of complex dimension nn. Dually, the complexified cotangent bundle T∗M⊗CT^*M \otimes \mathbb{C} decomposes as T∗1,0M⊕T∗0,1MT^{*1,0}M \oplus T^{*0,1}M, inducing a bigrading on the space of complex differential forms Λ∗(T∗M⊗C)=⨁p,qΛp,q(T∗M)\Lambda^* (T^*M \otimes \mathbb{C}) = \bigoplus_{p,q} \Lambda^{p,q} (T^*M), where Λp,q\Lambda^{p,q} consists of forms of type (p,q)(p,q). This bigrading allows the exterior derivative dd to decompose into components of bidegrees (1,0)(1,0), (0,1)(0,1), (2,−1)(2,-1), and (−1,2)(-1,2), denoted respectively as ∂\partial, ∂ˉ\bar{\partial}, μ\mu, and μˉ\bar{\mu}, so that d=∂+∂ˉ+μ+μˉd = \partial + \bar{\partial} + \mu + \bar{\mu}. The operators ∂:Λp,q→Λp+1,q\partial: \Lambda^{p,q} \to \Lambda^{p+1,q} and ∂ˉ:Λp,q→Λp,q+1\bar{\partial}: \Lambda^{p,q} \to \Lambda^{p,q+1}, known as the Dolbeault operators, satisfy ∂2=0\partial^2 = 0 and ∂ˉ2=0\bar{\partial}^2 = 0 on any almost complex manifold. However, without integrability of JJ, it does not hold in general that d=∂+∂ˉd = \partial + \bar{\partial}, as the Nijenhuis tensor contributes to the higher-degree components μ\mu and μˉ\bar{\mu}. The almost complex structure defines a reduction of the frame bundle of MM to the complex general linear group GL(n,C)\mathrm{GL}(n, \mathbb{C}), making the tangent bundle TMTM into a complex vector bundle of rank nn. Consequently, the Chern classes ck(TM)∈H2k(M;Z)c_k(TM) \in H^{2k}(M; \mathbb{Z}), k=0,…,nk = 0, \dots, n, serve as topological invariants of the almost complex structure. In particular, the first Chern class c1(TM)c_1(TM) equals the negative of the first Chern class of the canonical bundle KM=det⁡(T∗1,0M)K_M = \det(T^{*1,0}M), providing a characteristic class that encodes information about the determinant line bundle associated to the structure. The existence of an almost complex structure on a smooth manifold MM requires dim⁡M\dim M to be even, as J2=−IdJ^2 = -\mathrm{Id} implies the real dimension is twice the complex dimension nn. More generally, such structures correspond to sections of a principal GL(n,C)\mathrm{GL}(n, \mathbb{C})-bundle over MM, and obstructions to their existence lie in the groups Hk+1(M;πk(GL(n,C)))H^{k+1}(M; \pi_k(\mathrm{GL}(n, \mathbb{C}))) for relevant skeleta in a CW decomposition of MM. For spheres, almost complex structures exist on S2S^2 and S6S^6, but not on other even-dimensional spheres due to topological obstructions in these groups, nor on odd-dimensional spheres due to the dimension constraint.

Existence conditions

The existence of an almost complex structure on a smooth manifold MM of dimension 2n2n is equivalent to a reduction of the structure group of the TMTM from GL(2n,R)\mathrm{GL}(2n, \mathbb{R}) to GL(n,C)\mathrm{GL}(n, \mathbb{C}). For oriented manifolds, this corresponds to a reduction from SO(2n)\mathrm{SO}(2n) to U(n)\mathrm{U}(n), which can be analyzed using obstruction theory in the U(n)→BSO(2n)\mathrm{U}(n) \to \mathrm{BSO}(2n). A necessary condition for such a reduction is that MM is even-dimensional and orientable, meaning the first Stiefel-Whitney class satisfies w1(TM)=0w_1(TM) = 0. Additionally, the second Stiefel-Whitney class must admit an integral lift, i.e., the Bockstein homomorphism β(w2(TM))=0\beta(w_2(TM)) = 0 in H3(M;Z)H^3(M; \mathbb{Z}), which is the defining condition for MM to admit a Spinc\mathrm{Spin}^c structure. This equivalence holds because the inclusion U(n)↪Spinc(2n)\mathrm{U}(n) \hookrightarrow \mathrm{Spin}^c(2n) induces isomorphisms on low-dimensional homotopy groups relevant to the obstructions, making the existence of an almost complex structure topologically equivalent to the existence of a Spinc\mathrm{Spin}^c structure on TMTM. In dimensions up to 6, these conditions are often sufficient, with the primary obstruction being β(w2)=0\beta(w_2) = 0 for n>1n > 1. For compact manifolds, higher-dimensional obstructions arise from mismatches in characteristic classes, such as Pontryagin or Chern classes, preventing existence in certain cases. In dimension 4, a complete criterion is that there exists a cohomology class h∈H2(M;Z)h \in H^2(M; \mathbb{Z}) satisfying h2=2χ(M)+3σ(M)h^2 = 2\chi(M) + 3\sigma(M) and h≡w2(TM)(mod2)h \equiv w_2(TM) \pmod{2}, where χ(M)\chi(M) is the Euler characteristic and σ(M)\sigma(M) is the signature; this follows from the topological Noether formula relating Chern numbers to Hirzebruch signatures. For example, the connected sum CP2#CP2\mathbb{CP}^2 \# \mathbb{CP}^2 has χ=4\chi = 4 and σ=2\sigma = 2, so 2χ+3σ=142\chi + 3\sigma = 14, but its intersection form on H2H^2 (isomorphic to Z⊕Z\mathbb{Z} \oplus \mathbb{Z} with diagonal (1,1)) admits no class squaring to 14, hence no almost complex structure exists. The classification of almost complex structures on a given manifold is tied to cohomology via the associated characteristic classes: each such structure induces a complex vector bundle structure on TMTM, determining its Chern classes ck(TM,J)∈H2k(M;Z)c_k(TM, J) \in H^{2k}(M; \mathbb{Z}), which must satisfy Wu relations like ck≡w2k(mod2)c_k \equiv w_{2k} \pmod{2}. The moduli space of almost complex structures modulo diffeomorphisms often has components parameterized by these cohomology classes, reflecting the homotopy type of the classifying space BU(n)\mathrm{BU}(n). Examples of manifolds satisfying these conditions include Brieskorn varieties, which are smooth complex hypersurfaces in CPm\mathbb{CP}^{m} defined by equations ∑ziai=0\sum z_i^{a_i} = 0 with aia_i odd integers greater than 1. As complex submanifolds of the almost complex manifold CPm\mathbb{CP}^{m}, they inherit an almost complex structure by restricting the standard Fubini-Study structure on the ambient space. The Lefschetz hyperplane theorem ensures compatibility in homology up to the middle dimension, confirming the topological suitability for this induced structure.

Integrability

Integrability criteria

The integrability of an almost complex structure JJ on a smooth manifold MM is governed by the vanishing of the Nijenhuis tensor, a fundamental obstruction introduced in the study of such structures. The Nijenhuis tensor NJN_J is defined as the (1,2)(1,2)-tensor given by NJ(X,Y)=[JX,JY]−J[JX,Y]−J[X,JY]+[X,Y]N_J(X,Y) = [JX, JY] - J[JX, Y] - J[X, JY] + [X,Y] for all smooth vector fields X,YX, Y on MM. An almost complex structure JJ is integrable if and only if NJ=0N_J = 0 pointwise on MM. This condition admits an equivalent formulation in terms of the complexified tangent bundle. Decompose TM⊗C=T1,0M⊕T0,1MTM \otimes \mathbb{C} = T^{1,0}M \oplus T^{0,1}M, where T1,0MT^{1,0}M is the +i+i-eigensbundle of the complexification of JJ. The structure JJ is integrable if and only if T1,0MT^{1,0}M is involutive, meaning that the Lie bracket of any two sections of T1,0MT^{1,0}M remains in T1,0MT^{1,0}M. This equivalence follows from an application of the Frobenius theorem to the distribution defined by T1,0MT^{1,0}M. A key local consequence of integrability is the existence of holomorphic coordinates. If JJ is integrable, then for every point p∈Mp \in M, there exists a neighborhood UU of pp and complex coordinates z1,…,znz^1, \dots, z^n on UU (with zj=xj+iyjz^j = x^j + i y^j) such that JJ takes the standard complex form: J∂∂xk=∂∂ykJ \frac{\partial}{\partial x^k} = \frac{\partial}{\partial y^k} and J∂∂yk=−∂∂xkJ \frac{\partial}{\partial y^k} = -\frac{\partial}{\partial x^k} for k=1,…,nk = 1, \dots, n. Non-integrable examples illustrate the role of the Nijenhuis tensor. A standard construction on R4\mathbb{R}^4 involves modifying the usual complex structure on C2\mathbb{C}^2 by "twisting" the action on the second factor using a non-closed 1-form, resulting in NJ≠0N_J \neq 0.

Newlander–Nirenberg theorem

The Newlander–Nirenberg theorem asserts that an almost complex structure JJ on a smooth manifold MM is integrable—that is, its Nijenhuis tensor vanishes, NJ=0N_J = 0—if and only if, around every point of MM, there exist local coordinates in which JJ coincides with the standard complex structure induced by the identification R2n≅Cn\mathbb{R}^{2n} \cong \mathbb{C}^n. This equivalence establishes a precise relationship between the differential-geometric notion of integrability and the existence of a compatible complex manifold structure locally on MM. The proof, originally developed by Newlander and Nirenberg in 1957, relies on solving a Beltrami equation associated to the almost complex structure using theory to construct the required local holomorphic coordinate charts. Subsequent improvements, including a parametric version applicable to families of almost complex structures, were provided by Nijenhuis and Woolf in 1963, enhancing the theorem's scope for deformations and dependencies on parameters. For the global version, on paracompact manifolds—which include all smooth manifolds—the local holomorphic charts can be glued together via to form a maximal atlas, endowing MM with the structure of a . In the compact case, this atlas is finite, ensuring a well-defined global complex structure. However, counterexamples exist in non-smooth settings, such as CkC^k-integrable almost complex structures for finite k≥1k \geq 1, where the Nijenhuis tensor vanishes but no corresponding Ck+1C^{k+1}-holomorphic coordinates are available, highlighting the necessity of sufficient regularity for the theorem to hold. The theorem's implications are profound: complex manifolds are exactly those smooth manifolds admitting an integrable almost complex structure. A longstanding open problem concerns the 6-sphere S6S^6, which admits almost complex structures but whether any is integrable—thus making S6S^6 a complex manifold—remains unknown as of 2025.

Almost Hermitian Geometry

Compatible triples

A compatible triple on an almost complex manifold consists of an almost complex structure JJ, a Riemannian metric gg, and a 2-form ω\omega, where gg satisfies the compatibility condition g(JX,JY)=g(X,Y)g(JX, JY) = g(X, Y) for all tangent vectors X,YX, Y, ensuring JJ is orthogonal with respect to gg. The 2-form ω\omega is then defined by ω(X,Y)=g(JX,Y)\omega(X, Y) = g(JX, Y), which is non-degenerate as a consequence of the positive-definiteness of gg and the properties of JJ. This setup forms an almost Hermitian structure on the manifold, integrating the almost complex geometry with a compatible metric and associated fundamental form. The fundamental 2-form ω\omega is of type (1,1) with respect to JJ, meaning it maps type (1,0) vectors to their conjugates in a manner consistent with the eigenspaces of JJ. In this configuration, the triple (J,g,ω)(J, g, \omega) endows the manifold with a U(n)U(n)-structure, where the structure group of the reduces from GL(2n,R)GL(2n, \mathbb{R}) to the U(n)U(n). The non-degeneracy of ω\omega implies that the musical induced by ω\omega is invertible, allowing ω\omega to serve as a tool for studying the interplay between the metric and complex directions on the manifold. When JJ is integrable and ω\omega is closed, the triple defines a Kähler structure, but in the almost case, ω\omega need not be closed. Examples of compatible triples abound in standard geometric settings. On R2n\mathbb{R}^{2n} equipped with the standard almost complex structure J0J_0 (where J0(∂x)=∂yJ_0(\partial_x) = \partial_y and J0(∂y)=−∂xJ_0(\partial_y) = -\partial_x in coordinates), the Euclidean metric g0=dx2+dy2g_0 = dx^2 + dy^2 (extended componentwise) satisfies the compatibility g0(J0X,J0Y)=g0(X,Y)g_0(J_0 X, J_0 Y) = g_0(X, Y), and the induced ω0=dx∧dy\omega_0 = dx \wedge dy (again extended) forms the flat almost Hermitian structure. A non-Kähler example arises on the Hopf surface, obtained as C2∖{0}\mathbb{C}^2 \setminus \{0\} quotiented by the action (z1,z2)↦(2z1,2z2)(z_1, z_2) \mapsto (2z_1, 2z_2); it admits an integrable complex structure JJ and a compatible Hermitian metric gg (hence ω\omega non-degenerate), but lacks a closed ω\omega globally due to vanishing second .

Almost Kähler structures

An almost Kähler manifold is an almost Hermitian manifold (M2n,J,g)(M^{2n}, J, g) equipped with a closed fundamental 2-form ω(X,Y)=g(JX,Y)\omega(X, Y) = g(JX, Y), satisfying dω=0d\omega = 0. This closure condition renders ω\omega a symplectic form that is compatible with both the almost complex structure JJ and the Riemannian metric gg, thereby combining elements of symplectic and almost Hermitian . In the special case where JJ is integrable, making (M,J,g)(M, J, g) Hermitian, the almost Kähler condition dω=0d\omega = 0 implies that the Lee form θ\theta, defined by dω=θ∧ωd\omega = \theta \wedge \omega, must vanish (θ=0\theta = 0). This vanishing of the Lee form renders the metric balanced, as θ=0\theta = 0 is equivalent to the codifferential δω=0\delta \omega = 0 in this context, providing an obstruction to non-trivial conformal deformations within the class of Hermitian metrics. Notable subclasses of almost Kähler manifolds arise from refined conditions on the torsion. Nearly Kähler manifolds form a distinguished subclass, characterized by the of JJ satisfying (∇XJ)Y=−J(∇XJ)JY(\nabla_X J)Y = -J(\nabla_X J)JY for all vector fields X,YX, Y (i.e., ∇J\nabla J is skew-symmetric) and (∇XJ)X=0(\nabla_X J)X = 0 for all XX, ensuring a specific algebraic relation dω(X,Y,Z)=g((∇XJ)Y,Z)d\omega(X, Y, Z) = g((\nabla_X J)Y, Z). Quasi-Kähler manifolds, on the other hand, are defined by the condition ∂ω=0\partial \omega = 0, where ∂\partial is the Dolbeault operator induced by JJ, imposing a partial closure on the torsion components of ω\omega. The full Kähler manifolds emerge when JJ is integrable and dω=0d\omega = 0, satisfying both the Newlander-Nirenberg integrability and the symplectic closure. Prominent examples include the complex projective spaces CPn\mathbb{CP}^n with the Fubini-Study metric, which carry a standard Kähler structure where JJ is integrable and ω\omega is closed. Another example is the 6-sphere S6S^6, which admits a nearly Kähler structure derived from the multiplication of pure , yielding a non-integrable almost complex structure JJ compatible with a round metric gg and closed ω\omega, highlighting exceptional geometric features tied to division algebras.

Generalizations and Applications

Generalized almost complex structures

In generalized geometry, an almost complex structure is extended to the generalized tangent bundle TM⊕T∗MTM \oplus T^*M, providing a unified framework that interpolates between complex and symplectic geometries. A generalized almost complex structure Φ\Phi on a smooth manifold MM is defined as an of the real TM⊕T∗MTM \oplus T^*M satisfying Φ2=−Id\Phi^2 = -\mathrm{Id} and such that its +i+i-eigenspace LΦ⊂(TM⊕T∗M)⊗CL_\Phi \subset (TM \oplus T^*M) \otimes \mathbb{C} is isotropic with respect to the natural symmetric ⟨X+ξ,Y+η⟩=ξ(Y)+η(X)\langle X + \xi, Y + \eta \rangle = \xi(Y) + \eta(X) for sections X+ξ,Y+ηX + \xi, Y + \eta. This induces a neutral metric of (n,n)(n,n) on the bundle, where n=dim⁡Mn = \dim M, and isotropy ensures LΦL_\Phi has dimension nn and ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle vanishes on it. The standard almost complex structure JJ on TMTM embeds into this framework via the endomorphism ΦJ=(−J00Jt)\Phi_J = \begin{pmatrix} -J & 0 \\ 0 & J^t \end{pmatrix} acting on TM⊕T∗MTM \oplus T^*M, where JtJ^t is the (or musical ) induced by the pairing. Here, the +i+i-eigenspace LJL_J consists of sections X1,0+ξ0,1X^{1,0} + \xi^{0,1}, where X1,0X^{1,0} are holomorphic vectors and ξ0,1\xi^{0,1} anti-holomorphic forms, recovering the classical case when the structure is of maximal type nn. This correspondence highlights how generalized structures reduce to ordinary ones under appropriate projections. Integrability of Φ\Phi is defined analogously to the classical Nijenhuis condition but using the Courant algebroid structure on TM⊕T∗MTM \oplus T^*M. Specifically, Φ\Phi is integrable if LΦL_\Phi is closed under the Courant bracket [[X+ξ,Y+η]]C=[X,Y]+LXη−ιYdξ−12d(ιXη−ιYξ)[[X + \xi, Y + \eta]]_C = [X,Y] + \mathcal{L}_X \eta - \iota_Y d\xi - \frac{1}{2} d(\iota_X \eta - \iota_Y \xi), making LΦL_\Phi a Dirac structure. This condition ensures the existence of an atlas of local pure spinors defining the structure, generalizing the integrability of almost complex manifolds. Examples illustrate the breadth of this generalization. For a symplectic manifold with closed 2-form ω\omega, the structure arises from the pure spinor line bundle generated by eiωe^{i\omega}, yielding a generalized almost complex structure of type 0 where LωL_\omega is spanned by eiωe^{i\omega}-annihilated sections. In the complex case, including a closed BB-field (a real closed 2-form), the pure spinor eB+iωJe^{B + i \omega_J} defines Φ\Phi, blending Hermitian aspects while allowing type to vary from 0 to nn. Poisson structures, given by a bivector field π\pi satisfying [π,π]S=0[\pi, \pi]_S = 0 under the Schouten bracket, induce a generalized almost complex structure via the graph of π♯:T∗M→TM\pi^\sharp: T^*M \to TM, with type jumping along the rank locus of π\pi. In , almost complex structures play a crucial role by providing a compatible almost complex structure JJ that tames a given symplectic form ω\omega, meaning g(X,Y)=ω(X,JY)g(X,Y) = \omega(X, JY) defines a Riemannian metric and JJ satisfies J2=−IdJ^2 = -\mathrm{Id}. This compatibility enables the application of techniques analogous to the , which deforms symplectic forms within the same class while preserving the tamed structure, and extends the to local normal forms for symplectic manifolds equipped with such JJ. Calabi-Yau manifolds are compact Kähler manifolds equipped with an integrable almost complex structure and a Ricci-flat metric, satisfying the condition that the first vanishes, which ensures the existence of a unique Ricci-flat Kähler metric in each Kähler class. These structures are fundamental in , where they compactify extra dimensions to preserve , and in mirror symmetry, which posits a duality between pairs of Calabi-Yau threefolds exchanging complex structure moduli with Kähler moduli. The of almost complex structures compatible with a fixed symplectic form on a manifold is contractible, allowing the study of deformations within the symplectic category through invariants like Gromov-Witten invariants, which count pseudo-holomorphic curves for generic choices of JJ and provide obstructions or dimensions to this space. These invariants, defined via the virtual fundamental class of the of stable maps, remain unchanged under small deformations of JJ, thus parameterizing the symplectic deformations of almost complex structures. Almost complex structures find applications in G2G_2-manifolds, where the exceptional group G2⊂SO(7)G_2 \subset SO(7) induces an SU(3)SU(3)-structure on 6-dimensional links like S6S^6, linking nearly Kähler geometry on S6S^6 to broader G2G_2-holonomy constructions in 7 dimensions. In physics, generalized almost complex structures unify complex and symplectic geometries, facilitating T-duality transformations in string theory that relate geometries across dualities while preserving fluxes and brane configurations. A notable open problem persists regarding the integrability of almost complex structures on S6S^6, with no known integrable example as of 2025, despite various nearly complex constructions tied to G2G_2-structures and recent proposed constructions such as those by Etesi (2015, 2024).

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