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Almost complex manifold

An almost complex manifold is a smooth real manifold M of even dimension $2n equipped with a smooth tensor field J of type (1,1) on its tangent bundle TM such that J^2 = -\mathrm{Id}. This structure endows each tangent space T_pM with the properties of a complex vector space of dimension n, allowing the complexification TM \otimes \mathbb{C} to decompose into eigenspaces T^{1,0}M and T^{0,1}M corresponding to eigenvalues i and -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 J. Integrability is equivalent to the vanishing of the Nijenhuis tensor N_J, defined by N_J(X,Y) = [JX, JY] - J[JX, Y] - J[X, JY] + [X, Y] for vector fields X, Y, or alternatively, to the complex subbundle T^{0,1}M being closed under Lie brackets. When N_J = 0, local holomorphic coordinates exist, making (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 \mathbb{C}^n and Kähler manifolds, while S^2 and S^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 M is defined as a smooth section J of the bundle \mathrm{End}(TM) satisfying J^2 = -\mathrm{Id}_{TM}, where TM denotes the of M. Such a structure exists only if \dim M = 2n for some integer n \geq 1, as the condition J^2 = -\mathrm{Id} requires the real to be even. Moreover, the of J implies that M is , since J provides a consistent choice of orientation via the of tangent spaces into J-invariant planes. The almost complex structure J extends \mathbb{C}-linearly to the complexification T_{\mathbb{C}}M = TM \otimes_{\mathbb{R}} \mathbb{C}, yielding a direct sum decomposition T_{\mathbb{C}}M = T^{1,0}M \oplus T^{0,1}M, where T^{1,0}M is the eigenspace of J corresponding to the eigenvalue +i, and T^{0,1}M is the eigenspace for -i. This decomposition is unique and splits the complexified tangent bundle into holomorphic and anti-holomorphic components, each of complex dimension n. A dual decomposition arises on the complexified cotangent bundle T^*_{\mathbb{C}}M, with J acting on one-forms via the transpose, yielding (T^*)^{1,0}M and (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 TM from \mathrm{GL}(2n, \mathbb{R}) to \mathrm{GL}(n, \mathbb{C}). Ehresmann formalized this in his work on fiber bundles, showing that such a reduction endows the manifold with a \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 J, allowing the tangent spaces to be identified with \mathbb{C}^n in a smooth, varying manner across M.

Examples

The standard example of an almost complex manifold is the Euclidean space \mathbb{R}^{2n} equipped with the canonical almost complex structure J_0, obtained by identifying \mathbb{R}^{2n} with \mathbb{C}^n and defining J_0 as multiplication by i on each tangent space. This structure is integrable, making \mathbb{R}^{2n} biholomorphic to \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 \mathrm{SL}(n,\mathbb{C}), viewed as a real manifold of dimension $2(n^2-1), admits a left-invariant almost complex structure that is integrable by construction. Similarly, certain real semisimple Lie groups, such as \mathrm{SL}(3,\mathbb{R}), admit left-invariant complex structures, hence integrable almost complex structures. Among spheres, only S^2 and S^6 admit almost complex structures, as established by topological obstructions for higher even-dimensional spheres. The 2-sphere S^2 carries a standard almost complex structure via its identification with the \mathbb{CP}^1, induced by the S^1 \to S^3 \to S^2. For S^6, an almost complex structure arises from its embedding in the space of imaginary octonions, leveraging the octonionic Hopf fibration and related G_2-structures. Integrable almost complex manifolds include complex tori and Calabi–Yau manifolds. Complex tori, formed as quotients \mathbb{C}^n / \Lambda by a \Lambda, inherit an integrable almost complex structure from \mathbb{C}^n. Calabi–Yau manifolds, which are compact Kähler manifolds of complex dimension n 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 S^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 S^3 \times S^3, which is non-integrable and arises from its SU(2) \times SU(2) symmetry.

Properties and Topology

Differential topology

An almost complex structure J on a smooth manifold M of real dimension $2n endows the complexified tangent bundle TM \otimes \mathbb{C} with a \mathbb{C}-linear decomposition into eigenspaces T^{1,0}M \oplus T^{0,1}M, where T^{1,0}M = \{ v - i Jv \mid v \in TM \} is the +i-eigenspace and T^{0,1}M = \{ v + i Jv \mid v \in TM \} is the -i-eigenspace, each of complex dimension n. Dually, the complexified cotangent bundle T^*M \otimes \mathbb{C} decomposes as T^{*1,0}M \oplus T^{*0,1}M, inducing a bigrading on the space of complex differential forms \Lambda^* (T^*M \otimes \mathbb{C}) = \bigoplus_{p,q} \Lambda^{p,q} (T^*M), where \Lambda^{p,q} consists of forms of type (p,q). This bigrading allows the exterior derivative d to decompose into components of bidegrees (1,0), (0,1), (2,-1), and (-1,2), denoted respectively as \partial, \bar{\partial}, \mu, and \bar{\mu}, so that d = \partial + \bar{\partial} + \mu + \bar{\mu}. The operators \partial: \Lambda^{p,q} \to \Lambda^{p+1,q} and \bar{\partial}: \Lambda^{p,q} \to \Lambda^{p,q+1}, known as the Dolbeault operators, satisfy \partial^2 = 0 and \bar{\partial}^2 = 0 on any almost complex manifold. However, without integrability of J, it does not hold in general that d = \partial + \bar{\partial}, as the Nijenhuis tensor contributes to the higher-degree components \mu and \bar{\mu}. The almost complex structure defines a reduction of the frame bundle of M to the complex general linear group \mathrm{GL}(n, \mathbb{C}), making the tangent bundle TM into a complex vector bundle of rank n. Consequently, the Chern classes c_k(TM) \in H^{2k}(M; \mathbb{Z}), k = 0, \dots, n, serve as topological invariants of the almost complex structure. In particular, the first Chern class c_1(TM) equals the negative of the first Chern class of the canonical bundle 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 M requires \dim M to be even, as J^2 = -\mathrm{Id} implies the real dimension is twice the complex dimension n. More generally, such structures correspond to sections of a principal \mathrm{GL}(n, \mathbb{C})-bundle over M, and obstructions to their existence lie in the groups H^{k+1}(M; \pi_k(\mathrm{GL}(n, \mathbb{C}))) for relevant skeleta in a CW decomposition of M. For spheres, almost complex structures exist on S^2 and S^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 M of dimension $2n is equivalent to a of the structure group of the TM from \mathrm{GL}(2n, \mathbb{R}) to \mathrm{GL}(n, \mathbb{C}). For oriented manifolds, this corresponds to a from \mathrm{SO}(2n) to \mathrm{U}(n), which can be analyzed using obstruction theory in the \mathrm{U}(n) \to \mathrm{BSO}(2n). A necessary condition for such a reduction is that M is even-dimensional and orientable, meaning the first Stiefel-Whitney class satisfies w_1(TM) = 0. Additionally, the second Stiefel-Whitney class must admit an integral lift, i.e., the Bockstein homomorphism \beta(w_2(TM)) = 0 in H^3(M; \mathbb{Z}), which is the defining condition for M to admit a \mathrm{Spin}^c structure. This equivalence holds because the inclusion \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 \mathrm{Spin}^c structure on TM. In dimensions up to 6, these conditions are often sufficient, with the primary obstruction being \beta(w_2) = 0 for n > 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 \in H^2(M; \mathbb{Z}) satisfying h^2 = 2\chi(M) + 3\sigma(M) and h \equiv w_2(TM) \pmod{2}, where \chi(M) is the Euler characteristic and \sigma(M) is the signature; this follows from the topological Noether formula relating Chern numbers to Hirzebruch signatures. For example, the connected sum \mathbb{CP}^2 \# \mathbb{CP}^2 has \chi = 4 and \sigma = 2, so $2\chi + 3\sigma = 14, but its intersection form on H^2 (isomorphic to \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 TM, determining its Chern classes c_k(TM, J) \in H^{2k}(M; \mathbb{Z}), which must satisfy Wu relations like 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 \mathrm{BU}(n). Examples of manifolds satisfying these conditions include Brieskorn varieties, which are smooth complex hypersurfaces in \mathbb{CP}^{m} defined by equations \sum z_i^{a_i} = 0 with a_i odd integers greater than 1. As complex submanifolds of the almost complex manifold \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 J on a smooth manifold M is governed by the vanishing of the Nijenhuis tensor, a fundamental obstruction introduced in the study of such structures. The Nijenhuis tensor N_J is defined as the (1,2)-tensor given by N_J(X,Y) = [JX, JY] - J[JX, Y] - J[X, JY] + [X,Y] for all smooth vector fields X, Y on M. An almost complex structure J is integrable if and only if N_J = 0 pointwise on M. This condition admits an equivalent formulation in terms of the complexified tangent bundle. Decompose TM \otimes \mathbb{C} = T^{1,0}M \oplus T^{0,1}M, where T^{1,0}M is the +i-eigensbundle of the complexification of J. The structure J is integrable if and only if T^{1,0}M is involutive, meaning that the Lie bracket of any two sections of T^{1,0}M remains in T^{1,0}M. This equivalence follows from an application of the Frobenius theorem to the distribution defined by T^{1,0}M. A key local consequence of integrability is the existence of holomorphic coordinates. If J is integrable, then for every point p \in M, there exists a neighborhood U of p and complex coordinates z^1, \dots, z^n on U (with z^j = x^j + i y^j) such that J takes the standard form: J \frac{\partial}{\partial x^k} = \frac{\partial}{\partial y^k} and J \frac{\partial}{\partial y^k} = -\frac{\partial}{\partial x^k} for k = 1, \dots, n. Non-integrable examples illustrate the role of the Nijenhuis tensor. A standard construction on \mathbb{R}^4 involves modifying the usual structure on \mathbb{C}^2 by "twisting" the action on the second factor using a non-closed 1-form, resulting in N_J \neq 0.

Newlander–Nirenberg theorem

The Newlander–Nirenberg theorem asserts that an almost complex structure J on a smooth manifold M is integrable—that is, its Nijenhuis tensor vanishes, N_J = 0—if and only if, around every point of M, there exist local coordinates in which J coincides with the standard complex structure induced by the identification \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 M. 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 M 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 C^k-integrable almost complex structures for finite k \geq 1, where the Nijenhuis tensor vanishes but no corresponding C^{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 S^6, which admits almost complex structures but whether any is integrable—thus making S^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 J, a Riemannian metric g, and a 2-form \omega, where g satisfies the compatibility condition g(JX, JY) = g(X, Y) for all tangent vectors X, Y, ensuring J is orthogonal with respect to g. The 2-form \omega is then defined by \omega(X, Y) = g(JX, Y), which is non-degenerate as a consequence of the positive-definiteness of g and the properties of J. 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 \omega is of type (1,1) with respect to J, meaning it maps type (1,0) vectors to their conjugates in a manner consistent with the eigenspaces of J. In this , the triple (J, g, \omega) endows the manifold with a U(n)-structure, where the structure group of the reduces from GL(2n, \mathbb{R}) to the U(n). The non-degeneracy of \omega implies that the musical induced by \omega is invertible, allowing \omega to serve as a tool for studying the interplay between the metric and complex directions on the manifold. When J is integrable and \omega is closed, the triple defines a Kähler structure, but in the almost case, \omega need not be closed. Examples of compatible triples abound in standard geometric settings. On \mathbb{R}^{2n} equipped with the standard almost complex structure J_0 (where J_0(\partial_x) = \partial_y and J_0(\partial_y) = -\partial_x in coordinates), the Euclidean g_0 = dx^2 + dy^2 (extended componentwise) satisfies the compatibility g_0(J_0 X, J_0 Y) = g_0(X, Y), and the induced \omega_0 = dx \wedge dy (again extended) forms the flat almost Hermitian structure. A non-Kähler example arises on the Hopf surface, obtained as \mathbb{C}^2 \setminus \{0\} quotiented by the action (z_1, z_2) \mapsto (2z_1, 2z_2); it admits an integrable structure J and a compatible Hermitian g (hence \omega non-degenerate), but lacks a closed \omega globally due to vanishing second .

Almost Kähler structures

An almost Kähler manifold is an almost Hermitian manifold (M^{2n}, J, g) equipped with a closed fundamental 2-form \omega(X, Y) = g(JX, Y), satisfying d\omega = 0. This closure condition renders \omega a form that is compatible with both the almost complex structure J and the Riemannian metric g, thereby combining elements of and almost Hermitian . In the special case where J is integrable, making (M, J, g) Hermitian, the almost Kähler condition d\omega = 0 implies that the Lee form \theta, defined by d\omega = \theta \wedge \omega, must vanish (\theta = 0). This vanishing of the Lee form renders the metric balanced, as \theta = 0 is equivalent to the codifferential \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 J satisfying (\nabla_X J)Y = -J(\nabla_X J)JY for all vector fields X, Y (i.e., \nabla J is skew-symmetric) and (\nabla_X J)X = 0 for all X, ensuring a specific algebraic relation d\omega(X, Y, Z) = g((\nabla_X J)Y, Z). Quasi-Kähler manifolds, on the other hand, are defined by the condition \partial \omega = 0, where \partial is the Dolbeault induced by J, imposing a partial on the torsion components of \omega. The full Kähler manifolds emerge when J is integrable and d\omega = 0, satisfying both the Newlander-Nirenberg integrability and the . Prominent examples include the complex projective spaces \mathbb{CP}^n with the Fubini-Study metric, which carry a standard Kähler structure where J is integrable and \omega is closed. Another example is the 6-sphere S^6, which admits a nearly Kähler structure derived from the multiplication of pure , yielding a non-integrable almost complex structure J compatible with a round metric g and closed \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 \oplus T^*M, providing a unified framework that interpolates between complex and geometries. A generalized almost complex structure \Phi on a smooth manifold M is defined as an of the real TM \oplus T^*M satisfying \Phi^2 = -\mathrm{Id} and such that its +i-eigenspace L_\Phi \subset (TM \oplus T^*M) \otimes \mathbb{C} is isotropic with respect to the natural symmetric \langle X + \xi, Y + \eta \rangle = \xi(Y) + \eta(X) for sections X + \xi, Y + \eta. This induces a neutral metric of (n,n) on the bundle, where n = \dim M, and isotropy ensures L_\Phi has dimension n and \langle \cdot, \cdot \rangle vanishes on it. The standard almost complex structure J on TM embeds into this framework via the endomorphism \Phi_J = \begin{pmatrix} -J & 0 \\ 0 & J^t \end{pmatrix} acting on TM \oplus T^*M, where J^t is the (or musical ) induced by the pairing. Here, the +i-eigenspace L_J consists of sections X^{1,0} + \xi^{0,1}, where X^{1,0} are holomorphic vectors and \xi^{0,1} anti-holomorphic forms, recovering the classical case when the structure is of maximal type n. This correspondence highlights how generalized structures reduce to ordinary ones under appropriate projections. Integrability of \Phi is defined analogously to the classical Nijenhuis condition but using the Courant algebroid structure on TM \oplus T^*M. Specifically, \Phi is integrable if L_\Phi is closed under the Courant bracket [[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 L_\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 \omega, the structure arises from the pure spinor line bundle generated by e^{i\omega}, yielding a generalized almost complex structure of type 0 where L_\omega is spanned by e^{i\omega}-annihilated sections. In the complex case, including a closed B-field (a real closed 2-form), the pure spinor e^{B + i \omega_J} defines \Phi, blending Hermitian aspects while allowing type to vary from 0 to n. Poisson structures, given by a bivector field \pi satisfying [\pi, \pi]_S = 0 under the Schouten bracket, induce a generalized almost complex structure via the graph of \pi^\sharp: T^*M \to TM, with type jumping along the rank locus of \pi.

Symplectic and related applications

In , almost complex structures play a crucial role by providing a compatible almost complex structure J that tames a given form \omega, meaning g(X,Y) = \omega(X, JY) defines a Riemannian metric and J satisfies J^2 = -\mathrm{Id}. This compatibility enables the application of techniques analogous to the , which deforms forms within the same class while preserving the tamed structure, and extends the to local normal forms for symplectic manifolds equipped with such J. Calabi-Yau manifolds are compact Kähler manifolds equipped with an integrable almost 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 structure moduli with Kähler moduli. The of almost structures compatible with a fixed 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 J 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 J, thus parameterizing the symplectic deformations of almost structures. Almost complex structures find applications in G_2-manifolds, where the exceptional group G_2 \subset SO(7) induces an SU(3)-structure on 6-dimensional links like S^6, linking nearly Kähler geometry on S^6 to broader G_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 S^6, with no known integrable example as of 2025, despite various nearly complex constructions tied to G_2-structures and recent proposed constructions such as those by Etesi (2015, 2024).