Complex manifold

A complex manifold is a Hausdorff, second countable topological space locally homeomorphic to an open subset of $\mathbb{C}^n$Cn for some $n$n, equipped with a maximal atlas whose transition functions are biholomorphic maps.^[1] Equivalently, it is a smooth real manifold of dimension $2n$2n endowed with an integrable almost complex structure $J$J, meaning the Nijenhuis tensor vanishes, which by the Newlander–Nirenberg theorem admits a compatible holomorphic atlas.^[1] This complex structure enables the definition of holomorphic functions and maps on the manifold, which are those that are holomorphic in local coordinates.^[1] The complex dimension $n$n is locally constant, and the real dimension is thus $2n$2n, distinguishing complex manifolds from their real counterparts by the additional rigidity imposed by holomorphy.^[2] Complex manifolds form the foundation of complex geometry, bridging complex analysis, differential geometry, and algebraic geometry.^[3] They generalize Riemann surfaces (the case $n=1$n=1) and include important examples such as complex projective space $\mathbb{CP}^n$CPn, which is compact and serves as a model for projective varieties.^[1] Submanifolds inherit the complex structure, allowing for the study of zero sets of holomorphic functions, and compact complex submanifolds of $\mathbb{CP}^n$CPn are algebraic by Chow's theorem.^[1] A prominent subclass consists of Kähler manifolds, which admit a Hermitian metric whose associated Kähler form is closed, providing a symplectic structure compatible with the complex one; these are central to Hodge theory and have applications in theoretical physics, including mirror symmetry and string theory.^[3][2] Key tools for their study include Dolbeault cohomology, which measures obstructions to holomorphic extensions, and theorems like Kodaira's embedding, which embed compact Kähler manifolds into projective space using ample line bundles.^[2]

Definition and Construction

Formal definition

A complex manifold of complex dimension $n$n is defined as a second-countable, Hausdorff topological space $X$X that admits a maximal atlas of complex charts, where each chart $(U_i, \phi_i)$(Ui​,ϕi​) consists of an open set $U_i \subseteq X$Ui​⊆X and a homeomorphism $\phi_i: U_i \to V_i$ϕi​:Ui​→Vi​ onto an open subset $V_i \subseteq \mathbb{C}^n$Vi​⊆Cn, such that the transition maps $\phi_j \circ \phi_i^{-1}: \phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j)$ϕj​∘ϕi−1​:ϕi​(Ui​∩Uj​)→ϕj​(Ui​∩Uj​) are biholomorphic (i.e., holomorphic with holomorphic inverses) on their domains of definition.^[1] This structure ensures that $X$X is locally modeled on open sets in $\mathbb{C}^n$Cn, inheriting the holomorphic category for coordinate changes.^[2] The underlying topological manifold is required to be paracompact, a condition that follows from second-countability and Hausdorffness in the context of locally compact spaces like complex manifolds, enabling the existence of partitions of unity and facilitating global constructions in complex geometry.^[2] The complex dimension $n$n is well-defined and constant across the manifold, corresponding to a real dimension of $2n$, as each complex coordinate doubles into real and imaginary parts, making the tangent space at each point isomorphic to$,aseachcomplexcoordinatedoublesintorealandimaginaryparts,makingthetangentspaceateachpointisomorphicto\mathbb{C}^n$ as a complex vector space.^[1] The foundational concept of abstract Riemann surfaces was introduced by Hermann Weyl in 1913 in his book Die Idee der Riemannschen Fläche, providing a model that influenced the development of complex manifolds in higher dimensions.^[4]

Holomorphic atlases and transition functions

A holomorphic coordinate chart on a complex manifold $M$M of complex dimension $n$n consists of an open subset $U \subseteq M$U⊆M together with a biholomorphic map $\phi: U \to V$ϕ:U→V, where $V \subseteq \mathbb{C}^n$V⊆Cn is open.^[5] This means $\phi$ϕ is holomorphic and has a holomorphic inverse, allowing local identification of neighborhoods in $M$M with regions in complex Euclidean space while preserving the structure of holomorphic functions.^[2] Such charts provide a local complex coordinate system, where points in $U$U are assigned complex coordinates $z = (z_1, \dots, z_n) \in V$z=(z1​,…,zn​)∈V via $\phi$ϕ.^[1] A holomorphic atlas on $M$M is a collection of holomorphic coordinate charts $\{(U_\alpha, \phi_\alpha)\}_{\alpha \in A}${(Uα​,ϕα​)}α∈A​ that covers $M$M, meaning $\bigcup_{\alpha \in A} U_\alpha = M$⋃α∈A​Uα​=M.^[5] The charts must satisfy a compatibility condition: for any $\alpha, \beta \in A$α,β∈A with $U_\alpha \cap U_\beta \neq \emptyset$Uα​∩Uβ​=∅, the transition function $\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta)$ϕβ​∘ϕα−1​:ϕα​(Uα​∩Uβ​)→ϕβ​(Uα​∩Uβ​) is holomorphic.^[1] In local coordinates, if $z$z denotes the coordinates from the $\alpha$α-chart and $z'$z′ from the $\beta$β-chart, the transition is expressed as $$z' = f(z),$$z′=f(z), where $f: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta)$f:ϕα​(Uα​∩Uβ​)→ϕβ​(Uα​∩Uβ​) is a holomorphic map that is invertible with holomorphic inverse, ensuring biholomorphic compatibility.^[2] This condition guarantees that holomorphic functions and maps are well-defined independently of the choice of chart.^[5] Any holomorphic atlas can be enlarged by adding all compatible holomorphic charts, and by Zorn's lemma, it extends to a unique maximal holomorphic atlas.^[1] A maximal holomorphic atlas is one that cannot be properly extended while maintaining compatibility, and it uniquely determines the complex structure on $M$M.^[5] Two atlases define the same complex structure if their union is contained in a common maximal atlas.^[2]

Core Properties

Implications of complex structure

The complex structure on a manifold $M$M induces an almost complex structure $J: TM \to TM$J:TM→TM on its tangent bundle, characterized by the relation $J^2 = -\mathrm{Id}$J2=−Id, where $\mathrm{Id}$Id denotes the identity map. This $J$J acts as multiplication by $i$i on the $(1,0)$(1,0)-part and by $-i$−i on the $(0,1)$(0,1)-part of the complexified tangent bundle.^[6] The complexification $TM \otimes \mathbb{C}$TM⊗C decomposes into the direct sum $T^{1,0}M \oplus T^{0,1}M$T1,0M⊕T0,1M, where $T^{1,0}M = \{ v \in TM \otimes \mathbb{C} \mid Jv = iv \}$T1,0M={v∈TM⊗C∣Jv=iv} consists of holomorphic tangent vectors and $T^{0,1}M = \{ v \in TM \otimes \mathbb{C} \mid Jv = -iv \}$T0,1M={v∈TM⊗C∣Jv=−iv} consists of anti-holomorphic tangent vectors. This splitting reflects the integrability of the complex structure, enabling the definition of holomorphic vector fields as sections of $T^{1,0}M$T1,0M. The Newlander-Nirenberg theorem guarantees that, due to this integrability condition (the Nijenhuis tensor vanishing), local holomorphic coordinates exist around every point, making $M$M a genuine complex manifold rather than merely almost complex.^[7][6] The complex structure defines a sheaf of holomorphic functions $\mathcal{O}_M$OM​ on $M$M, consisting of functions that are holomorphic in local coordinates; global holomorphic functions and sections of holomorphic vector bundles arise as elements of these sheaves. For instance, holomorphic sections of line bundles provide meromorphic functions under suitable conditions, underpinning much of complex geometry.^[5] Finally, the holomorphic atlas underlying the complex structure determines a unique compatible smooth ($C^\infty$C∞) structure on $M$M, as holomorphic transition functions are infinitely differentiable, ensuring that any smooth atlas compatible with the complex one coincides with it. This uniqueness prohibits incompatible smooth or real-analytic structures that would contradict the holomorphy requirements.^[5]

Dimension and topological aspects

A complex manifold of complex dimension $n$n is a real differentiable manifold of dimension $2n$2n, as the local model is an open subset of $\mathbb{C}^n$Cn, which is diffeomorphic to $\mathbb{R}^{2n}$R2n.^[2][8] This doubling of dimension arises directly from the identification of the complex tangent space $T_p^{1,0}M$Tp1,0​M with the holomorphic vectors, while the full real tangent space $T_pM$Tp​M splits as $T_pM = T_p^{1,0}M \oplus \overline{T_p^{1,0}M}$Tp​M=Tp1,0​M⊕Tp1,0​M​, each of complex dimension $n$n.^[2] The complex structure imposes significant topological restrictions on the underlying real manifold. It must be even-dimensional, as the real dimension is always $2n$2n, and the tangent bundle is orientable due to the almost complex structure $J$J satisfying $J^2 = -\mathrm{id}$J2=−id, which induces a consistent orientation via the positive determinant of the transition maps in the holomorphic atlas.^[2][8][9] However, not every even-dimensional orientable real manifold admits a complex structure; for instance, the real projective plane $\mathbb{RP}^2$RP2 is non-orientable and thus cannot support one, highlighting that orientability is necessary but insufficient for integrability of the almost complex structure via the Newlander-Nirenberg theorem.^[8][9][2] Many complex manifolds exhibit strong connectivity properties, particularly simply connected ones, where the fundamental group $\pi_1(M) = 0$π1​(M)=0. For such manifolds, the first cohomology group $H^1(M, \mathbb{Z}) = 0$H1(M,Z)=0, ensuring that the natural map from global holomorphic functions to invertible ones is surjective, which facilitates unique extensions of holomorphic maps.^[2] A prominent class of complex manifolds are the Stein manifolds, which are holomorphically convex and have vanishing higher cohomology for coherent sheaves, enabling the solution of $\bar{\partial}$∂ˉ-equations globally.^[2][8] The topology of complex manifolds also influences their cohomology, though advanced tools like Hodge theory—decomposing de Rham cohomology into harmonic forms—are available only for the subclass of compact Kähler manifolds, where the Kähler form provides a compatible metric structure.^[2]

Canonical Examples

Riemann surfaces and curves

Riemann surfaces represent the one-dimensional case of complex manifolds, serving as the foundational examples in complex geometry. A Riemann surface is defined as a one-dimensional complex manifold, which is a connected Hausdorff space locally homeomorphic to the open complex plane $\mathbb{C}$C via holomorphic charts, with transition functions that are biholomorphic maps.^[10] This structure equips the surface with a complex analytic atlas, allowing the extension of holomorphic functions beyond the plane. Historically, the concept was introduced by Bernhard Riemann in his 1851 doctoral dissertation, where he developed the geometric foundations for the theory of functions of one complex variable, laying the groundwork for modern complex analysis.^[11] As topological objects, Riemann surfaces correspond to orientable two-dimensional manifolds equipped with a complex structure, which induces a compatible orientation and conformal metric. For a given complex structure, there is a unique maximal compatible atlas. This setting bridges complex analysis and differential topology, enabling the study of multi-valued functions like the square root or logarithm through branched coverings. The uniformization theorem provides a profound classification for simply connected Riemann surfaces, stating that every such surface is biholomorphic to one of three standard models: the open unit disk $\mathbb{D}$D, the complex plane $\mathbb{C}$C, or the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$C^=C∪{∞}.^[12] This result, first conjectured by Henri Poincaré and Felix Klein around 1882 and proved in 1907 by Poincaré and Paul Koebe, underscores the conformal rigidity of these spaces and implies that non-simply connected surfaces arise as quotients or coverings of these universal covers. For instance, hyperbolic surfaces (those uniformized by $\mathbb{D}$D) dominate for higher complexity, while $\mathbb{C}$C covers parabolic cases and $\hat{\mathbb{C}}$C^ the elliptic sphere.^[12] Canonical examples illustrate these principles. The Riemann sphere $\hat{\mathbb{C}}$C^ is constructed as the one-point compactification of $\mathbb{C}$C, endowed with charts $z$z on $\mathbb{C}$C and $w = 1/z$w=1/z near infinity, yielding holomorphic transition maps that define its complex manifold structure.^[13] This compact surface of genus zero serves as the prototype for elliptic uniformization. In contrast, the torus emerges as the quotient $\mathbb{C}/\Lambda$C/Λ, where $\Lambda$Λ is a lattice generated by two linearly independent complex numbers, inheriting a complex structure from the covering map $\mathbb{C} \to \mathbb{C}/\Lambda$C→C/Λ with deck transformations by lattice translations; this yields a compact Riemann surface of genus one, biholomorphic to an elliptic curve.^[14] For compact Riemann surfaces, classification proceeds via the genus $g$g, a topological invariant counting the number of "handles" or the Euler characteristic $\chi = 2 - 2g$χ=2−2g. Surfaces of genus zero are biholomorphic to the Riemann sphere, while genus one corresponds to tori, and higher genera $g \geq 2$g≥2 yield hyperbolic surfaces uniformized by $\mathbb{D}$D. The moduli space $\mathcal{M}_g$Mg​ parametrizes isomorphism classes of these surfaces, with dimension $3g - 3$3g−3 for $g \geq 2$g≥2, reflecting the degrees of freedom in deforming the complex structure while preserving biholomorphic equivalence; this space is itself a complex manifold, central to Teichmüller theory.^[15]

Complex projective spaces

Complex projective spaces $\mathbb{CP}^n$CPn serve as fundamental examples of compact complex manifolds, constructed as the quotient space $(\mathbb{C}^{n+1} \setminus \{0\}) / \mathbb{C}^*$(Cn+1∖{0})/C∗, where $\mathbb{C}^*$C∗ acts by scalar multiplication on nonzero vectors in $\mathbb{C}^{n+1}$Cn+1.^[1] Points in $\mathbb{CP}^n$CPn are represented by homogeneous coordinates $[z_0 : \cdots : z_n]$[z0​:⋯:zn​], with $[z_0, \dots, z_n] \sim [\lambda z_0, \dots, \lambda z_n]$[z0​,…,zn​]∼[λz0​,…,λzn​] for any $\lambda \in \mathbb{C}^*$λ∈C∗.^[1] This construction endows $\mathbb{CP}^n$CPn with a natural complex structure of dimension $n$n, making it a homogeneous complex manifold acted upon transitively by the unitary group $U(n+1)$U(n+1).^[2] To define the complex atlas, consider the standard open covers $U_i = \{ [z_0 : \cdots : z_n] \mid z_i \neq 0 \}$Ui​={[z0​:⋯:zn​]∣zi​=0} for $i = 0, \dots, n$i=0,…,n, each diffeomorphic to $\mathbb{C}^n$Cn via the affine charts $\phi_i: U_i \to \mathbb{C}^n$ϕi​:Ui​→Cn given by $\phi_i([z_0 : \cdots : z_n]) = (z_0/z_i, \dots, \hat{z_i}/z_i, \dots, z_n/z_i)$ϕi​([z0​:⋯:zn​])=(z0​/zi​,…,zi​^​/zi​,…,zn​/zi​).^[1] The transition maps between charts, say $\phi_i \circ \phi_j^{-1}$ϕi​∘ϕj−1​ for $i < j$i<j, are holomorphic functions of the form $(w_1, \dots, w_n) \mapsto (w_1/w_j, \dots, w_i/w_j, 1/w_j, w_{i+1}/w_j, \dots, w_n/w_j)$(w1​,…,wn​)↦(w1​/wj​,…,wi​/wj​,1/wj​,wi+1​/wj​,…,wn​/wj​), ensuring the atlas is compatible and defines a complex manifold structure.^[1] These charts cover $\mathbb{CP}^n$CPn entirely, as every point has at least one nonzero coordinate. Key topological properties include compactness, arising from the identification of $\mathbb{CP}^n$CPn with the quotient of the unit sphere $S^{2n+1}$S2n+1 by the $S^1$S1-action, and simple connectedness for $n \geq 1$n≥1.^[1][2] The homogeneity under $U(n+1)$U(n+1) reflects the transitive action via linear transformations $A \in U(n+1)$A∈U(n+1) mapping $\mapsto [Az]$↦[Az], preserving the complex structure.^[2] Additionally, $\mathbb{CP}^n$CPn admits the Fubini-Study metric, a natural Kähler metric invariant under this group action.^[2] In algebraic geometry, $\mathbb{CP}^n$CPn plays a central role as the classifying space for complex line bundles, where every holomorphic line bundle over $\mathbb{CP}^n$CPn is isomorphic to $O_{\mathbb{CP}^n}(d)$OCPn​(d) for some integer $d \in \mathbb{Z}$d∈Z, classified by the first Chern class.^[2] This structure underpins the embedding of projective varieties into projective spaces via ample line bundles.^[1]

Algebraic varieties and tori

Smooth complex algebraic varieties are defined as the common zero loci of finite collections of holomorphic polynomials in $\mathbb{C}^n$Cn or in projective space $\mathbb{CP}^n$CPn, equipped with the induced complex structure from the ambient space, where smoothness requires that the variety has no singular points, meaning the Jacobian matrix of the defining equations has maximal rank at every point.^[16] These varieties bridge algebraic geometry and complex analysis, as their points can be described both algebraically via polynomial equations and analytically via holomorphic functions.^[17] Complex tori, on the other hand, arise as quotients $\mathbb{C}^n / \Lambda$Cn/Λ, where $\Lambda$Λ is a discrete lattice subgroup of $\mathbb{C}^n$Cn of rank $2n$2n generated by $2n$2n linearly independent vectors over $\mathbb{R}$R, inheriting a natural complex structure from $\mathbb{C}^n$Cn via the quotient map.^[18] When such a torus admits a projective embedding into $\mathbb{CP}^m$CPm for some $m$m, it becomes an abelian variety, a special class of projective algebraic varieties that are also complex Lie groups.^[19] Representative examples include elliptic curves, which are smooth projective curves of genus one and serve as one-dimensional complex tori $\mathbb{C}/\Lambda$C/Λ, and K3 surfaces, which are compact smooth complex surfaces that qualify as Calabi-Yau manifolds.^[20][21] A key property is that smooth complex algebraic varieties are compact in the classical (Euclidean) topology if and only if they are projective, meaning they embed as closed subvarieties of some $\mathbb{CP}^m$CPm.^[22] In contrast, complex tori $\mathbb{C}^n / \Lambda$Cn/Λ with $n \geq 1$n≥1 are compact but not simply connected, as their fundamental group is isomorphic to $\mathbb{Z}^{2n}$Z2n, the abelianization of the lattice.^[18] Chow's theorem asserts that every compact complex submanifold of projective space $\mathbb{CP}^n$CPn is algebraic, i.e., it arises as the zero set of homogeneous polynomials.^[23]

Geometric Distinctions

Disc, plane, and polydisc comparisons

The unit disc $D = \{ z \in \mathbb{C} : |z| < 1 \}$D={z∈C:∣z∣<1} serves as a fundamental model domain in complex geometry, equipped with the Poincaré metric $\lambda_D(z) |dz| = \frac{2 |dz|}{1 - |z|^2}$λD​(z)∣dz∣=1−∣z∣22∣dz∣​, which induces a complete hyperbolic geometry of constant negative curvature $-1$−1.^[24] This metric arises from the invariant distance under the automorphism group of $D$D, highlighting its role in local uniformization and conformal mappings.^[25] In contrast, the complex plane $\mathbb{C}$C is a non-compact, simply connected domain without boundary, where holomorphic functions are entire and unbounded unless constant, as established by Liouville's theorem: any bounded entire function must be constant.^[26] This property underscores the global nature of $\mathbb{C}$C, distinguishing it from bounded domains like $D$D, and implies no biholomorphic equivalence between $D$D and $\mathbb{C}$C, since a biholomorphism would map bounded functions to bounded entire functions, yielding only constants by Liouville's theorem.^[26] The polydisc $D^n = D \times \cdots \times D$Dn=D×⋯×D ($n$n times) in $\mathbb{C}^n$Cn inherits a product structure, where holomorphic functions exhibit separate holomorphy in each coordinate, allowing independent analytic behavior along coordinate axes.^[27] This separability contrasts with more symmetric domains like the unit ball $B^n = \{ z \in \mathbb{C}^n : \|z\| < 1 \}$Bn={z∈Cn:∥z∥<1}, as there exists no biholomorphism between $D^n$Dn and $B^n$Bn for $n > 1$n>1; for instance, the automorphism group of $D^n$Dn fixing the origin is abelian, while that of $B^n$Bn is not, preventing such mappings.^[28] A key distinction in higher dimensions arises from Hartogs' theorem, which states that if $\Omega \subset \mathbb{C}^n$Ω⊂Cn ($n \geq 2$n≥2) is a bounded domain and $K \subset \Omega$K⊂Ω is compact such that $\Omega \setminus K$Ω∖K is connected, then every holomorphic function on $\Omega \setminus K$Ω∖K extends holomorphically to all of $\Omega$Ω.^[29] This extension phenomenon, exemplified in polydiscs, enables analytic continuation across compact singularities absent in one complex variable.^[29]

Stein vs. non-Stein manifolds

A Stein manifold is a complex manifold that admits a proper strictly plurisubharmonic exhaustion function, equivalently, it is holomorphically convex and strictly pseudoconvex in the sense that compact subsets have compact holomorphic convex hulls and the manifold separates points via holomorphic functions.^[30] This structure enables key approximation properties, such as the Oka-Weil theorem, which states that on any compact subset $K$K of a Stein manifold $X$X, any holomorphic function defined on a neighborhood of $K$K can be uniformly approximated on $K$K by holomorphic functions on the entire $X$X.^[31] Stein manifolds possess several distinguishing properties: they are non-compact, as the exhaustion function must be proper and bounded below without attaining a maximum; they admit exhausting plurisubharmonic functions that restrict subharmonically to holomorphic curves; and they satisfy Cartan's theorems A and B for coherent analytic sheaves.^[30] Cartan's theorem A asserts that the global sections of a coherent sheaf $\mathcal{F}$F on a Stein manifold $V$V generate $\mathcal{F}_x$Fx​ as an $\mathcal{O}_x$Ox​-module at every point $x \in V$x∈V, while theorem B guarantees that the higher cohomology groups $H^q(V, \mathcal{F}) = 0$Hq(V,F)=0 for $q \geq 1$q≥1.^[30] These theorems underpin the solvability of Cousin problems and the generation of global holomorphic sections. Representative examples of Stein manifolds include $\mathbb{C}^n$Cn for any $n$n, which serves as the prototypical model with its polydisc neighborhoods acting as local Stein spaces, and smooth affine algebraic varieties, which inherit Stein properties from their embedding in $\mathbb{C}^N$CN.^[30] In contrast, no compact complex manifold is Stein, as it cannot admit a proper plurisubharmonic exhaustion function without violating strict pseudoconvexity.^[30] Non-Stein manifolds, such as complex tori, fail these criteria: they lack holomorphic convexity and proper exhausting plurisubharmonic functions, and moreover, their holomorphic line bundles are generally non-trivial, preventing the full range of extension and approximation properties afforded by Stein spaces.^[30] This distinction highlights how Stein manifolds generalize the favorable analytic behavior of domains like polydiscs to global settings, while non-Stein examples like tori exhibit cohomological obstructions to such uniformity.^[30]

Related Structures

Almost complex structures

An almost complex structure on a smooth real manifold $M$M of even dimension $2n$2n is a smooth bundle endomorphism $J: TM \to TM$J:TM→TM satisfying $J^2 = -\mathrm{Id}_{TM}$J2=−IdTM​.^[32] This condition endows each tangent space $T_pM$Tp​M with the structure of a complex vector space of dimension $n$n, where multiplication by $i$i corresponds to application of $J$J.^[33] Such structures exist on even-dimensional smooth manifolds if the topological obstructions, which lie in certain cohomology groups, vanish; for example, they exist on $S^2$S2 and $S^6$S6 but not on $S^4$S4 or $S^8$S8, but they cannot exist on odd-dimensional manifolds because $J^2 = -\mathrm{Id}$J2=−Id requires the real dimension to be even. If $M$M is equipped with a Riemannian metric $g$g that is compatible with $J$J, meaning $g(JX, JY) = g(X, Y)$g(JX,JY)=g(X,Y) for all vector fields $X, Y$X,Y, then $(M, J, g)$(M,J,g) forms an almost Hermitian manifold.^[32] Compatibility ensures that $J$J is orthogonal with respect to $g$g, preserving the metric's inner product.^[33] The extent to which an almost complex structure $J$J fails to define a full complex structure is measured by the Nijenhuis tensor $N_J$NJ​, a tensor field of type $(2,1)$(2,1) defined by $$N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] + [X, Y]$$NJ​(X,Y)=[JX,JY]−J[JX,Y]−J[X,JY]+[X,Y] for vector fields $X, Y$X,Y on $M$M.^[34] This tensor is $C^\infty(M)$C∞(M)-bilinear and skew-symmetric in its arguments, capturing the non-commutativity of $J$J with the Lie bracket.^[35] A canonical example arises on $\mathbb{R}^{2n}$R2n, where $J$J is given by the standard complex multiplication: if coordinates are $(x_1, y_1, \dots, x_n, y_n)$(x1​,y1​,…,xn​,yn​), then $J(\partial/\partial x_k) = \partial/\partial y_k$J(∂/∂xk​)=∂/∂yk​ and $J(\partial/\partial y_k) = -\partial/\partial x_k$J(∂/∂yk​)=−∂/∂xk​.^[32] Among spheres, almost complex structures exist on $S^2$S2 (induced from the standard complex structure on $\mathbb{CP}^1$CP1) and on $S^6$S6 (constructed using the octonions, as shown by Kirchhoff in 1947). The concept of almost complex structures was introduced by Charles Ehresmann in 1947 to generalize complex manifolds without requiring integrability.^[36] Heinz Hopf contributed in 1947 by studying their existence on spheres, proving obstructions for dimensions like 4 and 8.

Integrability conditions

An almost complex structure $J$J on a smooth manifold $M$M is integrable if and only if its Nijenhuis tensor $N_J$NJ​ vanishes identically, i.e., $N_J = 0$NJ​=0.^[32] This condition arises from the application of the Frobenius theorem to the distribution defined by the $(1,0)$(1,0)-eigenspace of $J$J.^[37] The vanishing of $N_J$NJ​ ensures that the subbundle $T^{1,0}M = \{ X - i JX \mid X \in TM \otimes \mathbb{C} \}$T1,0M={X−iJX∣X∈TM⊗C} is involutive, meaning it is closed under the Lie bracket: $[\Gamma(T^{1,0}M), \Gamma(T^{1,0}M)] \subseteq \Gamma(T^{1,0}M)$[Γ(T1,0M),Γ(T1,0M)]⊆Γ(T1,0M).^[32] This involutivity allows $T^{1,0}M$T1,0M to define a complex tangent bundle, bridging the differential-geometric view of $J$J to the holomorphic structure on $M$M. When $N_J = 0$NJ​=0, the Newlander-Nirenberg theorem guarantees the local existence of holomorphic coordinates. Specifically, around every point in $M$M, there is a neighborhood with coordinates $z^1, \dots, z^n$z1,…,zn such that $J$J acts as multiplication by $i$i on the holomorphic tangent vectors $\partial/\partial z^j$∂/∂zj.^[7] The local holomorphic charts glue globally to form a holomorphic atlas because the transition functions preserve the integrable structure $J$J, yielding a full complex manifold structure on $M$M. An illustrative example is the Hopf surface, a compact complex surface diffeomorphic to $S^1 \times S^3$S1×S3, which admits an integrable almost complex structure despite its non-trivial topology that precludes it from being projective or Kähler.^[38]

Special Classes

Kähler manifolds

A Kähler manifold is a complex manifold $(M, J)$(M,J) equipped with a Hermitian metric $h$h such that the associated Kähler form $\omega \in \Omega^{1,1}(M)$ω∈Ω1,1(M) is closed, i.e., $d\omega = 0$dω=0.^[39] The Riemannian metric $g$g induced by $h$h satisfies $g(u,v) = h(u, Jv)$g(u,v)=h(u,Jv), and the fundamental (or Kähler) form is defined by $\omega(X,Y) = g(JX, Y)$ω(X,Y)=g(JX,Y) for vector fields $X, Y$X,Y.^[39] This structure ensures compatibility between the complex structure $J$J, the Riemannian metric $g$g, and the symplectic form $\omega$ω.^[39] The closedness of $\omega$ω implies that $(M, \omega)$(M,ω) is a symplectic manifold, with the symplectic structure being of type $(1,1)$(1,1) with respect to $J$J, meaning $\omega(JX, JY) = \omega(X,Y)$ω(JX,JY)=ω(X,Y) and $\omega(X, JX) > 0$ω(X,JX)>0 for $X \neq 0$X=0.^[39] On a Kähler manifold, the Levi-Civita connection $\nabla$∇ of $g$g is torsion-free and parallel to $J$J, satisfying $\nabla J = 0$∇J=0, which makes $\nabla$∇ the unique connection compatible with both $g$g and $J$J.^[39] Additionally, the Ricci curvature form $\rho$ρ, defined as $\rho = -i \partial \bar{\partial} \log \det(g_{j\bar{k}})$ρ=−i∂∂ˉlogdet(gjkˉ​), is a real closed $(1,1)$(1,1)-form representing the first Chern class $c_1(M, K_M) \in H^2(M, \mathbb{R})$c1​(M,KM​)∈H2(M,R).^[39] Prominent examples include the complex projective space $\mathbb{CP}^n$CPn endowed with the Fubini-Study metric, whose Kähler form is the generator of $H^2(\mathbb{CP}^n, \mathbb{R})$H2(CPn,R).^[40] Another example is a complex torus $\mathbb{C}^n / \Lambda$Cn/Λ, where $\Lambda$Λ is a lattice, equipped with the flat metric pulled back from the Euclidean metric on $\mathbb{C}^n$Cn.^[39] On a compact Kähler manifold, the Hodge theorem yields a decomposition of the cohomology groups: $$H^k(M, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(M),$$Hk(M,C)=p+q=k⨁​Hp,q(M), where $H^{p,q}(M) = \ker \bar{\partial} \cap \ker \partial \cap \Omega^{p,q}(M)$Hp,q(M)=ker∂ˉ∩ker∂∩Ωp,q(M), and the decomposition is orthogonal with respect to the $L^2$L2-inner product induced by the Kähler metric.^[41] This Hodge decomposition arises from the Kähler identities, which relate the Laplacians $\Delta_{\partial}$Δ∂​ and $\Delta_{\bar{\partial}}$Δ∂ˉ​ to the full de Rham Laplacian $\Delta_d$Δd​.^[41] The notion of a Kähler manifold was introduced by Erich Kähler in his 1933 paper, where he studied Hermitian metrics with closed fundamental forms on complex manifolds.^[42]

Calabi-Yau manifolds

Calabi-Yau manifolds are compact Kähler manifolds of complex dimension $n$n that admit a Ricci-flat Kähler metric, meaning the Ricci tensor satisfies $\mathrm{Ric} = 0$Ric=0, and possess a trivial canonical bundle, equivalently vanishing first Chern class $c_1 = 0$c1​=0. This structure ensures the existence of a unique Ricci-flat metric in any given Kähler class, as established by the Calabi-Yau theorem, which resolves Calabi's conjecture by proving that on a compact Kähler manifold with $c_1 = 0$c1​=0, there exists a Kähler metric with prescribed Ricci form equal to zero. These manifolds inherit the properties of Kähler manifolds, such as a closed Kähler form $\omega$ω, but specialize to Ricci-flat cases with profound implications in geometry and physics.^[43] The Calabi-Yau theorem, proved by Shing-Tung Yau in 1978, guarantees the existence and uniqueness of this Ricci-flat metric, providing a canonical geometric structure on such manifolds. Representative examples include complex tori, which are flat Calabi-Yau manifolds in any dimension; K3 surfaces, which form the complete class of simply connected Calabi-Yau manifolds in complex dimension 2; and, in complex dimension 3, the quintic hypersurface in $\mathbb{CP}^4$CP4 defined by the zero locus of a homogeneous degree-5 polynomial, a prototypical Calabi-Yau threefold with Euler characteristic -200. These examples illustrate the diversity of Calabi-Yau manifolds, from abelian varieties to hypersurfaces in projective space. A key property of Calabi-Yau manifolds is that their Ricci-flat Kähler metric induces a holonomy group contained in $\mathrm{SU}(n)$SU(n), reflecting the special unitary structure preserved by the trivial canonical bundle and ensuring supersymmetry in associated physical models.^[43] Another significant feature is the mirror symmetry conjecture, formulated in the 1990s by Greene, Plesser, and collaborators, which posits that Calabi-Yau manifolds appear in dual pairs $(X, \tilde{X})$(X,X~) where the Hodge numbers are interchanged, $h^{p,q}(X) = h^{n-p,q}(\tilde{X})$hp,q(X)=hn−p,q(X~), leading to isomorphisms between physical theories on each.90280-0) Although applicable in general dimension $n$n, Calabi-Yau manifolds are particularly studied for $n=3$n=3 due to their role in theoretical physics. In string theory, Calabi-Yau manifolds serve as internal spaces for compactifications that preserve $\mathcal{N}=1$N=1 supersymmetry in four dimensions, enabling realistic model building by determining the low-energy effective field theory through topological invariants like Hodge numbers.^[44] This application underscores their importance in unifying gravity with particle physics, where the $\mathrm{SU}(3)$SU(3) holonomy for threefolds ensures the correct number of supersymmetric generations.