The Almost Complex Operator

Introduction

An almost complex operator on a smooth manifold $M$ is a field $J$ of endomorphisms of the tangent bundle with $J^2 = -\mathrm{id}$ — the field form of the complex structure of a complex vector space, and an operator on the tangent spaces in its own right. It equips every tangent space with the structure of a complex vector space, and it splits the complexified tangent bundle into the $\pm i$ eigenbundles; the two projections onto those eigenbundles are the operators $$ \pi^{1,0} = \tfrac12(\mathrm{id} - iJ), \qquad \pi^{0,1} = \tfrac12(\mathrm{id} + iJ), $$ which are idempotent and conjugate. The operator $J$ alone is not enough to make $M$ a complex manifold: the projections need not be compatible with the Lie bracket, and the obstruction is a tensor, the Nijenhuis operator $N_J$, built from $J$ and the bracket. The integrability theorem of Newlander–Nijenhuis states that $N_J$ vanishes exactly when the almost complex structure comes from a holomorphic atlas. When a Hermitian metric $g$ is chosen with $g(JX,JY) = g(X,Y)$, the operator $J$ becomes an isometry of each tangent space, and the pair $(J,g)$ is an almost Hermitian structure whose fundamental form is $\Omega(X,Y) = g(JX,Y)$.

The article has four sections: the almost complex operator and the eigenprojections; the linear model, in which $J$ is always integrable; the Nijenhuis operator and the integrability criterion; and the compatibility with the chosen Hermitian metric. The almost complex structure, its type decomposition, the Nijenhuis tensor with the corpus normalisation, the integrability theorem and the examples are Hermitian Geometry and Almost Complex Structures; the article here reads the same structure as an operator, names its eigenprojections and gives the operator form of the integrability criterion. The exterior algebra and the Lie bracket are Differential Forms and The Lie Derivative; the Hermitian metric, the fundamental form and the Kähler condition are Kähler Geometry and Kähler Manifolds and the Hermitian Form. The complex structure of a linear space and its linearity are The Involution on a Complex Vector Space, later in this category.

Throughout, $M$ is a smooth manifold of even dimension $2n$, $J \in \Gamma(\operatorname{End}(TM))$ is an almost complex structure, $T_{\mathbb C}M = T^{1,0}M\oplus T^{0,1}M$ is the decomposition into the $+i$ and $-i$ eigenbundles of $J$, $\pi^{1,0}$ and $\pi^{0,1}$ are the eigenprojections, $[\,\cdot\,,\,\cdot\,]$ is the Lie bracket of vector fields, and $N_J$ is the Nijenhuis tensor.

The Almost Complex Operator and the Eigenprojections

Definition. An almost complex structure on $M$ is a smooth field $J$ of endomorphisms of $TM$ with $$ J^2 = -\mathrm{id} ; $$ such a field is also called an almost complex operator. A pair $(M,J)$ is an almost complex manifold. A map $F : (M,J)\to(N,J')$ is almost complex when $dF\circ J = J'\circ dF$.

Proposition (the eigenprojections). On the complexified tangent bundle the operator $J$ has the eigenvalues $\pm i$, and the projections onto the eigenbundles are $$ \pi^{1,0} = \tfrac12(\mathrm{id} - iJ), \qquad \pi^{0,1} = \tfrac12(\mathrm{id} + iJ), $$ which satisfy $$ (\pi^{1,0})^2 = \pi^{1,0}, \qquad (\pi^{0,1})^2 = \pi^{0,1}, \qquad \pi^{1,0}\pi^{0,1} = \pi^{0,1}\pi^{1,0} = 0, \qquad \pi^{1,0} + \pi^{0,1} = \mathrm{id} . $$ They are conjugate, $\overline{\pi^{1,0}} = \pi^{0,1}$, and the complexified tangent space is their direct sum.

Proof. From $J^2 = -\mathrm{id}$ the polynomial $x^2+1 = (x-i)(x+i)$ annihilates $J$, and the two factors are coprime over $\mathbb{C}$, so $T_{\mathbb C}M$ is the direct sum of the kernels. The identities are the same computation as for a linear operator with minimal polynomial $x^2+1$: $\pi^{1,0}\pi^{0,1} = \tfrac14(\mathrm{id}-iJ)(\mathrm{id}+iJ) = \tfrac14(\mathrm{id}+J^2) = 0$, and $\pi^{1,0}+\pi^{0,1} = \mathrm{id}$; conjugation reverses the sign of $J$ in the definition.

Remark (the operator and the reduction). An almost complex operator is the same datum as a reduction of the structure group of $TM$ from $GL(2n,\mathbb R)$ to $GL(n,\mathbb C)$, and the eigenprojections are the operators by which that reduction is expressed: a complex-linear frame is one whose complexification lies in $T^{1,0}M$. The choice of an almost complex structure is a choice of a field of operators, and it is one of the two structures the geometry of the category reads on the tangent spaces.

The Linear Model

Definition. On a complex vector space $V$ the structure of multiplication by $i$ is a real endomorphism $J$ with $J^2 = -\mathrm{id}$; the same construction with $V$ the tangent space at a point of $M$ gives the linear model of an almost complex structure, and on $M = \mathbb C^n$ with its standard coordinates the field $J$ is the constant operator of multiplication by $i$ on each tangent space.

Proposition (the linear model is integrable). On a complex vector space $V$, and on $\mathbb C^n$ with the standard structure, the almost complex operator $J$ satisfies $N_J = 0$ for the Nijenhuis operator of the next section, and $T^{1,0}V$ is the complex vector space $V$ itself, on which multiplication by $i$ acts as $J$ and the projections $\pi^{1,0},\pi^{0,1}$ are the two standard projections of the complexification $V\otimes\mathbb C = V \oplus \bar V$.

Proof. The Lie bracket on a vector space, read in constant coordinates, is zero; hence every term of $N_J$ vanishes. The identification $V\otimes_{\mathbb R}\mathbb C \cong V\oplus\bar V$ with the two projections is the standard form of the complexification, and multiplication by $i$ acts on $V$ by $J$ and on $\bar V$ by $-J$.

Remark. The linear model is the reason an almost complex structure is a "complex structure" at each point: the tangent space at every point is a complex vector space. Integrability is the separate question of whether these pointwise complex structures vary so as to come from a single holomorphic atlas; the local model $\mathbb C^n$ answers it affirmatively, and the obstruction for a general $J$ is the operator of the next section.

The Nijenhuis Operator and the Integrability Criterion

Definition. The Nijenhuis operator of an almost complex structure $J$ is the tensor $$ N_J(X, Y) = [JX, JY] - J[X, JY] - J[JX, Y] - [X, Y] \qquad (X, Y \in \mathfrak{X}(M)), $$ with no scalar factor.

Proposition (tensoriality and elementary symmetries). $N_J$ is a tensor of type $(1,2)$, that is it is $C^\infty(M)$-bilinear, it is antisymmetric, $N_J(Y,X) = -N_J(X,Y)$, and it satisfies $$ N_J(X, JX) = 0, \qquad N_J(JX, JY) = N_J(X, Y) . $$ When $J$ is integrable the tensor vanishes.

Proof. The $C^\infty$-bilinearity is the verification that all the second derivatives cancel, exactly as for the analogous computation in Hermitian Geometry and Almost Complex Structures, and it uses $J^2 = -\mathrm{id}$ and the Jacobi identity; antisymmetry is read from the definition, and $N_J(X,JX) = [JX,J(JX)] - J[X,J(JX)] - J[JX,JX] - [X,JX] = [JX,-X]-J[X,-X]-0-[X,JX] = 0$, since the first two terms cancel the last. The transformation $X\to JX$, $Y\to JY$ leaves $N_J$ unchanged by substitution and $J^2=-\mathrm{id}$.

Theorem (the integrability criterion). For an almost complex structure $J$ on $M$ the following are equivalent: (i) $N_J = 0$; (ii) the eigenbundle $T^{1,0}M$ is closed under the Lie bracket; (iii) $M$ carries a holomorphic atlas whose induced almost complex structure is $J$. The equivalence of (i) and (iii) is the theorem of Newlander–Nijenhuis, quoted; an almost complex structure with $N_J = 0$ is integrable and $(M,J)$ is a complex manifold.

Proof. The equivalence of (i) and (ii) is the computation that the bracket of two sections of $T^{1,0}M$ has a $(0,1)$-component equal to the corresponding value of $N_J$, so that $T^{1,0}M$ is bracket-closed exactly when $N_J$ vanishes; this is the operator form of the Newlander–Nijenhuis vanishing. The theorem that bracket-closedness is equivalent to the existence of a holomorphic atlas is quoted from Hermitian Geometry and Almost Complex Structures, where the Nijenhuis tensor is introduced and the filtration of the proof is deferred.

Example (an integrable and a non-integrable structure on $\mathbb R^4$). On $\mathbb R^4$ with coordinates $(x,y,z,t)$ the standard complex structure $J\partial_x = \partial_y$, $J\partial_y = -\partial_x$, $J\partial_z = \partial_t$, $J\partial_t = -\partial_z$ has $J^2 = -\mathrm{id}$ and $N_J = 0$, being the linear model. The structure $$ J\partial_x = \partial_y + z\,\partial_z, \quad J\partial_y = -\partial_x - z\,\partial_t, \quad J\partial_z = \partial_t, \quad J\partial_t = -\partial_z $$ also has $J^2 = -\mathrm{id}$, and a direct computation gives $$ N_J(\partial_x, \partial_z) = \partial_t \neq 0 , $$ so this almost complex structure is not integrable and $\mathbb R^4$ with it carries no holomorphic atlas compatible with $J$.

The Compatibility with the Chosen Hermitian Metric

Definition. A Hermitian metric $g$ on an almost complex manifold $(M,J)$ is a Riemannian metric with $$ g(JX, JY) = g(X, Y) \qquad (X, Y \in \mathfrak{X}(M)); $$ the pair $(J,g)$ is an almost Hermitian structure, and its fundamental form is $\Omega(X,Y) = g(JX,Y)$. The structure is almost Kähler when $\Omega$ is closed, Kähler when moreover $\nabla J = 0$ for the Levi-Civita connection.

Proposition (the operator is an isometry of each tangent space). When $g$ is Hermitian for $J$, the operator $J$ is orthogonal at every point, $J^{\mathsf T}J = \mathrm{id}$; the fundamental form is a real $(1,1)$-form, alternating, $\Omega(Y,X) = -\Omega(X,Y)$; and the complexified tangent bundle carries the positive-definite Hermitian form $g_{\mathbb C}(v,w) = g(v,\bar w)$ whose holomorphic part is the restriction to $T^{1,0}M$.

Proof. $g(JX,JY)=g(X,Y)$ is the orthogonality, and the transpose form $g(JX,Y)=g(JX,J(JY))... $ shows $\Omega$ is skew: $\Omega(Y,X)=g(JY,X)=g(JY,J(JX))=-g(Y,JX)=-\Omega(X,Y)$ using $g(J\cdot,J\cdot)=g$ and $g(JY,J(JX))=g(JY,-X)=-g(JY,X)$. The form $g_{\mathbb C}$ is Hermitian because $J$ is orthogonal, and its positivity is that of $g$; restricting to $T^{1,0}M$ gives the Hermitian form of the holomorphic tangent space. This is Hermitian Geometry and Almost Complex Structures and Kähler Geometry.

Remark (what is chosen). The almost complex operator $J$ is a choice and the Hermitian metric $g$ is a second, independent choice; the type, the eigenprojections and the Nijenhuis tensor depend on $J$ alone, while the isometry of $J$, the fundamental form and the comparison of $J$ with the Levi-Civita connection depend on the pair $(J,g)$. The existence of a $g$-compatible $J$ on a symplectic manifold, and the contractibility of the compatible almost complex structures, are Symplectic Geometry, earlier in this Part; the equality $N_J = 0$ is untouched by the metric, which is the sense in which integrability is a property of the operator $J$ alone.

Summary

An almost complex operator is a field $J$ of endomorphisms of $TM$ with $J^2 = -\mathrm{id}$; it makes each tangent space a complex vector space, splits the complexified tangent bundle into the eigenbundles $T^{1,0}M$, $T^{0,1}M$ of $\pm i$, and gives the idempotent conjugate eigenprojections $\pi^{1,0} = \tfrac12(\mathrm{id}-iJ)$ and $\pi^{0,1} = \tfrac12(\mathrm{id}+iJ)$. On a complex vector space, and on $\mathbb C^n$ with the standard structure, $N_J = 0$ and the structure is the linear model; in general the obstruction to integrability is the Nijenhuis operator $N_J(X,Y) = [JX,JY] - J[X,JY] - J[JX,Y] - [X,Y]$, a $(1,2)$-tensor, antisymmetric, vanishing on $(X,JX)$ and invariant under $(X,Y)\mapsto(JX,JY)$, and $N_J = 0$ is equivalent to the bracket-closedness of $T^{1,0}M$ and, by Newlander–Nijenhuis, to the existence of a holomorphic atlas. The non-integrable example $J\partial_x = \partial_y + z\partial_z$, $J\partial_y = -\partial_x - z\partial_t$, $J\partial_z = \partial_t$, $J\partial_t = -\partial_z$ on $\mathbb R^4$ has $J^2 = -\mathrm{id}$ and $N_J(\partial_x,\partial_z) = \partial_t\neq0$. A chosen Hermitian metric $g$ with $g(JX,JY)=g(X,Y)$ makes $J$ an isometry and defines the fundamental form $\Omega(X,Y)=g(JX,Y)$; the pair $(J,g)$ is almost Hermitian, almost Kähler when $d\Omega = 0$ and Kähler when $\nabla J = 0$. The type decomposition, the Nijenhuis tensor and the integrability theory are Hermitian Geometry and Almost Complex Structures; the metric and the Kähler condition are Kähler Geometry and Kähler Manifolds and the Hermitian Form.

Summary of Notation

Symbol	Meaning
$J$, $J^2 = -\mathrm{id}$	the almost complex operator
$T^{1,0}M$, $T^{0,1}M$	the $\pm i$ eigenbundles of $J$
$\pi^{1,0}$, $\pi^{0,1}$	eigenprojections, $\tfrac12(\mathrm{id}\mp iJ)$, idempotent and conjugate
$N_J(X,Y)$	the Nijenhuis operator, $[JX,JY]-J[X,JY]-J[JX,Y]-[X,Y]$
$g(JX,JY)=g(X,Y)$	the Hermitian compatibility of the metric
$\Omega(X,Y)=g(JX,Y)$	the fundamental form
$\nabla J = 0$	the Kähler condition