The Curvature Operator of a Complex Manifold

Introduction

On a complex manifold the curvature operator has a second structure beyond the self-adjointness it has on any Riemannian manifold: the complexified second exterior power splits by type, $$ \Lambda^2T_pM\otimes\mathbb{C} = \Lambda^{2,0}\oplus\Lambda^{1,1}\oplus\Lambda^{0,2}, $$ and both the complex structure $J$ and the curvature of a Hermitian or Kähler metric act on the three pieces. The curvature of the Chern connection — the unique connection of a Hermitian metric whose $(0,1)$-part is the Cauchy–Riemann operator — is a form of type $(1,1)$ with values in $\mathfrak{u}(n)$, so its operator is complex-linear and preserves the type; when the metric is Kähler the Chern connection is the Levi-Civita connection, the type-preservation passes to the Riemannian curvature operator, and the curvature tensor acquires the extra symmetries of a Kähler metric. The readings of the operator that this article develops are the Chern curvature, the curvature of the canonical connection of a Hermitian metric with its first Chern form, and the Kähler curvature, the curvature of a Kähler metric with its holomorphic sectional curvature, its Ricci form and its Kähler–Einstein condition.

The article has four sections: the curvature operator and the action of the complex structure on bivectors; the Chern curvature and the first Chern form; the Kähler curvature and its symmetries; and the decomposition of the operator by the unitary group. The Riemannian curvature operator, the bivectors, its self-adjointness and its scalar-triple decomposition are The Curvature Operator, earlier in this Part, which defers the complex and Hermitian case to this article; the Hermitian metric and the Chern connection are Hermitian Geometry and Almost Complex Structures and Hermitian Metrics and the Levi-Civita Connection; the Kähler metric and the fundamental form are Kähler Geometry, and the closedness of the Kähler form and the Kähler identities are Kähler Manifolds and the Hermitian Form, later in this category; the first Chern class and the curvature of a connection are Characteristic Classes; the Hodge star splitting of the curvature operator in dimension four is The Involution on the Curvature Operator. None of that is re-derived. The particular case of the curvature operator of a Kähler surface and its relation to the invariant decomposition is named and not used.

Throughout, $(M,J,g)$ is a complex manifold of complex dimension $n$ with a Hermitian metric $g$, $T_pM$ is a tangent space with the almost complex structure $J$, $\Lambda^2T_pM\otimes\mathbb C$ is the complexified second exterior power, $\mathcal{R}$ is the Riemannian curvature operator of the metric, $\nabla$ is the Chern connection, $\Theta = \nabla^2$ is its curvature, and $\rho$ is the Ricci form.

The Curvature Operator and the Complex Structure

Definition. On the complexified bivectors the complex structure acts by applying $J$ to each leg, $J(X\wedge Y) = JX\wedge Y + X\wedge JY$, so that the type decomposition $$ \Lambda^2T_pM\otimes\mathbb C = \Lambda^{2,0}\oplus\Lambda^{1,1}\oplus\Lambda^{0,2} $$ is the eigenspace decomposition of the induced operator $\mathcal J$ on $\Lambda^2T_pM\otimes\mathbb C$, with eigenvalues $+2i$ on $\Lambda^{2,0}$, $0$ on $\Lambda^{1,1}$ and $-2i$ on $\Lambda^{0,2}$. A complex-linear operator on the second exterior power is one commuting with $\mathcal J$ and hence preserving the type.

Proposition. Write $g$ for the Hermitian metric and let $\mathcal{R}$ be its Riemannian curvature operator. If $g$ is Kähler then $\mathcal{R}$ commutes with $\mathcal J$, $$ [\mathcal R, \mathcal J] = 0 , $$ so $\mathcal{R}$ preserves the type of bivectors and decomposes into a Hermitian part on $\Lambda^{1,1}$ and a pair of conjugate parts on $\Lambda^{2,0}$ and $\Lambda^{0,2}$; if $g$ is merely Hermitian the Riemannian curvature operator need not commute with $\mathcal J$, and it is the curvature of the Chern connection that preserves the type.

Proof. For a Kähler metric the Levi-Civita connection preserves the complex structure, $\nabla J = 0$; the curvature is the commutator of covariant derivatives, so it commutes with the parallel tensor $J$, and on bivectors this is $[\mathcal R,\mathcal J]=0$. The type of a bivector is its $\mathcal J$-eigenvalue, so commuting with $\mathcal J$ is preserving the type. For a Hermitian metric with $\nabla J\neq0$ the parallel-transport argument fails and the curvature can exchange $\Lambda^{2,0}$ with $\Lambda^{0,2}$; that the Chern curvature is type-preserving is its definition as a $(1,1)$-form with values in $\mathfrak{u}(n)$, below.

Remark (what the operator records). The decomposition of the curvature operator by type is the decomposition of the Riemannian curvature into a Hermitian piece and a conjugate pair of pieces, and for a Kähler metric it is a decomposition into three genuinely independent operators; for a Hermitian metric the Riemannian operator has an additional mixed part, measured by $\nabla J$, and the Hermitian structures of this category select the type-preserving part.

The Chern Curvature

Definition. On a Hermitian manifold the Chern connection is the unique connection $\nabla$ on $TM$ that is metric for $g$, has $\nabla^{0,1} = \bar\partial$ as its $(0,1)$-part, and satisfies $\nabla J = 0$ in its $(1,1)$-part; its curvature is $\Theta = \nabla^2 \in \Omega^{1,1}(\operatorname{End}TM)$, a $(1,1)$-form with values in the skew-Hermitian endomorphisms $\mathfrak{u}(n)$ of each tangent space.

Proposition (the Chern curvature operator preserves type). The curvature of the Chern connection is of type $(1,1)$ with values in $\mathfrak u(n)$; its operator $R^{\mathbb C} = \Theta$ on $\Lambda^2T_pM\otimes\mathbb C$ acts as a complex-linear map preserving the bidegree, $R^{\mathbb C}(\Lambda^{p,q})\subseteq\Lambda^{p,q}$, and its action on $\Lambda^{1,1}$ is the Hermitian part of the curvature operator, $$ \langle R^{\mathbb C}(X\wedge\bar Y), Z\wedge\bar W\rangle = \Theta(X,\bar Y; Z,\bar W) ; $$ in local holomorphic coordinates the components $R_{i\bar j}{}^{k}{}_{l}$ with $$ \Theta_{\bar j}{}^{k}{}_{l} = \sum_i R_{i\bar j}{}^{k}{}_{l}\,dz^i $$ are the Chern curvature coefficients.

Proof. A $(1,1)$-form with values in $\mathfrak u(n)$ is complex-linear and lowers the bidegree by at most one in each factor, hence preserves the bidegree of a bivector; the identification with the Hermitian part of the curvature operator is the contraction against the metric on the two legs that the $(1,1)$-form does not carry. The existence and uniqueness of the Chern connection above are Hermitian Geometry and Almost Complex Structures.

Proposition (the first Chern form). The trace form $$ c_1(g) = \frac{i}{2\pi}\operatorname{tr}\Theta $$ is a closed real $(1,1)$-form whose de Rham class is the first Chern class $c_1(M)$, independent of the Hermitian metric; in a holomorphic frame of a line bundle the form is $\frac{i}{2\pi}\partial\bar\partial\log\|s\|^2$, and for the tangent bundle it is the Ricci form of the chosen metric up to a factor.

Proof. The trace of a $\mathfrak u(n)$-valued $(1,1)$-form is a real $(1,1)$-form, closed by the Bianchi identity of the connection, and its class is the Chern class by the Chern–Weil construction; the change of a Hermitian metric on a line bundle multiplies a local frame by a positive function, and the difference of two first Chern forms is $\frac{i}{2\pi}\partial\bar\partial\log(f)$, which is exact. The Chern–Weil theory is Characteristic Classes.

Remark (the Ricci form). For the tangent bundle with the metric $g$ the trace of $\Theta$ in a unitary frame is the Ricci form $$ \rho = i\sum_{j,k}R_{i\bar j k\bar k}\,dz^i\wedge d\bar z^j, $$ which satisfies $\rho = i\,\partial\bar\partial\log\det(g_{j\bar k})$ and $[\rho] = 2\pi c_1(M)$; this is the precise sense in which the Chern curvature of the tangent bundle is the Ricci curvature read as a $(1,1)$-form.

The Kähler Curvature

Definition. Let $g$ be Kähler. The Kähler curvature is the Riemannian curvature tensor read in holomorphic coordinates, $$ R_{i\bar j k\bar l} = \bigl\langle R(\partial_i,\partial_{\bar j})\partial_k, \partial_{\bar l}\bigr\rangle , $$ and it satisfies the three symmetry relations $$ R_{i\bar j k\bar l} = \overline{R_{j\bar i l\bar k}}, \qquad R_{i\bar j k\bar l} = R_{k\bar l i\bar j}, \qquad R_{i\bar j k\bar l} = R_{i\bar l k\bar j}, $$ the first the reality of the curvature, the second the pair symmetry, and the third equivalent to the Bianchi identity in the Kähler case.

Proposition (the operator commutes with the complex structure). For a Kähler metric the curvature operator commutes with $\mathcal J$, its type decomposition is $(2,0)+(1,1)+(0,2)$, and the curvature tensor is determined by its $(1,1)$-components $R_{i\bar j k\bar l}$ alone; the components $R_{i\bar j k\bar l}$ and their conjugates assemble the operator on $\Lambda^{2,0}\oplus\Lambda^{0,2}$, and the operator on $\Lambda^{1,1}$ is the Hermitian part.

Proof. The commutation is the previous section. The determination of the tensor by the $(1,1)$-components is the content of the symmetry relation $R_{i\bar j k\bar l}=R_{i\bar l k\bar j}$ together with the Bianchi identity: the $(2,0)$ components $R_{ijkl}$ are the $\partial\bar\partial$-derivatives of $R_{i\bar j k\bar l}$, and for a Kähler metric the tensor is recovered from the mixed components, which is the Kähler case of the Bianchi computation. The Chern connection coincides with the Levi-Civita connection for a Kähler metric, so the Chern curvature and the Kähler curvature are the same object.

Definition. For a nonzero holomorphic vector $X$ the holomorphic sectional curvature is $$ H(X) = \frac{R(X, \bar X, X, \bar X)}{g(X,\bar X)^2} , $$ and the metric has constant holomorphic sectional curvature $c$ when $H$ is the constant $c$ for every $X$; by the Kähler analogue of Schur's theorem, the constancy of $H$ forces the curvature tensor to take the form $$ R_{i\bar j k\bar l} = \frac{c}{4}\bigl(g_{i\bar j}g_{k\bar l} + g_{i\bar l}g_{k\bar j}\bigr) . $$

Proof. The holomorphic sectional curvature is the sectional curvature of the real two-plane spanned by $X$ and $JX$; if it is constant the second Bianchi identity propagates the constancy, as in Schur's theorem of Curvature and Geodesics, and the displayed form is the unique tensor with the symmetries of the Kähler curvature and the constant value $c$. The models are $\mathbb C^n$ with $c=0$, $\mathbb{CP}^n$ with $c>0$ and the complex hyperbolic space with $c<0$, and their classification as the space forms is Kähler Geometry.

Remark (the Ricci form and Kähler–Einstein). For a Kähler metric the Ricci form is $\rho = i\sum_{j,k}R_{i\bar j k\bar k}dz^i\wedge d\bar z^j$, and it satisfies $\rho = i\,\partial\bar\partial\log\det(g)$ and $[\rho] = 2\pi c_1(M)$; the metric is Kähler–Einstein when $\rho = \lambda\,\Omega$ for the Kähler form $\Omega$, equivalently $R_{i\bar j} = \lambda g_{i\bar j}$, and then the scalar curvature is the constant $2n\lambda$. The constant-holomorphic-sectional-curvature metrics are the Kähler–Einstein metrics with the strongest rigidity, and the existence problem for the general Kähler–Einstein metric, and its obstruction, belong to Kähler Geometry and beyond.

The Decomposition by the Unitary Group

Definition. The space of algebraic curvature tensors at a point is a representation of the orthogonal group $O(2n)$ and, for a Kähler metric, of the unitary group $U(n)$ acting on the complexified tangent space; an invariant decomposition is a decomposition into subrepresentations, and the irreducible parts of the curvature operator are the isotypic components.

Proposition (the unitary decomposition). For a Kähler metric the space of algebraic curvature tensors decomposes under $U(n)$ into the scalar part, the traceless-Ricci part, the Bochner part and the Weyl part, $$ \mathcal R = \mathcal R_{\text{scalar}} + \mathcal R_{\text{Ric}} + \mathcal R_{\text{Bochner}} + \mathcal R_{\text{Weyl}} , $$ and the first three are determined by the scalar curvature, the traceless Ricci form and the $(2,0)$-part of the curvature, while the Weyl part is the conformally invariant remainder; in complex dimension two the Weyl part splits further by the Hodge star into the self-dual and anti-self-dual halves.

Proof. The decomposition is the restriction to $U(n)$ of the $O(2n)$-decomposition of The Curvature Operator; the subrepresentations are the scalar, the traceless-Ricci, the Bochner (the $(2,0)$-type part) and the Weyl tensors, and the traceless-Ricci part is the traceless part of the Ricci tensor, the scalar part its trace. The four-dimensional splitting of the Weyl part is The Involution on the Curvature Operator. Constant holomorphic sectional curvature is exactly the vanishing of the last three parts, which is the operator form of the classification of the Kähler space forms.

Summary

On a complex manifold the complexified second exterior power splits into the pieces of type $(2,0)$, $(1,1)$ and $(0,2)$, and the complex structure $\mathcal J$ acts on them with eigenvalues $2i,0,-2i$. The Riemannian curvature operator commutes with $\mathcal J$, $[\mathcal R,\mathcal J]=0$, exactly for a Kähler metric, so that it preserves the type; for a Hermitian metric the type-preserving curvature is that of the Chern connection, a $(1,1)$-form with values in $\mathfrak u(n)$ whose trace $c_1(g)=\frac{i}{2\pi}\operatorname{tr}\Theta$ is the first Chern form, of class $c_1(M)$, and whose contraction is the Ricci form $\rho = i\,\partial\bar\partial\log\det g$ with $[\rho]=2\pi c_1(M)$. For a Kähler metric the Chern connection is the Levi-Civita connection, the curvature tensor satisfies $R_{i\bar j k\bar l}=\overline{R_{j\bar i l\bar k}}=R_{k\bar l i\bar j}=R_{i\bar l k\bar j}$, it is determined by the mixed components, and the holomorphic sectional curvature $H(X)=R(X,\bar X,X,\bar X)/g(X,\bar X)^2$, when constant, forces $R_{i\bar j k\bar l}=\frac{c}{4}(g_{i\bar j}g_{k\bar l}+g_{i\bar l}g_{k\bar j})$ with $\mathbb C^n$, $\mathbb{CP}^n$ and the complex hyperbolic space as the models. Under the unitary group the operator decomposes into the scalar, traceless-Ricci, Bochner and Weyl parts, the Kähler space forms being those with only the scalar part. The Riemannian operator and its $O(2n)$-decomposition are The Curvature Operator; the complex and Hermitian structures are Hermitian Geometry and Almost Complex Structures, Hermitian Metrics and the Levi-Civita Connection and Kähler Geometry; the Chern classes are Characteristic Classes; the four-dimensional splitting is The Involution on the Curvature Operator.

Summary of Notation

Symbol	Meaning
$\Lambda^{2,0}\oplus\Lambda^{1,1}\oplus\Lambda^{0,2}$	the type decomposition of the complexified bivectors
$\mathcal J$	the complex structure induced on bivectors
$\mathcal R$	the Riemannian curvature operator
$\nabla$, $\Theta=\nabla^2$	the Chern connection and its curvature
$R_{i\bar j k\bar l}$	the Kähler curvature components
$\rho=i\,\partial\bar\partial\log\det g$	the Ricci form, $[\rho]=2\pi c_1(M)$
$H(X)=R(X,\bar X,X,\bar X)/g(X,\bar X)^2$	holomorphic sectional curvature
$\mathcal R=\mathcal R_{\text{scalar}}+\mathcal R_{\text{Ric}}+\mathcal R_{\text{Bochner}}+\mathcal R_{\text{Weyl}}$	the unitary decomposition