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E-valued Dolbeault operators and their L2 structure

Definition

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for density integration and completion. Let $X$ be a complex manifold of dimension $m \geq 1$ (Complex manifolds, holomorphic vector bundles and Hermitian metrics) with a Hermitian metric $g$ , and let $E \to X$ be a holomorphic vector bundle of rank $r$ with a Hermitian metric $h$ (Complex manifolds, holomorphic vector bundles and Hermitian metrics). Fix integers $p, q \geq 0$ . In a holomorphic chart with coordinates $z = (z_{1}, \dots, z_{m})$ and a holomorphic frame of $E$ , an $E$ -valued $(p, q)$ -form is an expression $s = \sum_{∣ I ∣ = p} \sum_{∣ J ∣ = q} s_{I J} \otimes d z^{I} \land d {\overset{ˉ}{z}}^{J},$ with $s_{I J}$ smooth local sections of $E$ and increasing multi-indices $I, J$ (Bigraded complex forms and the Dolbeault operators); such expressions agree on overlaps because holomorphic changes of coordinates preserve bidegree and holomorphic frame changes are complex-linear. The resulting smooth global sections of the bundle $Λ^{p, q} T^{*} X \otimes E$ are denoted $A^{p, q} (X, E)$ ; the bundle is zero, and so is $A^{p, q} (X, E)$ , when $p > m$ or $q > m$ .

The Dolbeault operator is the first-order operator ${\overset{ˉ}{\partial}}_{E} : A^{p, q} (X, E) \to A^{p, q + 1} (X, E), {\overset{ˉ}{\partial}}_{E} s = \sum_{I, J} \sum_{k = 1}^{m} {\overset{ˉ}{\partial}}_{z_{k}} s_{I J} \otimes d {\overset{ˉ}{z}}_{k} \land d z^{I} \land d {\overset{ˉ}{z}}^{J},$ with the Wirtinger operators ${\overset{ˉ}{\partial}}_{z_{k}}$ (Wirtinger operators in $C^{m}$ , Bigraded complex forms and the Dolbeault operators) acting componentwise on the local sections $s_{I J}$ . It is independent of the holomorphic chart and frame, and ${\overset{ˉ}{\partial}}_{E}^{2} = 0$ (The d, partial and dbar identities); both assertions are proved below.

Equip $Λ^{p, q} T^{*} X \otimes E$ with the Hermitian metric induced by $g$ and $h$ (Riemannian metrics induce metrics on dual tensor and exterior bundles and Complex manifolds, holomorphic vector bundles and Hermitian metrics). For compactly supported smooth $s, t \in A^{p, q} (X, E)$ the $L^{2}$ pairing $⟨ s, t ⟩_{L^{2}} = \int_{X} ⟨ s, t ⟩ d μ_{g}$ uses the Riemannian volume density $μ_{g}$ (Riemannian volume density, Integral of a compactly supported smooth density) and is a complex inner product in the first-variable-linear convention (The complex $L^{2}$ pairing on equivalence classes); its completion $L^{2} (X, Λ^{p, q} T^{*} X \otimes E)$ is a complex Hilbert space by the completion argument below, using The metric completion of a normed space carries a unique compatible Banach-space structure. By Formal adjoints of differential operators on metric vector bundles the operator ${\overset{ˉ}{\partial}}_{E}$ has a formal adjoint ${\overset{ˉ}{\partial}}_{E}^{*} : A^{p, q + 1} (X, E) \to A^{p, q} (X, E)$ , the unique smooth first-order operator with $⟨ {\overset{ˉ}{\partial}}_{E} s, t ⟩_{L^{2}} = ⟨ s, {\overset{ˉ}{\partial}}_{E}^{*} t ⟩_{L^{2}}$ for all compactly supported smooth forms.

The Dolbeault Laplacian $Δ_{{\overset{ˉ}{\partial}}_{E}} = {\overset{ˉ}{\partial}}_{E} {\overset{ˉ}{\partial}}_{E}^{*} + {\overset{ˉ}{\partial}}_{E}^{*} {\overset{ˉ}{\partial}}_{E}$ is a formally symmetric Laplace-type second-order operator (Second-order Laplace-type operators on metric vector bundles) preserving each bidegree $(p, q)$ , with $⟨ Δ_{{\overset{ˉ}{\partial}}_{E}} s, s ⟩_{L^{2}} = ∥ {\overset{ˉ}{\partial}}_{E} s ∥_{2}^{2} + ∥ {\overset{ˉ}{\partial}}_{E}^{*} s ∥_{2}^{2} \geq 0$ for compactly supported smooth $s$ . Its principal symbol is $σ_{2} (Δ_{{\overset{ˉ}{\partial}}_{E}}) (x, ξ) = ∣ ξ^{0, 1} ∣_{g}^{2} i d$ , where $ξ^{0, 1}$ is the $(0, 1)$ -part of $ξ \in T_{x}^{*} X$ ; in particular the principal form is smooth, and it is positive definite on nonzero real covectors.

Facts & Assumptions

Given: Countable Choice, the complex manifold $X$ with Hermitian metric $g$ , the holomorphic bundle $E$ with Hermitian metric $h$ , and the data of the Definition; density integration and completion use this choice assumption, while the local operator constructions do not.

[F1]

Complex forms on an open set decompose uniquely by bidegree, $\overset{ˉ}{\partial}$ is the $(p, q + 1)$ -component of $d$ , and ${\overset{ˉ}{\partial}}^{2} = 0$ ; a holomorphic change of coordinates has complex-linear Jacobian and preserves bidegrees (Bigraded complex forms and the Dolbeault operators, The d, partial and dbar identities, Holomorphic maps $C^{m} \to C^{n}$ and the complex Jacobian matrix).

[F2]

The Wirtinger operators are $\partial_{z_{k}} = \frac{1}{2} (\partial_{x_{k}} - i \partial_{y_{k}})$ and $\partial_{{\overset{ˉ}{z}}_{k}} = \frac{1}{2} (\partial_{x_{k}} + i \partial_{y_{k}})$ , and the chain rule computes the derivatives of composites (Wirtinger operators in $C^{m}$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F3]

A Riemannian metric induces smooth metrics on dual, tensor and exterior bundles; on increasing orthonormal wedge monomials it is the determinant pairing and has value one, and the tangent bundle of a complex manifold carries the complex structure $J$ with the Hermitian compatibility $g (J v, J w) = g (v, w)$ (Riemannian metrics induce metrics on dual tensor and exterior bundles, Complex manifolds, holomorphic vector bundles and Hermitian metrics).

[F4]

Scalar complex $L^{2}$ has the first-variable-linear integral inner product and is Hilbert under Countable Choice; scalar Hölder bounds its pairing (The complex $L^{2}$ pairing on equivalence classes, $L^{2}$ with the integral pairing is a Hilbert space, Complex Holder, Minkowski, and the quotient norm). The bundle pairing and its completion are verified in step 1.2. Countable Choice also supplies the density-integral convention of the Definition (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Integral of a compactly supported smooth density).

[F5]

A smooth first-order operator between finite-rank metric bundles over a boundaryless manifold with a smooth positive density has a unique formal adjoint, with the local formula and the leading-coefficient rule of that lemma (Formal adjoints of differential operators on metric vector bundles).

[F6]

Laplace type, formal symmetry and the principal symbol $σ_{2}$ are as in Second-order Laplace-type operators on metric vector bundles; in particular for an operator with second-order part $- \sum a^{i j} \partial_{i} \partial_{j}$ the symbol is the positive definite form $a^{i j} ξ_{i} ξ_{j}$ .

[F7]

Every finite-dimensional real or complex inner product space has an orthonormal basis, so at each point one may compute in a unitary basis of the cotangent space (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans).

[F8]

Under Countable Choice a normed space has a Banach completion by Cauchy classes, with termwise vector operations and norm $∥ [s_{j}] ∥ = \lim_{j} ∥ s_{j} ∥$ (The metric completion of a normed space carries a unique compatible Banach-space structure).

Proof

1.1F1F2givenalgebra

Gluing of ${\overset{ˉ}{\partial}}_{E}$ . First let $θ_{α}, θ_{β}$ be holomorphic frames of $E$ with transition matrix $T$ , a holomorphic function with values in ${G L}_{r} (C)$ (Complex manifolds, holomorphic vector bundles and Hermitian metrics). For a local section $u$ , the Leibniz rule for $\overset{ˉ}{\partial}$ ([F1]) and $\overset{ˉ}{\partial} T = 0$ give $\overset{ˉ}{\partial} (T u) = \overset{ˉ}{\partial} T \land u + T \overset{ˉ}{\partial} u = T \overset{ˉ}{\partial} u$ , so the local $(p, q + 1)$ -forms computed in two frames agree. Next let $φ$ be a holomorphic transition between two holomorphic charts and write $g = f \circ φ$ for a coefficient function. Because $φ$ is holomorphic, ${\overset{ˉ}{\partial}}_{z_{k}^{'}} φ_{j} = 0$ and ${\overset{ˉ}{\partial}}_{z_{k}^{'}} {\overset{ˉ}{φ}}_{j} = \overline{\partial_{z_{k}^{'}} φ_{j}}$ ([F1], [F2]); the chain rule [F2] therefore gives ${\overset{ˉ}{\partial}}_{z_{k}^{'}} g = \sum_{j} \overline{\partial_{z_{k}^{'}} φ_{j}} (\partial_{{\overset{ˉ}{z}}_{j}} f) (φ)$ . Since $φ^{*} (d {\overset{ˉ}{z}}_{j}) = \sum_{k} \overline{\partial z_{j} / \partial z_{k}^{'}} d {\overset{ˉ}{z}}_{k}^{'}$ , multiplying by $d {\overset{ˉ}{z}}_{k}^{'}$ and summing reproduces the pullback of $\sum_{j} (\partial_{{\overset{ˉ}{z}}_{j}} f) d {\overset{ˉ}{z}}_{j}$ ; the holomorphic factors $d z^{I}$ pull back to $d z^{' I}$ -combinations, so bidegrees are preserved and $φ^{*} \overset{ˉ}{\partial} = \overset{ˉ}{\partial} φ^{*}$ . Hence the local operators agree on overlaps and glue to a global smooth first-order operator ${\overset{ˉ}{\partial}}_{E}$ that is independent of the atlas and frames.

1.2F3F4F8givenalgebra

The integral pairing on compactly supported smooth bundle sections is sesquilinear and conjugate symmetric by the fibre pairing and linearity of density integration. It is positive definite: a nonzero section has positive squared norm on a coordinate neighbourhood, hence positive integral. Fibrewise Cauchy–Schwarz follows by expanding $∥ s - λ t ∥^{2} \geq 0$ and minimizing over $λ$ ; the same expansion for the integral pairing gives $∣ ⟨ s, t ⟩_{L^{2}} ∣ \leq ∥ s ∥_{2} ∥ t ∥_{2}$ . Its norm therefore admits the Banach completion [F8]. For Cauchy sequences $s_{j}, t_{j}$ , the estimate $∣ ⟨ s_{j}, t_{j} ⟩ - ⟨ s_{l}, t_{l} ⟩ ∣ \leq ∥ s_{j} - s_{l} ∥_{2} ∥ t_{j} ∥_{2} + ∥ s_{l} ∥_{2} ∥ t_{j} - t_{l} ∥_{2}$ makes their scalar pairings Cauchy and their limit independent of equivalent sequences. This limit is sesquilinear and conjugate symmetric, and its diagonal is $\lim_{j} ∥ s_{j} ∥_{2}^{2} = ∥ [s_{j}] ∥^{2}$ . Thus it is a positive-definite inner product inducing the complete norm, so the completion is Hilbert. This also covers the zero bundle.

2.1step 1.1F1

Vanishing of the square and bidegree action. In each holomorphic chart ${\overset{ˉ}{\partial}}^{2} = 0$ coefficientwise by [F1], and the gluing of step 1.1 identifies these local squares with ${\overset{ˉ}{\partial}}_{E}^{2}$ ; hence ${\overset{ˉ}{\partial}}_{E}^{2} = 0$ globally. By construction ${\overset{ˉ}{\partial}}_{E}$ raises the second bidegree by one and preserves the first; when $p > m$ or $q > m$ the bundles and hence both sides vanish by the degree convention.

3.1step 2.1F3F5

Formal adjoint and its bidegree. The operator ${\overset{ˉ}{\partial}}_{E}$ is a smooth first-order linear differential operator between the finite-rank Hermitian bundles $Λ^{p, q} T^{*} X \otimes E$ and $Λ^{p, q + 1} T^{*} X \otimes E$ over $X$ , whose density $μ_{g}$ is smooth and positive, so [F5] supplies a unique smooth first-order formal adjoint ${\overset{ˉ}{\partial}}_{E}^{*}$ , characterized by the displayed pairing identity. It lowers the second bidegree by one: for a compactly supported smooth form $t$ of bidegree $(p, q + 1)$ and $s$ of bidegree $(p^{'}, q^{'})$ , the pairing $⟨ {\overset{ˉ}{\partial}}_{E} s, t ⟩_{L^{2}}$ is zero unless $(p^{'}, q^{'} + 1) = (p, q + 1)$ by step 2.1 and orthogonality of distinct bidegrees, so ${\overset{ˉ}{\partial}}_{E}^{*} t$ has no component in any bidegree other than $(p, q)$ .

4.1step 3.1F4F6algebra

Formal symmetry and energy. Let $s, t$ be compactly supported smooth $E$ -valued forms. Using the defining adjoint identities of step 3.1, $⟨ Δ_{{\overset{ˉ}{\partial}}_{E}} s, t ⟩_{L^{2}} = ⟨ {\overset{ˉ}{\partial}}_{E} {\overset{ˉ}{\partial}}_{E}^{*} s, t ⟩_{L^{2}} + ⟨ {\overset{ˉ}{\partial}}_{E}^{*} {\overset{ˉ}{\partial}}_{E} s, t ⟩_{L^{2}} = ⟨ {\overset{ˉ}{\partial}}_{E}^{*} s, {\overset{ˉ}{\partial}}_{E}^{*} t ⟩_{L^{2}} + ⟨ {\overset{ˉ}{\partial}}_{E} s, {\overset{ˉ}{\partial}}_{E} t ⟩_{L^{2}} .$ Applying the pairing property of [F4] to each summand shows the right side is conjugate-symmetric in $s, t$ , so $⟨ Δ_{{\overset{ˉ}{\partial}}_{E}} s, t ⟩_{L^{2}} = \overline{⟨ Δ_{{\overset{ˉ}{\partial}}_{E}} t, s ⟩_{L^{2}}}$ ; taking $s = t$ gives $⟨ Δ_{{\overset{ˉ}{\partial}}_{E}} s, s ⟩_{L^{2}} = ∥ {\overset{ˉ}{\partial}}_{E} s ∥_{2}^{2} + ∥ {\overset{ˉ}{\partial}}_{E}^{*} s ∥_{2}^{2} \geq 0$ . Thus $Δ_{{\overset{ˉ}{\partial}}_{E}}$ is formally symmetric and nonnegative in the sense of [F6], and by step 3.1 it preserves every bidegree $(p, q)$ .

5.1step 2.1step 3.1F3F6F7algebra∎

Principal symbol and Laplace type. Fix a point and compute in a chart and a unitary basis ([F7]); write $ξ^{0, 1}$ for the $(0, 1)$ -part of a real covector $ξ$ . In real coordinates the coefficient vectors of ${\overset{ˉ}{\partial}}_{E}$ multiply to $ε (ξ^{0, 1})$ , and those of ${\overset{ˉ}{\partial}}_{E}^{*}$ to $- ε (ξ^{0, 1})^{*}$ , where $ε$ is exterior multiplication and $*$ the metric adjoint; the metric adjoint of $ε (η)$ deletes each factor of a wedge monomial pairing with $η$ , with alternating signs, so the deletion check on unitary wedge monomials gives $ε (η) ε (η)^{*} + ε (η)^{*} ε (η) = ∣ η ∣^{2} i d$ , including the holomorphic $p$ -factors and the bundle factor. Therefore the second-order coefficient of $Δ_{{\overset{ˉ}{\partial}}_{E}}$ is the scalar $∣ ξ^{0, 1} ∣_{g}^{2}$ , that is, $- \sum a^{i j} \partial_{i} \partial_{j}$ with $a^{i j} ξ_{i} ξ_{j} = ∣ ξ^{0, 1} ∣_{g}^{2}$ ; this form is smooth in $x$ and positive definite on nonzero real $ξ$ , since $ξ^{0, 1} = 0$ forces the conjugate $(1, 0)$ -part $ξ^{1, 0} = 0$ and hence $ξ = 0$ . By [F6] the operator $Δ_{{\overset{ˉ}{\partial}}_{E}}$ is therefore Laplace type with the displayed principal symbol, and all assertions of the Definition are established.

Depends on

Used by

Harmonic representatives for compact de Rham and Dolbeault complexes Theorem

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106 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean-Pierre Demailly, Complex Analytic and Differential Geometry; Ch. V §10–§11, (10.1)–(10.2) and Definitions (11.1), (11.3), printed pp. 266–268: (0,1)-connections, holomorphic vector bundles, the canonical d'' connection and the Dolbeault complex; Ch. I §3.C, printed pp. 23–25: bidegree and d = d' + d'' (standard reference, not scraped)
Christian Schnell, Complex Manifolds, MAT 545 Fall 2024; Class 7, Definition 7.9 and the decomposition d = ∂ + ∂̄ (standard reference, not scraped)
Christian Schnell, Complex Manifolds, MAT 545 Fall 2024; Class 9, Definition 9.4 and the following local Hermitian formulas (standard reference, not scraped)