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The Dolbeault Laplacian has finite-dimensional kernel and closed range

Statement

Assume the Axiom of Choice (The Axiom of Choice). It is inherited through the construction of the boundary and Green operators in Dolbeault green operator is compact on the orthogonal complement of the kernel. It supplies Dependent Choice for the closed-range theorem and Countable Choice for the Hilbert orthogonal-decomposition theorem (The Axiom of Countable Choice ( ${A C}_{ω}$ ), AC implies DC implies countable choice). Let $X$ be a nonempty compact Riemann surface, let $E \to X$ be a holomorphic line bundle with Hermitian metric $h$ , and let $g$ be a compatible Riemannian metric. Use the Hilbert spaces and block Dolbeault Laplacian $Δ^{''}$ from The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface. Put $H : = \ker Δ^{''} \subset L^{2} : = L_{0}^{2} \oplus L_{1}^{2}$ . Let $B : L^{2} \to L^{2}$ and $G : H^{⊥} \to H^{⊥}$ be the compact boundary and Green operators from Dolbeault green operator is compact on the orthogonal complement of the kernel; in particular, $B$ is compact and self-adjoint, $\ker (I - B) = H$ , and $(I + Δ^{''}) B = I on L^{2}, B (I + Δ^{''}) u = u (u \in dom Δ^{''}) .$ The Green identities are $Δ^{''} G f = f$ for $f \in H^{⊥}$ and $G Δ^{''} u = u$ for $u \in dom Δ^{''} \cap H^{⊥}$ . The space $H^{2}$ below is the finite-chart Sobolev space of Gårding estimates for the Dolbeault Laplacian on a compact Riemann surface.

Finite-dimensional harmonic space. $H$ is finite dimensional.
Closed range and orthogonal decomposition. The range $Δ^{''} (dom Δ^{''})$ is closed and equals $H^{⊥}$ , so $L^{2} = H \oplus Δ^{''} (dom Δ^{''})$ orthogonally. The range of $Δ^{''}$ on each summand $L_{q}^{2}$ , $q = 0, 1$ , is closed in that summand.
Quantitative inverse. There is a constant $c > 0$ such that $∥ Δ^{''} u ∥_{L^{2}} \geq c ∥ u ∥_{L^{2}} (u \in H^{⊥} \cap dom Δ^{''}) .$ Moreover, $Δ^{''}$ restricts to a topological isomorphism $H^{⊥} \cap dom Δ^{''} ⟶ H^{⊥}$ with inverse $G$ . On its domain use the graph norm $∥ u ∥_{H^{2}, Δ} : = ∥ u ∥_{H^{2}} + ∥ Δ^{''} u ∥_{L^{2}} .$

Facts & Assumptions

Given: The compact Riemann surface and supplied metrics; the maximal Dolbeault complex, its total Hilbert space, and its block Laplacian; the boundary and Green operators of the preceding lemma; and full AC.

[F1]

The total space $L_{0}^{2} \oplus L_{1}^{2}$ is a complex Hilbert space and hence a Banach space (Hilbert space).

[F2]

The Laplacian is the nonnegative self-adjoint block operator $Δ_{0}^{''} \oplus Δ_{1}^{''}$ on the direct-sum composition domain (The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface).

[F3]

$B$ is compact, bounded and self-adjoint, $\ker (I - B) = H$ , and $(I + Δ^{''}) B = I$ on $L^{2}$ while $B (I + Δ^{''}) = I$ on $dom Δ^{''}$ . The Green operator $G$ is bounded into $H^{2}$ , has range $dom Δ^{''} \cap H^{⊥}$ , and satisfies both Green identities (Dolbeault green operator is compact on the orthogonal complement of the kernel).

[F4]

If $K$ is compact on a normed space, then $\ker (I - K)$ is finite dimensional (Kernel of identity minus compact is finite dimensional).

[F5]

If $K$ is compact on a Banach space, then $ran (I - K)$ is closed under DC (Range of identity minus compact is closed).

[F6]

For a bounded self-adjoint operator $T$ on a Hilbert space, $⟨ T x, y ⟩ = ⟨ x, T y ⟩$ for all $x, y$ ; the identity operator is self-adjoint (Self-adjoint, positive, unitary and normal operators).

[F7]

For a subset $S$ of a Hilbert space, $S^{⊥}$ consists of the vectors orthogonal to every element of $S$ ; every closed subspace of a Hilbert space has a unique orthogonal decomposition (Orthogonality and the orthogonal complement, Orthogonal decomposition by a closed subspace).

[F8]

The finite-chart $H^{2}$ norm is the Sobolev norm in the preceding Gårding theorem; the inclusion $H^{2} ↪ L^{2}$ is continuous, and $G : H^{⊥} \to H^{2}$ is bounded (Gårding estimates for the Dolbeault Laplacian on a compact Riemann surface, The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface, Dolbeault green operator is compact on the orthogonal complement of the kernel).

[F9]

Full AC implies DC and Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), AC implies DC implies countable choice).

Proof

technique · Use compact-perturbation facts for $I-B$ and the exact Green identities for $\Delta''$

1.1F1F3F4

By [F3], $H = \ker (I - B)$ . Apply the compact-kernel lemma [F4] to $B$ on the normed space $L^{2}$ from [F1]. Thus $H$ is finite dimensional.

1.2F1F3F5F9algebra

Set $A : = I - B$ . For $f \in L^{2}$ , $(I + Δ^{''}) B f = f$ , so $(I - B) f = Δ^{''} B f$ . For $u \in dom Δ^{''}$ , $B (I + Δ^{''}) u = u$ , so $(I - B) (I + Δ^{''}) u = (I + Δ^{''}) u - u = Δ^{''} u$ . Both containments give $ran Δ^{''} = ran A$ . The space $L^{2}$ is Banach by [F1], and AC supplies DC for [F5]; hence this range is closed.

1.3F3F8algebra

If $H^{⊥} = {0}$ , the displayed estimate holds with $c = 1$ . Otherwise boundedness of $G : H^{⊥} \to H^{2}$ and the continuous inclusion $H^{2} ↪ L^{2}$ give some $M > 0$ with $∥ G f ∥_{L^{2}} \leq M ∥ f ∥_{L^{2}}$ for all $f \in H^{⊥}$ . For $u \in H^{⊥} \cap dom Δ^{''}$ , the Green identity gives $u = G Δ^{''} u$ , whence $∥ u ∥_{L^{2}} \leq M ∥ Δ^{''} u ∥_{L^{2}}$ . Taking $c = M^{- 1}$ proves the estimate in this case.

2.1F3F6F7F9step 1.2

The operator $A = I - B$ is bounded and self-adjoint by [F3, F6]. If $y ⊥ ran A$ , then $⟨ x, A y ⟩ = ⟨ A x, y ⟩ = 0$ for every $x \in L^{2}$ , so $A y = 0$ ; conversely, $A y = 0$ implies $y ⊥ ran A$ . Therefore $(ran A)^{⊥} = \ker A = H$ by [F3]. Apply [F7] to the closed subspace $ran A$ from step 1.2: $L^{2} = ran A \oplus H$ . Every vector in $ran A$ is orthogonal to $H$ ; if $z \in H^{⊥}$ and $z = r + h$ is this decomposition, then $h = z - r \in H \cap H^{⊥} = {0}$ . Thus $ran A = H^{⊥}$ , and step 1.2 gives $ran Δ^{''} = H^{⊥}$ . AC supplies the theorem's countable-choice hypothesis by [F9].

2.2F2step 1.2

By [F2], $Δ^{''}$ is block diagonal and its domain is the direct sum of the two block domains, so its range is $ran Δ_{0}^{''} \oplus ran Δ_{1}^{''}$ . If a sequence in either block range converges in that $L_{q}^{2}$ summand, embed it in $L_{0}^{2} \oplus L_{1}^{2}$ with zero in the other component. Closedness of the total range from step 1.2 puts the limit in the total range with that other component still zero, hence in the same block range. Both degreewise ranges are closed.

3.1F3F8∎

The two Green identities make $Δ^{''}$ and $G$ inverse bijections between $H^{⊥} \cap dom Δ^{''}$ and $H^{⊥}$ . By [F8], $∥ G f ∥_{H^{2}} \leq C ∥ f ∥_{L^{2}}$ ; together with $∥ Δ^{''} G f ∥_{L^{2}} = ∥ f ∥_{L^{2}}$ this bounds $G$ into the graph norm $∥ \cdot ∥_{H^{2}, Δ}$ . The forward map is continuous in that graph norm by its definition, so this is a topological isomorphism.

Source notes

Demailly's Ch. VI §2 (2.4), (2.6)–(2.8) states the finite-dimensional kernel, closed smooth range and orthogonal decomposition for a smooth elliptic differential operator on a compact manifold; its proof uses the Gårding and Rellich results and cites Hörmander for elliptic PDE facts. Ch. VI §7 (7.1)–(7.2) states the smooth Chern-Dolbeault decomposition and finite-dimensional Dolbeault cohomology, without giving its proof there. The present proof establishes the Hilbert-domain range and inverse claims from the preceding Green-operator lemma and the compact-perturbation suppliers; it does not import Demailly's smooth decomposition as a Hilbert-domain argument.

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Sources

Jean-Pierre Demailly, Complex Analytic and Differential Geometry (author manuscript, Universite Grenoble Alpes) (standard reference, not scraped)
Jean-Pierre Demailly, Complex Analytic and Differential Geometry (author manuscript, Universite Grenoble Alpes) (standard reference, not scraped)