ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Dolbeault cohomology is independent of hermitian metric

Example

Assume the Axiom of Choice (The Axiom of Choice), inherited through the Hodge-decomposition and finiteness items used below. Let $Λ = Z ω_{1} \oplus Z ω_{2}$ have real-linearly independent periods and keep the flat torus $X = C / Λ$ with its trivial holomorphic line bundle $E = X \times C$ and flat metric, and let $h_{0}$ be the constant Hermitian metric $h_{0} (1, 1) = 1$ and $h_{1}$ the Hermitian metric $h_{1} (1, 1) = e^{φ}$ for a nonconstant smooth real function $φ$ on $X$ . Then:

The harmonic spaces differ: with $h_{0}$ , the harmonic $(0, 1)$ -forms are $C \cdot d \overset{ˉ}{z}$ , while with $h_{1}$ they are $C \cdot e^{- φ} d \overset{ˉ}{z}$ , so the harmonic representative of the class $[d \overset{ˉ}{z}]$ is $d \overset{ˉ}{z}$ for $h_{0}$ and the suitable multiple of $e^{- φ} d \overset{ˉ}{z}$ for $h_{1}$ .
The Dolbeault cohomology does not depend on the metric: $H^{0, 1} (X, E)$ and $H^{0, 0} (X, E)$ are computed from ${\overset{ˉ}{\partial}}_{E}$ alone, and the harmonic-representative isomorphisms for $h_{0}$ and for $h_{1}$ show $H^{0, 1} (X, E) ≅ C (dimension 1), H^{0, 0} (X, E) = C,$ for both choices.
Consequently the numbers $h^{0, 0} (X, E)$ and $h^{0, 1} (X, E)$ are invariants of the holomorphic line bundle; the dependence on the Hermitian metric lies entirely in the choice of harmonic representative.

Here $H^{0, q} (E)$ denotes the harmonic space $H^{0, q} (E) = \ker Δ_{q}^{''}$ with its smooth representatives, while $H^{0, q} (X, E)$ denotes smooth Dolbeault cohomology.

Facts & Assumptions

Given: A rank-two lattice $Λ \subset C$ , its flat compact torus, the trivial holomorphic bundle, weights $ψ_{0} = 1$ , $ψ_{1} = e^{φ}$ with smooth nonconstant real $φ$ , and full AC. Write $d A = d x d y$ , $A = \int_{X} d A > 0$ and $J = \int_{X} e^{- φ} d A > 0$ .

[F1]

Smooth degree-one harmonic forms are exactly those with zero Hilbert adjoint; the adjoint on a smooth form in a flat chart is ${\overset{ˉ}{\partial}}_{h}^{*} (u d \overset{ˉ}{z}) = - 2 ψ^{- 1} \partial_{z} (ψ u)$ . Harmonic forms are smooth (The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface, The Dolbeault adjoint and Laplacian: local formulas and ellipticity, Elliptic regularity for Dolbeault harmonic forms).

[F2]

A bounded entire function is constant. A smooth function satisfies $\partial_{\overset{ˉ}{z}} f = 0$ exactly when it is holomorphic (Liouville's theorem: every bounded entire function is constant, Holomorphic line bundles and meromorphic sections on a Riemann surface).

[F3]

Smooth degree-one forms split orthogonally into harmonic and smooth exact forms; every Dolbeault class has one harmonic representative. The cohomology groups are the smooth Dolbeault kernel and quotient, and their dimensions are finite (Hodge decomposition for Dolbeault forms on a compact Riemann surface, Dolbeault cohomology of a compact riemann surface is finite dimensional).

[F4]

The first-variable-linear degree-one pairing in flat charts is $⟨ u d \overset{ˉ}{z}, v d \overset{ˉ}{z} ⟩_{L^{2}} = 2 \int_{X} ψ u \overset{ˉ}{v} d A$ (Hermitian metric and $L^{2}$ pairing on a compact Riemann surface).

Verification

Given: The data in the Example and Facts.

1.1F1F2givenalgebra

Translation charts on $X$ have identity derivatives, so $d \overset{ˉ}{z}$ is a global nonvanishing form and every smooth $(0, 1)$ -form is $u d \overset{ˉ}{z}$ with a smooth lattice-periodic coefficient on the cover. For any positive smooth weight $ψ$ , [F1] says it is harmonic precisely when $\partial_{z} (ψ u) = 0$ . Then $\overline{ψ u}$ lifts to an entire function, bounded because it is periodic and bounded on the closed fundamental parallelogram. By [F2] it is constant. Conversely $u = c / ψ$ is smooth and satisfies that adjoint equation, hence is harmonic. Thus the two spaces are $C d \overset{ˉ}{z}$ and $C e^{- φ} d \overset{ˉ}{z}$ ; they differ because their equality would force the positive function $e^{- φ}$ to be constant. Holomorphic functions on this torus likewise lift to bounded entire functions and are constant.

2.1F3F4step 1.1givenalgebra

Set $b = e^{- φ} d \overset{ˉ}{z}$ . By [F4], $⟨ d \overset{ˉ}{z}, b ⟩_{h_{1}} = 2 A$ and $⟨ b, b ⟩_{h_{1}} = 2 J$ . Hence the orthogonal projection of $d \overset{ˉ}{z}$ onto $C b$ is $(A / J) b$ in the first-variable-linear convention. By [F3] the projection differs from $d \overset{ˉ}{z}$ by a smooth exact form, so the harmonic representative of $[d \overset{ˉ}{z}]$ for $h_{1}$ is precisely $(A / J) e^{- φ} d \overset{ˉ}{z}$ . For $h_{0}$ it is $d \overset{ˉ}{z}$ itself. In particular the class is nonzero, since its harmonic representative is nonzero.

3.1F2F3step 1.1step 2.1given∎

The smooth operator ${\overset{ˉ}{\partial}}_{E}$ is determined by the holomorphic transitions, so the same vector space $\ker {\overset{ˉ}{\partial}}_{E}$ in degree zero and the same quotient $Ω^{0, 1} (E) / {\overset{ˉ}{\partial}}_{E} Ω^{0, 0} (E)$ in degree one define the cohomology for both metrics. The harmonic isomorphisms of [F3] and step 1.1 therefore give $H^{0, 0} (X, E) = C$ and $\dim H^{0, 1} (X, E) = 1$ for either metric. For any compact Riemann surface and fixed holomorphic bundle the same kernel/quotient observation proves that $h^{0, 0}, h^{0, 1}$ are invariant under changing either supplied metric; only the harmonic representative can change. Full AC is inherited through [F1] and [F3]; the explicit projection uses no additional choice.

Source notes

Demailly’s Dolbeault results in §7 apply to a holomorphic Hermitian bundle; the flat-connection de Rham decomposition in §3.3 is contextual and is not used for a varying Hermitian weight. The verification above uses proved local operator interfaces and gives the concrete calculation itself.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

125 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 (author manuscript, Universite Grenoble Alpes) (standard reference, not scraped)