DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Dual and Hom vector bundles

Definition

Let $E \to M$ and $F \to M$ be smooth vector bundles over the same base.

The dual bundle $E^{*} \to M$ has fibre $E_{p}^{*}$ over $p \in M$ .

The Hom bundle $Hom (E, F) \to M$ has fibre $Hom (E_{p}, F_{p})$ over $p \in M$ .

The next theorem equips these fibrewise constructions with smooth vector-bundle structures.

Depends on

Used by

Hermitian metric and L² pairing on a compact Riemann surface Definition
Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
The annihilator bundle of a distribution Definition
The holomorphic line bundle associated to a divisor Definition
The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface Definition
Smoothing continuous families of formal immersions Lemma
Bundle maps over f are sections of the pulled-back Hom bundle Proposition
Sections of Hom are the same as smooth fibrewise linear maps Proposition
Vector-bundle curvature is an endomorphism-valued two-form Proposition
Dual and Hom transition functions define smooth bundles Theorem
Harmonic star duality for line bundle valued dolbeault cohomology Theorem
Serre duality on a compact Riemann surface Theorem

Dependency tree · two levels

6 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)