DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Determinant line and its projector

Definition

Let $M$ be a smooth manifold, let $r \geq 1$ , and let $P : M \to M_{r} (C)$ be a $C^{2}$ orthogonal projector of constant rank $m$ , with $1 \leq m \leq r$ . Its image bundle is $E_{P} = {(x, v) : P (x) v = v}$ , with the projected connection $\nabla s = P d s$ of C2 projector bundles and Kato transport. The determinant line bundle is the complex line bundle

$\det E_{P} : = ⋀^{m} E_{P} \subset M \times ⋀^{m} C^{r} .$

Give $⋀^{m} C^{r}$ the Hermitian inner product for which $e_{i_{1}} \land \dots \land e_{i_{m}}$ , $i_{1} < \dots < i_{m}$ , is an orthonormal basis, where $(e_{1}, \dots, e_{r})$ is the standard basis of $C^{r}$ . The determinant-line projector at $x$ is $Q (x) : = ⋀^{m} P (x)$ ; it is the orthogonal projection onto $⋀^{m} im P (x)$ .

Definition references

C2 projector bundles and Kato transport

The graded exterior algebra $Λ V$

Increasing-index wedges of a basis form a basis of $Λ^{k} V$

Depends on

Dependency tree · 2 levels

Sources

Keith Conrad, Exterior Powers