nLab tautological equivariant line bundle

Idea

A tautological equivariant line bundle is an equivariant tautological line bundle over a projective G-space.

Definition

Let $G$ be a finite group (or maybe a compact Lie group) and let $V$ be a $G$-linear representation over some topological ground field $k$, with $k P(V)$ its projective G-space.

Then the corresponding tautological equivariant line bundle $\mathcal{L}_V$ is the $k^\ast$-fiber bundle which is associated to the canonical $k^\times$-principal bundle over projective G-space:

$$\array{ \mathcal{L}_V & \coloneqq & \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times } & \overset{ [v,z] \mapsto \big( [v], v \cdot z \big) }{\hookrightarrow} & \frac{ V \setminus \{0\} }{ k^\times } \times V \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ \frac{id \times pt}{ k^\times } }} \\ k P(V) &=& \frac{ (V \setminus \{0\}) \times \ast }{ k^\times } }$$

and equipped with the induced $G$-action through $V$ (which passes to the quotient spaces since the $k$-multiplication action commutes with it, by linearity).

Here

• $k^\times \,\coloneqq\, k \setminus \{0\}$ is the group of units of $k$;

• $\frac{(-) \times (-)}{k^\times}$ denotes the quotient space of a product space by the diagonal action;

• and $k^\ast$ is $k$ equipped with the dual $k^\times$-action

  $$\array{ k^\times \times k^\ast &\longrightarrow& k^\ast \\ (g,z) &\mapsto& z/g }$$

• so that, for $z \neq 0$,

  $$[v,z] \;=\; [v, z \cdot 1] \;=\; [v \cdot z, 1] \;\;\in\;\; \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times }$$

Related concepts

• resolution of singularities

• equivariant complex-oriented cohomology theory