G-linearizations of invertible sheaves on a complex G-variety

Let $G$ be a complex affine algebraic group acting algebraically on a classical complex variety $X$ (Classical complex affine algebraic actions and rational modules, Classical algebraic prevarieties, regular maps, and varieties), and let $L$ be an invertible sheaf on $X$ with total space $p:L\to X$ (Invertible sheaves), the total space being obtained by gluing $U\times {\mathbb{A}}^1$ over local frames of $L$ using their invertible regular transition functions, and the projection being a morphism of locally ringed spaces (Morphisms of locally ringed spaces). Write $\sigma :G\times X\to X$ for the action.

A $G$-linearization of $L$ is an algebraic $G$-action $m:G\times L\to L$ on the total space such that

• $p(m(g,\mathrm{\ell }))=gp(\mathrm{\ell })$ for all $g\in G$, $\mathrm{\ell }\in L$, and
• for every $g\in G$ and $x\in X$ the fibre map $L_x\to L_{gx}$, $\mathrm{\ell }\mapsto g\mathrm{\ell }$, is $\mathbb{C}$-linear.

A $G$-linearized invertible sheaf is an invertible sheaf together with a linearization; the pair is written $(L,m)$. Equivalently, a linearization is an isomorphism $\phi :{\sigma }^{\ast }L\to {\mathrm{p}\mathrm{r}}_2^{\ast }L$ of sheaves on $G\times X$ satisfying the cocycle identity

${\mathrm{p}\mathrm{r}}_{23}^{\ast }\phi \circ ({\mathrm{i}\mathrm{d}}_G\times \sigma {)}^{\ast }\phi =(m_G\times {\mathrm{i}\mathrm{d}}_X{)}^{\ast }\phi ,$

where $m_G:G\times G\to G$ is the multiplication; at $(g,h,x)$ both sides map the fibre $L_{ghx}$ to $L_x$. The isomorphism $\phi$ sends a vector over $gx$ to its translate by $g^{-1}$ over $x$, so its inverse recovers the action on total spaces.

For any algebraic character $\chi :G\to {\mathbf{G}}_m={\mathbb{C}}^{\times }$ the twist $(L,m{)}_{\chi }$ multiplies the fibre action by $\chi (g)$ and is again a linearization of the same invertible sheaf. In particular linearizations are not unique, the trivial action admits the trivial linearization of ${\mathcal{O}}_X$ and its twists, and for a finite-dimensional rational $G$-module $V$ the induced action on the tautological line bundle linearizes ${\mathcal{O}}_{\mathbf{P}(V)}(-1)$, and its dual linearizes ${\mathcal{O}}_{\mathbf{P}(V)}(1)$.

The two main theorems of this page always assume that a linearization is given; no general existence of linearizations is claimed here.