On linearization of line bundles. (English) Zbl 1342.14012

Brion, Michel

J. Math. Sci., Tokyo 22, No. 1, 113-147 (2015).

From the introduction: “Linearization of line bundles in the presence of algebraic group actions is a basic notion of geometric invariant theory; it also has applications to the local properties of such actions. For example, given an action of a connected linear algebraic group \(G\) on a normal variety \(X\) over a field \(k\), and a line bundle \(L\) on \(X\), some positive power \(L^{\otimes n}\) admits a \(G\)-linearization (as shown by D. Mumford et al. [Geometric invariant theory. 3rd enl. ed. Berlin: Springer-Verlag (1993; Zbl 0797.14004), Corollary 1.6] when \(X\) is proper, and by H. Sumihiro [J. Math. Kyoto Univ. 15, 573–605 (1975; Zbl 0331.14008), Theorem. 1.6] in a more general setting of group schemes; when \(k\) is algebraically closed of characteristic \(0\), we may take for \(n\) the order of the Picard group of \(G\) as shown by F. Knop et al. [in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63–75 (1989; Zbl 0722.14032)]). It can be inferred that \(X\) is covered by \(G\)-stable Zariski open subsets \(U_i\), equivariantly isomorphic to \(G\)-stable subvarieties of projectivizations of finite-dimensional \(G\)-modules; if \(G\) is a split torus, then the \(U_i\) may be taken affine (see [H. Sumihiro, J. Math. Kyoto Univ. 14, 1–28 (1974; Zbl 0277.14008), Corollary 2], [H. Sumihiro, J. Math. Kyoto Univ. 15, 573–605 (1975; Zbl 0331.14008), Theorem. 3.8, Corollary 3.11], [F. Knop et al., in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63–75 (1989; Zbl 0722.14032), Theorem 1.1]).
In this article, we show that the above results on linearization of line bundles and the local properties of algebraic group actions hold under weaker assumptions than normality, if the Zariski topology is replaced with the étale topology. For simplicity, we state our main result in the case where \(k\) is algebraically closed:
Theorem 1.1. Let \(X\) be a variety equipped with an action of a connected linear algebraic group \(G\).

(i)

If \(X\) is seminormal, then there exists a torsor \(\pi : Y \to X\) under the character group of \(G\), and a positive integer \(n\) (depending only on \(G\)) such that \(\pi^*(L^{\otimes n})\) is \(G\)-linearizable for any line bundle \(L\) on \(X\).

(ii)

If in addition \(X\) is quasi-projective, then it admits an equivariant étale covering by \(G\)-stable subvarieties of projectivizations of finite-dimensional \(G\)-modules.

(iii)

If \(G\) is a torus and \(X\) is quasi-projective, then \(X\) admits an equivariant étale covering by affine varieties.

In Section 2, we gather preliminary results on the Picard group of linear algebraic groups, and on the equivariant Picard group \(\mathrm{Pic}^G(X)\) which classifies \(G\)-linearized line bundles on a \(G\)-scheme \(X\); these results are variants of those in [H. Sumihiro, J. Math. Kyoto Univ. 15, 573–605 (1975; Zbl 0331.14008); F. Knop et al., in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 77–87 (1989; Zbl 0705.14005); F. Knop et al., in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63–75 (1989; Zbl 0722.14032)]. In particular, when \(X\) is reduced, we obtain an exact sequence \[ \mathrm{Pic}^G(X){\varphi}{\longrightarrow}\mathrm{Pic}(X){\psi}{\longrightarrow}\mathrm{Pic}(G \times X)/p_2^*\mathrm{Pic}(X), \] where \(\varphi\) denotes the forgetful map, and the obstruction map \(\psi\) arises from the pull-back under the action morphism \(G \times X \to X\) (see Proposition 2.10). Finally, we show that the obstruction group \(\mathrm{Pic}(G \times X)/p_2^* \mathrm{Pic}(X)\) is \(n\)-torsion if \(X\) is normal, where \(n\) is a positive integer depending only on \(G\) (Theorem 2.14).
The obstruction group is studied further in Section 3. We construct an injective map \(c : H^1_{et}(X,\hat{G}) \to \mathrm{Pic}(G \times X)/p_2^*\mathrm{Pic}(X)\), where the left-hand side denotes the first étale cohomology group with coefficients in the character group of \(G\) (viewed as an étale sheaf); recall that this cohomology group classifies \(\hat{G}\)-torsors over \(X\).
In Section 4, we present several applications of our analysis of the obstruction group. We first show that linearizability is preserved under algebraic equivalence (Proposition 4.1). Then we obtain a version of Theorem 1.1 over an arbitrary base field (Theorems 4.4, 4.7 and 4.8). Finally, we show that the seminormality assumption in Theorem 1.1 (i) and (ii) may be suppressed in prime characteristics (Subsection 4.3).”

Reviewer: Ronan Terpereau (Bonn)

MSC:

14C20	Divisors, linear systems, invertible sheaves
20G15	Linear algebraic groups over arbitrary fields
14C22	Picard groups
14L24	Geometric invariant theory
14M05	Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)

Keywords:

line bundle; linearization; seminormal

Citations:

Zbl 0797.14004; Zbl 0331.14008; Zbl 0722.14032; Zbl 0277.14008; Zbl 0705.14005