Homogeneous projective bundles over abelian varieties.

*(English)*Zbl 1315.14057A \({\mathbb P}^{n-1}\) bundle (or Brauer-Severi variety) on \(P\) an abelian variety \(X\) over an algebraically closed field \(k\) is said to be homogeneous if it is isomorphic to its pullback under any translation in \(X\). S. Mukai characterised the vector bundles \(V\) such that \({\mathbb P}V\) is homogeneous: they are the semihomogeneous bundles studied in [J. Math. Kyoto Univ. 18, 239–272 (1978; Zbl 0417.14029)]. This paper studies the general case.

The basic fact needed is that such a \(P\) corresponds to an exact sequence \[ 1\longrightarrow H \longrightarrow G \longrightarrow X \longrightarrow 1 \] and a faithful representation \(H\hookrightarrow {\text{PGL}}_n\), in which \(G\) is an antiaffine (i.e., without nontrivial global functions) connected commutative group scheme. Antiaffine group schemes are well understood, so what is needed is a study of commutative subschemes of \({\text{PGL}}_n\). This is carried out here: the essential ingredients are the theta groups \(\widetilde H\subset {\text{GL}}_n\) that cover such groups, and in particular Heisenberg groups. This leads to a natural notion of irreducibility for homogeneous projective bundles (the group scheme of bundle automorphisms should be finite).

The paper then goes on to characterise homogeneous irreducible bundles (among projective bundles generally) as those for which all the cohomology of the adjoint vector bundle vanishes, and classifies them: they correspond to finite subgroups of \(\hat X\) equipped with an alternating pairing with values in \({\mathbb G}_m\). This then allows one to characterise the projectivisations of vector bundles (among homogeneous, not necessarily irreducible, projective varieties). This section of the paper concludes with some remarks about the Brauer group of \(X\): to quote the paper “in loose terms, the Brauer group is generated by homogeneous bundles and the relations arise from (bundles coming from) vector bundles”.

The final two sections deal with some examples and with self-dual projective bundles. The examples considered arise from spaces of algebraically equivalent effective divisors on some variety: these turn out to be homogeneous in the case of divisors on an abelian variety, but not in the case of a curve of genus \(\geq 2\). Self-dual projective bundles correspond to principal bundles for the projectivised orthogonal or symplectic groups: these are studied here from a geometric point of view.

There is a mild restriction on the characteristic \(p\geq 0\) of \(k\), which it would be interesting but non-trivial to remove: we need to assume that \(n\) is not a multiple of \(p\) (and for the self-dual bundles, that \(p\neq 2\)). This is needed because in the contrary case there are more complicated group subschemes of \({\text{PGL}}_n\).

The basic fact needed is that such a \(P\) corresponds to an exact sequence \[ 1\longrightarrow H \longrightarrow G \longrightarrow X \longrightarrow 1 \] and a faithful representation \(H\hookrightarrow {\text{PGL}}_n\), in which \(G\) is an antiaffine (i.e., without nontrivial global functions) connected commutative group scheme. Antiaffine group schemes are well understood, so what is needed is a study of commutative subschemes of \({\text{PGL}}_n\). This is carried out here: the essential ingredients are the theta groups \(\widetilde H\subset {\text{GL}}_n\) that cover such groups, and in particular Heisenberg groups. This leads to a natural notion of irreducibility for homogeneous projective bundles (the group scheme of bundle automorphisms should be finite).

The paper then goes on to characterise homogeneous irreducible bundles (among projective bundles generally) as those for which all the cohomology of the adjoint vector bundle vanishes, and classifies them: they correspond to finite subgroups of \(\hat X\) equipped with an alternating pairing with values in \({\mathbb G}_m\). This then allows one to characterise the projectivisations of vector bundles (among homogeneous, not necessarily irreducible, projective varieties). This section of the paper concludes with some remarks about the Brauer group of \(X\): to quote the paper “in loose terms, the Brauer group is generated by homogeneous bundles and the relations arise from (bundles coming from) vector bundles”.

The final two sections deal with some examples and with self-dual projective bundles. The examples considered arise from spaces of algebraically equivalent effective divisors on some variety: these turn out to be homogeneous in the case of divisors on an abelian variety, but not in the case of a curve of genus \(\geq 2\). Self-dual projective bundles correspond to principal bundles for the projectivised orthogonal or symplectic groups: these are studied here from a geometric point of view.

There is a mild restriction on the characteristic \(p\geq 0\) of \(k\), which it would be interesting but non-trivial to remove: we need to assume that \(n\) is not a multiple of \(p\) (and for the self-dual bundles, that \(p\neq 2\)). This is needed because in the contrary case there are more complicated group subschemes of \({\text{PGL}}_n\).

Reviewer: G. K. Sankaran (Bath)