Homogeneous projective bundles over abelian varieties

Abstract

We consider projective bundles (or Brauer-Severi varieties) over an abelian variety which are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative group schemes; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties. IntroductionThe main objects of this article are the projective bundles (or Brauer-Severi varieties) over an abelian variety X which are homogeneous, i.e., isomorphic to their pull-backs under all translations. Among these bundles, the projectivizations of vector bundles are well understood by work of Mukai (see [22]). Indeed, the vector bundles with homogeneous projectivization are exactly the semi-homogeneous vector bundles of [loc. cit.]. Those that are simple (i.e., their global endomorphisms are just scalars) admit several remarkable characterizations; for example, they are all obtained as direct images of line bundles under isogenies. Moreover, every indecomposable semi-homogeneous vector bundle is the tensor product of a unipotent bundle and of a simple semi-homogeneous bundle.In this article, we obtain somewhat similar statements for the structure of homogeneous projective bundles. We build on the results of our earlier paper [9] about homogeneous principal bundles under an arbitrary algebraic group; here we consider of course the projective linear group PGL n . In loose terms, the approach of [loc. cit.] reduces the classification of homogeneous bundles to that of commutative subgroup schemes of PGL n . The latter, carried out in Section 2, is based on the classical construction of Heisenberg groups and their irreducible representations.2010 Mathematics Subject Classification: Primary 14K05; Secondary 14F22, 14J60, 14L30. 1In Section 3, we introduce a notion of irreducibility for homogeneous projective bundles, which is equivalent to the group scheme of bundle automorphisms being finite. (The projectivization of a semi-homogeneous vector bundle E is irreducible if and only if E is simple). We characterize those projective bundles that are homogeneous and irreducible, by the vanishing of all the cohomology groups of their adjoint vector bundle (Proposition 3.7). Also, we show that the homogeneous irreducible bundles are classified by the pairs (H, e), where H is a finite subgroup of the dual abelian variety, and e : H × H → G m a non-degenerate alternating bilinear pairing (Proposition 3.1). Finally, we obtain a characterization of those homogeneous projective bundles that are projectivizations of vector bundles, first in the irreducible case (Proposition 3.10; it states in loose terms that the pairing e originates from a line bundle on X) and then in the general case (Theorem 3.11).The irreducible homogeneous projective bundles over an elliptic curve are exactly the projectivizations of indecomposable vector bundles wi...

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“…Part (i) follows readily from Theorem 3.1 of [Brion 2012a], and (ii) from Proposition 3.6 of the same reference.…”

“…For later use, we now present some general results on self-dual bundles; we omit their (easy) proofs, which can be found in the arXiv version of this article [Brion 2012b]. …”

“…In this article, we obtain somewhat similar statements for the structure of homogeneous projective bundles. We build on the results of [Brion 2012a] about homogeneous principal bundles under an arbitrary algebraic group; here we consider of course the projective linear group PGL n . In loose terms, the approach of our earlier paper reduces the classification of homogeneous bundles to that of commutative subgroup schemes of PGL n .…”

The Coherent Cohomology Ring of an Algebraic Group

“…Moreover, each coherent E-sheaf F on A has a finite increasing filtration with subquotients being the structure sheaf O A , i.e., F is the sheaf of local sections of a unipotent vector bundle. In fact, this yields an equivalence from Coh E (A) to the category Uni(A) of unipotent vector bundles on A (see [Br12,Rem. 3.13(ii)]).…”

Refinements to Mumford’s theta and adelic theta groups

“…Some more recent examples include the work of Gross-Popescu, [7, Example 2.10, p. 349], where theta groups and their higher weight representation theory play a role in their study of syzygies, work of Nakamura, [16], where theta group schemes are used to compactify the moduli scheme of abelian schemes over Spec Z[ζ N , 1/N], and work of Oprea, [19, §2], where theta groups and their relation to semi-homogeneous vector bundles are used to study Verlinde bundles. Even more recently Brion has considered theta groups associated to Brauer-Severi varieties over abelian varieties, [3], while Shin has extended Mumford's work by constructing theta and adelic theta groups associated to line bundles on abelian schemes, [21].…”