Abstract
Let \(X\) be an abelian variety defined over an algebraically closed field \(k\). We consider theta groups associated to simple, semi-homogenous vector bundles of separable type on \(X\). We determine the structure and representation theory of these groups. In doing so we relate work of Mumford, Mukai, and Umemura. We also consider adelic theta groups associated to line bundles on \(X\). After reviewing Mumford’s construction of these groups we determine functorial properties which they enjoy and then realize the Neron–Severi group of \(X\) as a subgroup of the cohomology group \(\mathrm {H}^2(\mathrm {V}(X), k^{\times })\).
Résumé
Soit \(X\) une variété abélienne définie sur un corps algébriquement clos \(k\). Nous étudions des groupes thêta associés aux faisceaux localement libres sur \(X\) qui sont simples, semi-homogènes de type séparable. Nous déterminons la structure et la théorie des représentations de ces groupes. Il s’avère que nous relions des travaux de Mumford, Mukai, et Umemura. En plus, nous étudions des groupes thêta adéliques associés à des fibrés vectoriels de rang \(1\) sur \(X\). Après avoir examiné la construction de Mumford de ces groupes, nous déterminons certaines de leurs propriétés fonctorielles. Enfin nous montrons que le groupe de Néron-Severi de \(X\) peut s’identifier avec un sous-groupe du groupe de cohomologie \(\mathrm {H}^2(\mathrm {V}(X), k^\times )\).
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Notes
Throughout we employ the (somewhat non-standard) terminology of [8]: if \(A\) and \(C\) are groups and \(1\rightarrow A \rightarrow B \rightarrow C \rightarrow 1\) is a short exact sequence of groups, then we say that \(B\) is an extension of \(A\) by \(C\).
References
Birkenhake, C., Lange, H.: Complex Abelian Varieties, 2nd edn. Springer, Berlin (2004)
Bosch, S., Lutkebohmert, W., Raynaud, M.: Neron Models. Springer, Berlin (1990)
Brion, M.: Homogeneous projective bundles over abelian varieties. Algebra Number Theory 7(10), 2475–2510 (2013)
Goren, E.Z.: Quasi-symmetric line bundles on Abelian varieties. Max Planck Inst., p. 67 (1995) (preprint)
Grieve, N.: Topics related to vector bundles on abelian varieties. Ph.D. thesis, Queen’s University (2013)
Grieve, N.: Index conditions and cup-product maps on Abelian varieties. Int. J. Math. 25(4), 1450036 (2014). (p. 31)
Gross, M., Popescu, S.: Equations of \((1, d)\)-polarized abelian surfaces. Math. Ann. 310(2), 333–377 (1998)
MacLane, S.: Homology. Springer, Berlin (1995)
Mukai, S.: Semi-homogeneous vector bundles on an Abelian variety. J. Math. Kyoto Univ. 18(2), 239–272 (1978)
Mumford, D.: On the equations defining abelian varieties. I. Invent. Math. 1, 287–354 (1966)
Mumford, D.: On the equations defining abelian varieties. II. Invent. Math. 3, 75–135 (1967)
Mumford, D.: On the equations defining abelian varieties. III. Invent. Math. 3, 215–244 (1967)
Mumford, D.: Varieties defined by quadratic equations. In: Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), pp. 29–100. Edizioni Cremonese, Rome (1970)
Mumford, D.: Tata lectures on theta. Birkhäuser Boston Inc., III (2007)
Mumford, D.: Abelian varieties. Published for the Tata Institute of Fundamental Research, Bombay. Corrected reprint of the second (1974) edition (2008)
Nakamura, I.: Stability of degenerate abelian varieties. Invent. Math. 136, 659–715 (1999)
Oda, T.: The first de Rham cohomology group and Dieudonne modules. Ann. Sc. Ecole Norm. Sup. Paris 4(2), 63–135 (1969)
Mumford, D.: Vector bundles on an elliptic curve. Nagoya Math. J. 43, 41–72 (1971)
Oprea, D.: The Verlinde bundles and the semihomogeneous Wirtinger duality. J. Reine Angew. Math. 654, 181–217 (2011)
Serre, J.P.: Linear Representations of Finite Groups. Springer, New York (1977)
Shin, S.W.: Abelian varieties and Weil representations. Algebra Number Theory 6(8), 1719–1772 (2012)
Umemura, H.: Some results in the theory of vector bundles. Nagoya Math. J. 52, 97–128 (1973)
Umemura, H.: On a certain type of vector bundles over an Abelian variety. Nagoya Math. J. 64, 31–45 (1976)
Acknowledgments
The author thanks his Ph.D. adviser Mike Roth for useful discussions. The author also acknowledges that the final writing of this work benefited from conversations with Eyal Goren, Jacques Hurtubise, and criticisms and suggestions given by the referee.
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Grieve, N. Refinements to Mumford’s theta and adelic theta groups. Ann. Math. Québec 38, 145–167 (2014). https://doi.org/10.1007/s40316-014-0024-0
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DOI: https://doi.org/10.1007/s40316-014-0024-0