Very ample line bundles on quasi-abelian varieties

2001 ◽  
Vol 236 (1) ◽  
pp. 191-200 ◽  
Author(s):  
Shigeharu Takayama
Keyword(s):  
Abelian Varieties ◽  
Line Bundles

scholarly journals Index conditions and cup-product maps on Abelian varieties

2014 ◽  
Vol 25 (04) ◽  
pp. 1450036 ◽  
Author(s):  
Nathan Grieve
Keyword(s):  
Asymptotic Behavior ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
General Setting ◽  
Index Theorem ◽  
Line Bundles ◽  
One Dimensional ◽  
Cup Product ◽  
Product Maps

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behavior of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one-dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

scholarly journals Positivity properties of metrics and delta-forms

2019 ◽  
Vol 2019 (752) ◽  
pp. 141-177 ◽  
Author(s):  
Walter Gubler ◽  
Klaus Künnemann
Keyword(s):  
Algebraic Variety ◽  
Abelian Varieties ◽  
Toric Varieties ◽  
Convex Geometry ◽  
Piecewise Smooth ◽  
Line Bundles ◽  
Canonical Metrics ◽  
On Line

Abstract In previous work, we have introduced δ-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern δ-forms. In this paper, we investigate positivity properties of δ-forms and δ-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang’s semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.

Some remarks on ample line bundles on abelian varieties

1987 ◽  
Vol 57 (2) ◽  
pp. 225-238 ◽  
Author(s):  
Akira Ohbuchi
Keyword(s):  
Abelian Varieties ◽  
Line Bundles

A Note on the Global Generation of Primitive Line Bundles on Abelian Varieties

2006 ◽  
Vol 117 (1) ◽  
pp. 133-135
Author(s):  
Luis Fuentes García
Keyword(s):  
Abelian Varieties ◽  
Line Bundles

scholarly journals Local deformations of isolated singularities associated with negative line bundles over abelian varieties

1979 ◽  
Vol 75 ◽  
pp. 41-70
Author(s):  
Hideo Omoto ◽  
Shigeo Nakano
Keyword(s):  
Complex Manifold ◽  
Real Hypersurface ◽  
Abelian Varieties ◽  
Analytic Space ◽  
Line Bundles ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Universality Theorem ◽  
Pseudo Convex ◽  
Local Deformations

Let V be an analytic space with an isolated singularity p. In [1] M. Kuranishi approached the problem of deformations of isolated singularities (c.f. [2] and [3]) as follows; Let M be a real hypersurface in the complex manifold V − {p}. Then one has the induced CR-structure °T″(M) on M by the inclusion map i: M→ V − {p} (c.f. Def. 1.6). Then deformations of the isolated singularity (V, p) give rise to ones of the induced CR-structure °T″(M). He established in §9 in [1] the universality theorem for deformations of the induced CR-structure °T″(M)9 when M is compact strongly pseudo-convex (Def. 1.5) of dim M ≧ 5. Form this theorem we can know CR-structures on M which appear in deformations of °T″(M).

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