On the Quotient Variety of an Abelian Variety

On the Quotient Variety of an Abelian Variety.

Algebraic Cycles on an Abelian Variety

It is shown that to every Q-linear cycle α modulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle α modulo rational equivalence on A lying above α, characterised by a condition on the spaces of cycles generated by α on products of A with itself. The assignment α 7→ α respects the algebraic operations and pullback and push forward along homomorphism...

On the cone of curves of an abelian variety

As usual denote by NE(X) the cone of curves on X, i.e. the convex cone in N1(X) generated by the effective 1-cycles. The closed cone of curves NE(X) is the closure of NE(X) in N1(X). One knows by the Cone Theorem [4] that it is rational polyhedral whenever c1(X) is ample. For varieties X such that c1(X) is not ample, however, it is in general difficult to determine the structure of NE(X), since...

The Quotient of a Complete Symmetric Variety

We study the quotient of a completion of a symmetric variety G/H under the action of H . We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points. 2000 Math. Subj. Class. 14L30, 14L24, 14M17.

The Splitting of Reductions of an Abelian Variety

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction Av of A modulo v splits up to isogeny. Assuming the Mumford–Tate conjecture for A and possibly increasing the field K, we will show that Av is isogenous to the m-th power of an absolutely simple abelian variety for all places v of K away from a set of density 0, wher...