$L^{2}$-metrics, projective flatness and families of polarized abelian varieties

Families of Abelian Varieties

Projective Normality of Abelian Varieties

We show that ample line bundles L on a g-dimensional simple abelian variety A, satisfying h0(A,L) > 2g · g!, give projective normal embeddings, for all g ≥ 1.

Canonical Metrics and Stability of Projective Varieties

The two most common ways to parameterise subvarieties (or subschemes) of P are through Hilbert points and Chow points. For brevity we shall only consider on the latter which are roughly defined as follows. Suppose X ⊂ P is smooth of dimension n. Then a general linear subspace of L ⊂ P of dimension N − n− 1 will not intersect X. However a general one parameter family of such subspaces clearly do...

Families of Canonically Polarized Varieties

Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relat...

Finite Flatness of Torsion Subschemes of Hilbert-Blumenthal Abelian Varieties

Let E be a totally real number field of degree d over Q. We give a method for constructing a set of Hilbert modular cuspforms f1, . . . , fd with the following property. Let K be the fraction field of a complete dvr A, and let X/K be a Hilbert-Blumenthal abelian variety with multiplicative reduction and real multiplication by the ring of integers of E. Suppose n is an integer such that n divide...