ar X iv : m at h / 06 12 24 1 v 1 [ m at h . L O ] 9 D ec 2 00 6 Filtration - equivalent א 1 - separable abelian groups of cardinality א

ar X iv : m at h / 06 12 24 2 v 1 [ m at h . A P ] 9 D ec 2 00 6 Optimal Uniform Elliptic Estimates for the Ginzburg - Landau System

We reconsider the elliptic estimates for magnetic operators in two and three dimensions used in connection with Ginzburg-Landau theory. Furthermore we discuss the so-called blow-up technique in order to obtain optimal estimates in the limiting cases.

ar X iv : m at h / 06 05 44 4 v 1 [ m at h . N T ] 1 6 M ay 2 00 6 ABELIAN VARIETIES OVER CYCLIC FIELDS

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. Unless K ⊂ F̄p for some p, the rank of A over K is infinite.

ar X iv : h ep - t h / 06 12 12 8 v 1 1 3 D ec 2 00 6 Riemannian Geometry of Noncommutative Surfaces

ar X iv : m at h / 02 12 21 6 v 1 [ m at h . A G ] 1 6 D ec 2 00 2 Nodes and the Hodge conjecture

The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities.

ar X iv : h ep - t h / 03 12 24 4 v 1 1 9 D ec 2 00 3 FIAN / TD / 07 – 03 On Sp ( 2 M ) Invariant Green Functions

Explicit form of two-point and three-point Sp (2M) invariant Green functions is found.