Multiplicity estimates and degeneration

Abstract

In this paper we prove a multiplicity estimate, which is best possible up to a multiplicative constant, in which the set of points is connected to an action of SL₂(Z) on the torus ${{\bf G}_m^2({\bf C})}$. This result is motivated by the construction, due to Roy, of a non-trivial auxiliary function that could be used to study the points on the Grassmannian whose coordinates are logarithms of algebraic numbers: to make use of this construction, only a zero estimate connected to the action of GL _m (Z) on ${\wedge^k {\bf C}^m}$ is missing. The result we prove is essentially analogous to it. The proof is based on the fact that a zero (or multiplicity) estimate can be derived from a lower bound for a Seshadri constant. Then a degeneration argument is used: inside a family any such lower bound holds on an open subset, so proving it for sufficiently many special cases yields it for almost all cases.