Estimated transversality and rational maps

Abstract

In this paper, we address a question of Donaldson’s on the best estimate that can be achieved for the transversality of an asymptotically holomorphic sequence of sections of increasing powers of a line bundle over an integral symplectic manifold. More specifically, we find an upper bound for the transversality of $n + 1$ such sequences of sections over a $2n$-dimensional symplectic manifold. In the simplest case of $S\sp 2$, we also relate the problem to a well-known question in potential theory (namely, that of finding logarithmic equilibrium points), thus establishing an experimental lower bound for the transversality.