AN ANALYSIS OF THE MODELS L[T2n]

We analyze the models L[T-2n], where T-2n is a tree on omega x kappa(1)(2n+1) projecting to a universal Pi(1)(2n) set of reals, for n > 1. Following Hjorth's work on L[T-2], we show that under Det((Pi) under tilde (1)(2n)), the models L[T-2n] are unique, that is they do not depend of the choice of the tree T-2n. This requires a generalization of the Kechris-Martin theorem to all pointclasses Pi(1)(2n+1). We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of M-2n+1(#). We then show that the GCH holds in L[T-2n], for every n < omega, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy.