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Contractions versus extensions of holonomy maps of foliations and applications
Julio Reb elo Consider a singular holomorphic foliation F on a pro jective surface. Over the leaves of F we can define real tra jectories over which the parallel transport of F yields a contraction for a natural way of measuring the distance between two leaves. These real tra jectories can also be seen as a singular real foliation denoted by H and, among the possible singularities of H, there are "sources" (resp. "sinks") corresponding to a point of local maximum (resp. minimum) for the distance between two leaves of F . In turn, these types of singularities lead to the notion of endpoints for a given tra jectory. It is then natural to consider foliations H containing a tra jectory of "infinite length" and foliations H all of whose tra jectories are of "finite length". In the first case we are going to show the presence of a suitable "contraction" (modulo ramification and a few other minor issues). In the remaining case, we are going to provide some general statements concerning the possibility of extending holonomy maps of among different transverse sections for F . We shall then indicate some applications of these results.

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