Harmonic bundles on noncompact curves

The purpose of this paper is to extend the correspondence between Higgs bundles and local systems [2,5,6,7, 13, 17, 19,20,21] to the case when X is a noncom pact algebraic curve. The basic result is that there is a class of analytic objects on X which will be put in one-to-one correspondences with two different classes of algebraic geometric objects on the completion X. The analytic objects are harmonic bundles on X satisfying a growth condition at the punctures which we call tameness. The two types of algebraic objects are Higgs bundles and g-x-modules, both with regular singularities at the punctures, together with additional data of filtrations of the fibers over the punctures. The filtered regular g-x-modules also correspond to topological objects, local systems (i.e. representations of the fundamental group), together with filtrations assigned to the punctures. For the algebraic or topological objects, there is an

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