Abstract
Take a smooth divisor in a compact Kähler manifold. Given a stable Higgs bundle with “logarithmic structure” over the divisor (this means that over the divisor the bundle has a parabolic structure and the Higgs field has a logarithmic singularity), we solve the Hermite-Einstein problem for a Kähler metric of Poincaré type around the divisor. For appropriate Chern numbers, this gives a “logarithmic” integrable connection. We solve also the inverse problem, so that we get a complete correspondence between logarithmic Higgs bundles and logarithmic integrable connections, generalizing Simpson's correspondence for curves. The correspondance has a nice specialization between the induced objects over the divisor. Finally we identify the natural cohomologies on both sides with L 2 cohomology.
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