HIGGS BUNDLES AND INTEGRABLE CONNECTIONS - THE LOGARITHMIC CASE (SMOOTH DIVISOR)

Take a smooth divisor in a compact Kahler manifold. Given a stable Hig gs bundle with ''logarithmic structure'' over the divisor (this means that over the divisor the bundle has a parabolic structure and the Hig gs field has a logarithmic singularity), we solve the Hermite-Einstein problem for a Kahler metric of Poincare type around the divisor. For appropriate Chern numbers, this gives a ''logarithmic'' integrable con nection. We solve also the inverse problem, so that we get a complete correspondence between logarithmic Higgs bundles and logarithmic integ rable connections, generalizing Simpson's correspondence for curves. T he correspondance has a nice specialization between the induced object s over the divisor. Finally we identify the natural cohomologies on bo th sides with L(2) cohomology.