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Higgs bundles and integrable connections. The logarithmic case (smooth divisor). (Fibrés de Higgs et connexions intégrables: Le cas logarithmique (diviseur lisse).) (French) Zbl 0876.53043
Author’s 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 also solve 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 correspondence has a nice specialization between the induced objects over the divisor. Finally, we identify the natural cohomologies on both sides with \( L^2 \) cohomology”.

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53C05 Connections (general theory)
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