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Geometry of cohomology support loci for local systems. I. (English) Zbl 0923.14010
Let $$X$$ be a Zariski open subset of a compact Kähler manifold. In this paper, the author studies the set $$\Sigma^k(X)$$ of one-dimensional local systems on $$X$$ with nonvanishing $$k$$-th cohomology. He shows that under certain conditions $$(X$$ compact, $$X$$ has a smooth compactification with trivial first Betti number, or $$k=1)$$, $$\Sigma^k(X)$$ is a union of translates of sets of the form $$f^*H(T,C^*)$$, where $$f:X\to T$$ is a holomorphic map to a complex Lie groups which is an extension of a compact complex torus by a product of $$C^*$$’s (these correspond to semiabelian varieties in the algebraic category). This generalizes earlier work of Beauville, Green, Lazarsfeld, Simpson and the author in the compact case. The main novelty lies in the proofs which involve consideration of Higgs fields with logarithmic poles. While a completely satisfactory theory of such objects is still lacking, the author is able to work out what he needs in the rank one case by borrowing ideas from mixed Hodge theory.

##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 58A14 Hodge theory in global analysis
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