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Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves. (English) Zbl 0759.14016

Summary: Let \(X\) be a compact Kähler manifold. The set \(\text{char}(X)\) of one- dimensional complex valued characters of the fundamental group of \(X\) forms an algebraic group. Consider the subset of \(\text{char}(X)\) consisting of those characters for which the corresponding local system has nontrivial cohomology in a given degree \(d\). This set is shown to be a union of finitely many components that are translates of algebraic subgroups of \(\text{char}(X)\). When the degree \(d\) equals 1, it is shown that some of these components are pullbacks of the character varieties of curves under holomorphic maps. As a corollary, it is shown that the number of equivalence classes (under a natural equivalence relation) of holomorphic maps, with connected fibers, of \(X\) onto smooth curves of a fixed genus \(>1\) is a topological invariant of \(X\). In fact it depends only on the fundamental group of \(X\).

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

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