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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves

Author: Donu Arapura
Journal: Bull. Amer. Math. Soc. 26 (1992), 310-314
MSC (2000): Primary 14C22; Secondary 14C30, 14F35, 14F40, 14J05
MathSciNet review: 1129312
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Abstract: Let X be a compact Kähler manifold. The set $ \operatorname{char}(X)$ of one-dimensional complex valued characters of the fundamental group of X forms an algebraic group. Consider the subset of $ \operatorname{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 $ \operatorname{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.

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Article copyright: © Copyright 1992 American Mathematical Society

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