Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves
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- Bull. Amer. Math. Soc. 26 (1992), 310-314 Request permission
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.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 26 (1992), 310-314
- MSC (2000): Primary 14C22; Secondary 14C30, 14F35, 14F40, 14J05
- DOI: https://doi.org/10.1090/S0273-0979-1992-00283-5
- MathSciNet review: 1129312