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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dehn twists and invariant classes


Author: Eugene Z. Xia
Journal: Proc. Amer. Math. Soc. 140 (2012), 1173-1183
MSC (2010): Primary 14D05, 20F34, 55N20
Published electronically: September 27, 2011
MathSciNet review: 2869103
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Abstract: A degeneration of compact Kähler manifolds gives rise to a monodromy action on the Betti moduli space

$\displaystyle \mathsf {H}^1(X, G) = \Hom (\pi _1(X),G)/G$

over smooth fibres with a complex algebraic structure group $ G$ that is either abelian or reductive. Assume that the singularities of the central fibre are of normal crossing. When $ G = \mathbb{C}$, the invariant cohomology classes arise from the global classes. This is no longer true in general. In this paper, we produce large families of locally invariant classes that do not arise from global ones for reductive $ G$. These examples exist even when $ G$ is abelian as long as $ G$ contains multiple torsion points. Finally, for general $ G$, we make a new conjecture on local invariant classes and produce some suggestive examples.

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Additional Information

Eugene Z. Xia
Affiliation: Department of Mathematics, National Cheng Kung University and National Center for Theoretical Sciences, Tainan 701, Taiwan
Email: ezxia@ncku.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11279-6
PII: S 0002-9939(2011)11279-6
Received by editor(s): July 15, 2010
Received by editor(s) in revised form: December 25, 2010
Published electronically: September 27, 2011
Additional Notes: The author gratefully acknowledges partial support by the National Science Council, Taiwan, with grants 96-2115-M-006-002 and 97-2115-M-006-001-MY3.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.