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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Dehn twists and invariant classes
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by Eugene Z. Xia PDF
Proc. Amer. Math. Soc. 140 (2012), 1173-1183 Request permission

Abstract:

A degeneration of compact Kähler manifolds gives rise to a monodromy action on the Betti moduli space \[ \mathsf {H}^1(X, G) = \textrm {Hom}(\pi _1(X),G)/G\] 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
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1173-1183
  • MSC (2010): Primary 14D05, 20F34, 55N20
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11279-6
  • MathSciNet review: 2869103