## Dehn twists and invariant classes

HTML articles powered by AMS MathViewer

- 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.## References

- Jørgen Ellegaard Andersen,
*Fixed points of the mapping class group in the $\textrm {SU}(n)$ moduli spaces*, Proc. Amer. Math. Soc.**125**(1997), no. 5, 1511–1515. MR**1376748**, DOI 10.1090/S0002-9939-97-03788-X - C. H. Clemens,
*Degeneration of Kähler manifolds*, Duke Math. J.**44**(1977), no. 2, 215–290. MR**444662** - Clifford J. Earle and Patricia L. Sipe,
*Families of Riemann surfaces over the punctured disk*, Pacific J. Math.**150**(1991), no. 1, 79–96. MR**1120713** - William M. Goldman,
*Invariant functions on Lie groups and Hamiltonian flows of surface group representations*, Invent. Math.**85**(1986), no. 2, 263–302. MR**846929**, DOI 10.1007/BF01389091 - William M. Goldman,
*Topological components of spaces of representations*, Invent. Math.**93**(1988), no. 3, 557–607. MR**952283**, DOI 10.1007/BF01410200 - William M. Goldman,
*Ergodic theory on moduli spaces*, Ann. of Math. (2)**146**(1997), no. 3, 475–507. MR**1491446**, DOI 10.2307/2952454 - Larsen, Michael, Rigidity in the invariant theory of compact groups. arXiv:math/0212193.
- Alexander Lubotzky and Andy R. Magid,
*Varieties of representations of finitely generated groups*, Mem. Amer. Math. Soc.**58**(1985), no. 336, xi+117. MR**818915**, DOI 10.1090/memo/0336 - D. Mumford, J. Fogarty, and F. Kirwan,
*Geometric invariant theory*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR**1304906**, DOI 10.1007/978-3-642-57916-5 - Chris A. M. Peters and Joseph H. M. Steenbrink,
*Mixed Hodge structures*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer-Verlag, Berlin, 2008. MR**2393625** - Doug Pickrell and Eugene Z. Xia,
*Ergodicity of mapping class group actions on representation varieties. I. Closed surfaces*, Comment. Math. Helv.**77**(2002), no. 2, 339–362. MR**1915045**, DOI 10.1007/s00014-002-8343-1 - A. Ramanathan,
*Stable principal bundles on a compact Riemann surface*, Math. Ann.**213**(1975), 129–152. MR**369747**, DOI 10.1007/BF01343949 - Wilfried Schmid,
*Variation of Hodge structure: the singularities of the period mapping*, Invent. Math.**22**(1973), 211–319. MR**382272**, DOI 10.1007/BF01389674 - Joseph Steenbrink,
*Limits of Hodge structures*, Invent. Math.**31**(1975/76), no. 3, 229–257. MR**429885**, DOI 10.1007/BF01403146 - Yen-Lung Tsai and Eugene Z. Xia,
*Non-abelian local invariant cycles*, Proc. Amer. Math. Soc.**135**(2007), no. 8, 2365–2367. MR**2302557**, DOI 10.1090/S0002-9939-07-08843-0

## 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