Dehn twists and invariant classes

Xia, Eugene

doi:10.1090/S0002-9939-2011-11279-6

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.

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