|
Non-abelian local invariant cycles
Author(s):
Yen-lung
Tsai;
Eugene
Z.
Xia
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2365-2367.
MSC (2000):
Primary 14D05, 20F34, 55N20
Posted:
March 22, 2007
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let  be a degeneration of Kähler manifolds. The local invariant cycle theorem states that for a smooth fiber of the degeneration, any cohomology class, invariant under the monodromy action, comes from a global cohomology class. Instead of the classical cohomology, one may consider the non-abelian cohomology. This note demonstrates that the analogous non-abelian version of the local invariant cycle theorem does not hold if the first non-abelian cohomology is the moduli space (universal categorical quotient) of the representations of the fundamental group.
References:
-
- 1.
- Jørgen Ellegaard Andersen, Fixed points of the mapping class group in the
 moduli spaces, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1511-1515. MR 1376748 (97j:58014) - 2.
- C. H. Clemens, Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), no. 2, 215-290. MR 0444662 (56:3012)
- 3.
- Vik. S. Kulikov and P. F. Kurchanov, Complex algebraic varieties: periods of integrals and Hodge structures, Algebraic geometry, III, Encyclopaedia Math. Sci., vol. 36, Springer, Berlin, 1998, pp. 1-217, 263-270. MR 1602375
- 4.
- Alexander Lubotzky and Andy R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985), no. 336, xi+117. MR 0818915 (87c:20021)
- 5.
- David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin, 1965.MR 0214602 (35:5451)
- 6.
- Ulf Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11 (1977), no. 189, xv+144. MR 0466149 (57:6030)
- 7.
- Joseph P. Previte and Eugene Z. Xia, Topological dynamics on moduli spaces. I, Pacific J. Math. 193 (2000), no. 2, 397-417.MR 1755824 (2001e:53090)
- 8.
- -, Topological dynamics on moduli spaces. II, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2475-2494 (electronic). MR 1885660 (2002k:53167)
- 9.
- Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319. MR 0382272 (52:3157)
- 10.
- Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229-257. MR 0429885 (55:2894)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
14D05, 20F34, 55N20
Retrieve articles in all Journals with MSC
(2000):
14D05, 20F34, 55N20
Additional Information:
Yen-lung
Tsai
Affiliation:
Department of Mathematical Sciences, National Chengchi University, Taipei 116, Taiwan
Email:
yenlung@math.nccu.edu.tw
Eugene
Z.
Xia
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Email:
ezxia@ncku.edu.tw
DOI:
10.1090/S0002-9939-07-08843-0
PII:
S 0002-9939(07)08843-0
Received by editor(s):
December 6, 2004
Received by editor(s) in revised form:
April 18, 2006
Posted:
March 22, 2007
Additional Notes:
Tsai is partially supported by the National Center for Theoretical Sciences, Hsinchu, Taiwan; Xia gratefully acknowledges partial support by National Science Council Taiwan grant NSC 93-2115-M-006-002.
Communicated by:
Michael Stillman
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org