zbMATH — the first resource for mathematics

The deformation theory of representations of fundamental groups of compact Kähler manifolds. (English) Zbl 0678.53059
Let \(\Gamma\) be the fundamental group of a compact Kähler manifold M and G a real algebraic Lie group. The set \({\mathcal R}={\mathcal R}(\Gamma,G)\) of the representations \(\Gamma\) \(\to G\) is naturally an affine variety. In the present paper the authors study the local structure of \({\mathcal R}\) at a point \(\rho\) under various conditions. It is proved that \({\mathcal R}\) is locally isomorphic to a cone defined by a finite number of quadratic equations in \(Z^ 1(\Gamma,{\mathfrak g}_{ad \rho})\) of the space of Lie algebra valued 1-cocycles. These equations are determined by the cup product induced by the Lie product (the obstruction map). This includes the following cases: (i) G is compact, (ii) \(\rho\) is the monodromy of a variation of Hodge structure on M, and (iii) G is the automorphism group of a Hermitian symmetric space with certain condition.
This deformation problem of \(\rho\) is equivalent as groupoids, to the deformation problem of flat connections of an associated principal bundle P. This latter in turn is equivalent to that of ad P valued exterior differential forms on M associated with the connection on P.
Reviewer: E.Horikawa

53C55 Global differential geometry of Hermitian and Kählerian manifolds
14D15 Formal methods and deformations in algebraic geometry
58H15 Deformations of general structures on manifolds
Full Text: DOI Numdam EuDML
[1] Arms, J., Marsden, J., andMoncrief, V., Symmetry and bifurcation of momentum mappings,Commun. Math. Phys.,78 (1981), 455–478. · Zbl 0486.58008 · doi:10.1007/BF02046759
[2] Artin, M., On solutions to analytic equations,Inv. Math.,5 (1968), 277–291. · Zbl 0172.05301 · doi:10.1007/BF01389777
[3] Atiyah, M. F. andBott, R., The Yang-Mills equations over a compact Riemann surface,Phil. Trans. Roy. Soc. London,A 308 (1982), 523–615.
[4] Chern, S. S., Geometry of characteristic classes, inProceedings of the Thirteenth Biennial Seminar, Canad. Math. Cong., Montreal (1972), 1–40.
[5] Corlette, K., Flat G-bundles with canonical metrics,J. Diff. Geo. (to appear). · Zbl 0676.58007
[6] Corlette, K., Gauge theory and representations of Kähler groups, inThe Geometry of Group Representations (Proceedings of Amer. Math. Soc. Summer Conference 1987, Boulder, Colorado),Contemp. Math. (to appear).
[7] Corlette, K.,Rigid monodromy representations (in preparation). · Zbl 0731.53061
[8] Deligne, P., Griffiths, P. A., Morgan, J. W. andSullivan, D., Rational homotopy type of compact Kähler manifolds,Inv. Math.,29 (1975), 245–274. · Zbl 0312.55011 · doi:10.1007/BF01389853
[9] Donaldson, S. K., Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles,Proc. Lond. Math. Soc.,50 (1985), 1–26. · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1
[10] Goldman, W. M., Representations of fundamental groups of surfaces, inGeometry and Topology, Proceedings, University of Maryland 1983–1984,J. Alexander andJ. Harer (eds.),Lecture Notes in Mathematics,1167, Berlin-Heidelberg-New York, Springer-Verlag (1985), 95–117.
[11] Goldman, W. M., Topological components of spaces of representations,Inv. Math. (to appear). · Zbl 0655.57019
[12] Goldman, W. M., andMillson, J. J., Local rigidity of discrete groups acting on complex hyperbolic space,Inv. Math.,88 (1987), 495–520. · Zbl 0627.22012 · doi:10.1007/BF01391829
[13] Goldman, W. M. andMillson, J. J., Deformations of flat bundles over Kähler manifolds, inGeometry and Topology, Manifolds, Varieties and Knots,C. McCrory andT. Shifrin (eds.),Lecture Notes in Pure and Applied Mathematics,105, Marcel Dekker, New York-Basel (1987), 129–145.
[14] Goldman, W. M. andMillson, J. J., Differential graded Lie algebras and singularities of level sets of momentum mapping, (submitted for publication).
[15] Greub, W., Halperin, S. andVanstone, R.,Connections, Curvature and Cohomology, Vol. II, Pure and Applied Mathematics,47, New York-London, Academic Press (1973).
[16] Griffiths, P. A. et al., Topics in Transcendental Algebraic Geometry, Ann. of Math. Studies,106 (1984), Princeton, New Jersey, Princeton Univ. Press. · Zbl 0528.00004
[17] Gunning, R. C.,Complex Analytic Varieties: the Local Parametrization Theorem, Mathematical Notes, Princeton University Press (1970). · Zbl 0213.35904
[18] Gunning, R. C. andRossi, H.,Analytic functions of several complex variables, Englewood Cliffs, New Jersey, Prentice-Hall (1965). · Zbl 0141.08601
[19] Jacobson, N.,Basic Algebra II, San Francisco, W. H. Freeman and Company (1980). · Zbl 0441.16001
[20] Johnson, D. andMillson, J., Deformation spaces associated to compact hyperbolic manifolds, inDiscrete Groups in Geometry and Analysis, Papers in Honor of G. D. Mostow on His Sixtieth Birthday,R. Howe (ed.),Progress in Mathematics,67, Boston-Basel-Stuttgart, Birkhäuser (1987), 48–106.
[21] Kobayashi, S.,Differential Geometry of Holomorphic Vector Bundles, Princeton University Press and Mathematical Society of Japan (1987). · Zbl 0708.53002
[22] Kobayashi, S. andNomizu, K.,Foundations of Differential Geometry, Volume 1, Interscience Tracts in Pure and Applied Mathematics,15 (1963), New York-London, John Wiley & Sons.
[23] Kunz, E.,Introduction to Commutative Algebra and Algebraic Geometry (1985), Birkhäuser Boston, Inc. · Zbl 0563.13001
[24] Lubotzky, A. andMagid, A.,Varieties of representations of finitely generated groups, Memoirs A.M.S.,336 (vol. 5) (1985).
[25] Morgan, J. W. andShalen, P. B., Valuations, trees and degenerations of hyperbolic structures I,Ann. Math.,120 (1984), 401–476. · Zbl 0583.57005 · doi:10.2307/1971082
[26] Mumford, D.,Introduction to Algebraic Geometry, Harvard University lecture notes. · Zbl 0114.13106
[27] Nadel, A. M.,Singularities and Kodaira dimension of the moduli space of flat Hermitian Yang-Mills connections, Harvard University preprint. · Zbl 0652.32017
[28] Nijenhuis, A. andRichardson, R. W., Cohomology and deformation of algebraic structures,Bull. A.M.S.,70 (1964), 406–411. · Zbl 0138.26301 · doi:10.1090/S0002-9904-1964-11117-4
[29] Nijenhuis, A. andRichardson, R. W., Cohomology and deformations in graded Lie algebras,Bull. A.M.S.,72 (1966), 1–29. · Zbl 0136.30502 · doi:10.1090/S0002-9904-1966-11401-5
[30] Nomizu, K., On the cohomology ring of compact homogeneous spaces of nilpotent Lie groups,Ann. Math.,59 (1954), 531–538. · Zbl 0058.02202 · doi:10.2307/1969716
[31] Schlessinger, M., Functors of Artin rings,Trans. A.M.S.,130 (1968), 208–222. · Zbl 0167.49503 · doi:10.1090/S0002-9947-1968-0217093-3
[32] Schlessinger, M. andStasheff, J.,Deformation theory and rational homotopy type, University of North Carolina preprint, 1979.
[33] Simpson, C. T.,Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, Princeton University preprint. · Zbl 0669.58008
[34] Simpson, C. T.,Higgs bundles and local systems, Princeton University preprint. · Zbl 0814.32003
[35] Steenrod, N. E.,The topology of fiber bundles, Princeton Mathematical Series 14 (1951), Princeton New Jersey, Princeton University Press. · Zbl 0054.07103
[36] Toledo, D., Representations of surface groups in PSU (1,n) with nonvanishing characteristic number,J. Diff. Geo. (to appear).
[37] Uhlenbeck, K. andYau, S. T., On the existence of Hermitian-Yang-Mills connections in stable vector bundles,Comm. Pure and Appl. Math.,39 (1986), 257–293. · Zbl 0615.58045 · doi:10.1002/cpa.3160390714
[38] Wells, R. O.,Differential Analysis on Complex Manifolds, Graduate Texts in Math.,65 (1980), Berlin-Heidelberg-New York, Springer-Verlag. · Zbl 0435.32004
[39] Whitney, H.,Complex Analytic Varieties (1972), Addison-Wesley Inc., Reading, Massachusetts. · Zbl 0265.32008
[40] Zucker, S., Hodge theory with degenerating coefficients,Ann. Math.,109 (1979), 415–476. · Zbl 0446.14002 · doi:10.2307/1971221
[41] Zucker, S., Locally homogeneous variations of Hodge structure,L’Ens. Math.,27 (1981), 243–276. · Zbl 0584.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.