## 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 $$Z^1(\Gamma,\mathfrak g_{\operatorname{ad} \rho})$$ 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 $$\operatorname{ad} P$$ valued exterior differential forms on $$M$$ associated with the connection on $$P$$.

### MSC:

 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14D15 Formal methods and deformations in algebraic geometry 58H15 Deformations of general structures on manifolds
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### References:

 [1] Arms, J., Marsden, J., andMoncrief, V., Symmetry and bifurcation of momentum mappings,Commun. Math. Phys.,78 (1981), 455–478. · Zbl 0486.58008 [2] Artin, M., On solutions to analytic equations,Inv. Math.,5 (1968), 277–291. · Zbl 0172.05301 [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 [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 [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 [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 [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 [29] Nijenhuis, A. andRichardson, R. W., Cohomology and deformations in graded Lie algebras,Bull. A.M.S.,72 (1966), 1–29. · Zbl 0136.30502 [30] Nomizu, K., On the cohomology ring of compact homogeneous spaces of nilpotent Lie groups,Ann. Math.,59 (1954), 531–538. · Zbl 0058.02202 [31] Schlessinger, M., Functors of Artin rings,Trans. A.M.S.,130 (1968), 208–222. · Zbl 0167.49503 [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 [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 [41] Zucker, S., Locally homogeneous variations of Hodge structure,L’Ens. Math.,27 (1981), 243–276. · Zbl 0584.14003
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