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Singularity of the varieties of representations of lattices in solvable Lie groups. (English) Zbl 1347.22008
Let $$\Gamma$$ be a lattice in a simply connected solvable Lie group $$G$$. Then the manifold $$G/\Gamma$$ is an aspherical manifold with fundamental group $$\Gamma$$. The purpose of this work is the study of the analytic germ $$(R(\Gamma,A),1)$$ at the trivial representation $$1$$ for $$A$$ a linear algebraic group. This can be studied by the differential graded algebra (DGA) $$A^\ast (G/\Gamma)$$ of differential forms of $$G/\Gamma$$. Let $$\mathfrak g$$ denote a solvable Lie algebra. Then the nilshadow $$\mathfrak u$$ is uniquely determined by $$\mathfrak g$$. Then one has a sub-DGA $$A^\ast_\Gamma \subset A^\ast (G/\Gamma) \ltimes \mathbb C$$, which induces an isomorphism in cohomolgy and such that $$A^\ast_\Gamma$$ can be regarded as a sub-DGA of $$\Lambda \mathfrak u^\ast \otimes \mathbb C$$. Assume that $$\mathfrak u$$ is $$\nu$$-step naturally graded and that $$A$$ is a linear algebraic group. Then a main theorem states that $$(R(\Gamma,A),1)$$ is cut out by polynomial equations of degree at most $$\nu+1$$.

##### MSC:
 22E25 Nilpotent and solvable Lie groups 32B10 Germs of analytic sets, local parametrization 22E40 Discrete subgroups of Lie groups 17B55 Homological methods in Lie (super)algebras
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##### References:
 [1] 1. C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology27 (1988) 513-518. genRefLink(16, ’S1793525316500114BIB001’, ’10.1016%252F0040-9383%252888%252990029-8’); [2] 2. L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications. Part I. Basic Theory and Examples. Cambridge Studies in Advanced Mathematics, Vol. 18 (Cambridge Univ. Press, 1990). · Zbl 0704.22007 [3] 3. K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures, Forum Math.12 (2000) 77-96. · Zbl 0946.22009 [4] 4. A. Dimca and S. Papadima, Nonabelian cohomology jump loci from an analytic viewpoint, Commun. Contemp. Math.16 (2014) 1350025. [Abstract] · Zbl 1315.14006 [5] 5. N. Dungey, A. F. M. ter Elst and D. W. Robinson, Analysis on Lie Groups with Polynomial Growth (Birkhäuser, 2003). genRefLink(16, ’S1793525316500114BIB005’, ’10.1007%252F978-1-4612-2062-6’); [6] 6. W. M. Goldman and J. J. Millson, The deformation theory of representations of fundamental groups of compact Khler manifolds, Inst. Hautes tudes Sci. Publ. Math. No.67 (1988) 43-96. genRefLink(16, ’S1793525316500114BIB006’, ’10.1007%252FBF02699127’); · Zbl 0678.53059 [7] 7. W. M. Goldman and J. J. Millson, The homotopy invariance of the Kuranishi space, Illinois J. Math.34 (1990) 337-367. · Zbl 0707.32004 [8] 8. A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I8 (1960) 289-331. · Zbl 0099.18003 [9] 9. H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom.93 (2013) 269-298. · Zbl 1373.53069 [10] 10. H. Kasuya, The Frolicher spectral sequences of certain solvmanifolds, J. Geom. Anal.25 (2015) 317-328. genRefLink(16, ’S1793525316500114BIB010’, ’10.1007%252Fs12220-013-9429-2’); · Zbl 1309.22009 [11] 11. H. Kasuya, de Rham and Dolbeault cohomology of solvmanifolds with local systems, Math. Res. Lett.21 (2014) 781-805. genRefLink(16, ’S1793525316500114BIB011’, ’10.4310%252FMRL.2014.v21.n4.a10’); · Zbl 1314.17010 [12] 12. M. S. Raghunathan, Discrete Subgroups of Lie Groups (Springer-Verlag, 1972). genRefLink(16, ’S1793525316500114BIB012’, ’10.1007%252F978-3-642-86426-1’); · Zbl 0254.22005 [13] 13. H. Sawai, A construction of lattices on certain solvable Lie groups, Topology Appl.154 (2007) 3125-3134. genRefLink(16, ’S1793525316500114BIB013’, ’10.1016%252Fj.topol.2007.08.006’); · Zbl 1138.53065 [14] 14. C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. No.75 (1992) 5-95. genRefLink(16, ’S1793525316500114BIB014’, ’10.1007%252FBF02699491’);
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