Singularity of the varieties of representations of lattices in solvable Lie groups. (English) Zbl 1347.22008

Kasuya, Hisashi

J. Topol. Anal. 8, No. 2, 273-285 (2016).

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 \(\nu+1\).

Reviewer: Gabriela Paola Ovando (Rosario)

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

Keywords:

variety of representations; Kuranishi space of differential graded Lie algebra; solvmanifold

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’);