×

Vanishing theorems for representation homology and the derived cotangent complex. (English) Zbl 1436.14006

Summary: Let \(G\) be a reductive affine algebraic group defined over a field \(k\) of characteristic zero. We study the cotangent complex of the derived \(G\)-representation scheme \( \mathrm{DRep}_G(X)\) of a pointed connected topological space \(X\). We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of \(\mathrm{DRep}_G(X)\) to the representation homology \(\mathrm{HR}_\ast(X,G) := \pi_\ast {\mathcal O}[\mathrm{DRep}_G(X)] \) to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in \(\mathbb{R}^3\) and generalized lens spaces. In particular, for any finitely generated virtually free group \(\Gamma\), we show that \(\mathrm{HR}_i(\mathrm{B}\Gamma, G) = 0 \) for all \( i > 0 \). For a closed Riemann surface \(\Sigma_g \) of genus \( g \ge 1 \), we have \(\mathrm{HR}_i(\Sigma_g, G) = 0 \) for all \( i > \dim G \). The sharp vanishing bounds for \(\Sigma_g\) actually depend on the genus: we conjecture that if \(g = 1\), then \( \mathrm{HR}_i(\Sigma_g, G) = 0 \) for \(i > \mathrm{rank}\ G \), and if \(g \ge 2\), then \(\mathrm{HR}_i(\Sigma_g, G) = 0 \) for \(i > \dim\mathcal{Z}(G) \), where \( \mathcal{Z}(G) \) is the center of \(G\). We prove these bounds locally on the smooth locus of the representation scheme \(\mathrm{Rep}_G[\pi_1(\Sigma_g)]\) in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined \(K\)-theoretic virtual fundamental class for \(\mathrm{DRep}_G(X)\) in the sense of I. Ciocan-Fontanine and M. Kapranov [Geom. Topol. 13, No. 3, 1779–1804 (2009; Zbl 1159.14002)]. We give a new “Tor formula” for this class in terms of functor homology.

MSC:

14A30 Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.)
14A20 Generalizations (algebraic spaces, stacks)
14D20 Algebraic moduli problems, moduli of vector bundles
14L24 Geometric invariant theory
57M07 Topological methods in group theory
14F17 Vanishing theorems in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry
14L17 Affine algebraic groups, hyperalgebra constructions

Citations:

Zbl 1159.14002
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] 10.1007/978-3-642-51449-4
[2] 10.4171/JEMS/729 · Zbl 1406.16014
[3] 10.1090/conm/607/12078
[4] 10.1016/j.aim.2013.06.020 · Zbl 1291.14006
[5] 10.1515/crelle-2014-0001 · Zbl 1428.16006
[6] 10.1007/978-3-540-38117-4
[7] 10.1016/0040-9383(72)90024-9 · Zbl 0202.22803
[8] 10.1007/978-1-4684-9327-6
[9] 10.2140/gt.2009.13.1779 · Zbl 1159.14002
[10] 10.1007/978-3-319-31580-5_15 · Zbl 1432.22011
[11] 10.2307/2006973 · Zbl 0529.57005
[12] 10.2307/2152819 · Zbl 0838.19001
[13] 10.1016/B978-044481779-2/50003-1
[14] 10.1007/978-3-0348-8707-6
[15] 10.1016/0001-8708(84)90040-9 · Zbl 0574.32032
[16] 10.1007/BFb0075218
[17] 10.2307/1993575 · Zbl 0106.25703
[18] 10.1090/conm/436/08410
[19] 10.1007/978-1-4899-6664-3_3
[20] 10.2307/1970006 · Zbl 0091.36901
[21] 10.2307/1970042 · Zbl 0091.36902
[22] 10.2307/2372755 · Zbl 0109.16201
[23] 10.1016/S0022-4049(99)00109-7 · Zbl 0972.18012
[24] 10.1016/j.jalgebra.2005.05.012 · Zbl 1072.14060
[25] 10.1007/978-3-662-11389-9
[26] 10.1090/memo/0336
[27] 10.1515/9781400830558 · Zbl 1175.18001
[28] ; May, Simplicial objects in algebraic topology. Simplicial objects in algebraic topology. Van Nostrand Mathematical Studies, 11, (1967) · Zbl 0165.26004
[29] 10.1007/s10240-013-0054-1 · Zbl 1328.14027
[30] 10.1016/S0012-9593(00)00107-5 · Zbl 0957.18004
[31] 10.1112/plms/pdn004 · Zbl 1162.55007
[32] 10.2140/gt.2013.17.1417 · Zbl 1274.14003
[33] 10.1016/j.aim.2013.01.009 · Zbl 1328.14028
[34] 10.1007/BFb0097438 · Zbl 0168.20903
[35] 10.2307/1970725 · Zbl 0191.53702
[36] ; Quillen, Applications of categorical algebra, 65, (1970)
[37] ; Richardson, Compositio Math., 38, 311, (1979)
[38] 10.1016/S0022-4049(96)00058-8 · Zbl 0888.55010
[39] ; Shalen, Handbook of geometric topology, 955, (2002)
[40] 10.1090/S0002-9947-2012-05448-1 · Zbl 1291.14022
[41] ; Thomas, J. Homotopy Relat. Struct., 3, 359, (2008)
[42] 10.1215/S0012-7094-92-06817-7 · Zbl 0813.19002
[43] 10.4171/EMSS/4 · Zbl 1314.14005
[44] ; Toën, Proceedings of the International Congress of Mathematicians, 769, (2014)
[45] 10.1016/j.aim.2004.05.004 · Zbl 1120.14012
[46] 10.1090/memo/0902
[47] 10.1017/CBO9781139644136
[48] 10.2307/1970495 · Zbl 0192.12802
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.