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- W2783455690 abstract "Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {rm 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 $ {rm DRep}_G(X) $ to the representation homology $ {rm HR}_*(X,G) := pi_*{mathcal O}[{rm 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 f.g. virtually free group $ Gamma $, we show that $, {rm HR}_i({rm B}Gamma, G) = 0 ,$ for all $ i > 0 $. For a closed Riemann surface $Sigma_g $ of genus $ g ge 1 $, we have $, {rm HR}_i(Sigma_g, G) = 0 ,$ for all $ i > dim G $. The sharp vanishing bounds for $ Sigma_g $ depend actually on the genus: we conjecture that if $ g = 1 $, then $, {rm HR}_i(Sigma_g, G) = 0 ,$ for $ i > {rm rank},G $, and if $ g ge 2 $, then $, {rm HR}_i(Sigma_g, G) = 0 ,$ 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 $ {rm 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 $ {rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology." @default.
- W2783455690 created "2018-01-26" @default.
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- W2783455690 date "2019-02-06" @default.
- W2783455690 modified "2023-09-26" @default.
- W2783455690 title "Vanishing theorems for representation homology and the derived cotangent complex" @default.
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- W2783455690 doi "https://doi.org/10.2140/agt.2019.19.281" @default.
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