## 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

Zbl 1159.14002
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