Address:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Departamento Matemática, IST, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
E-mail:
indranil@math.tifr.res.in
carlos.florentino@tecnico.ulisboa.pt
Abstract: Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\,\subset \, G$ be a maximal compact subgroup. Let $X$, $Y$ be irreducible smooth complex projective varieties and $f\colon X\rightarrow Y$ an algebraic morphism, such that $\pi _1(Y)$ is virtually nilpotent and the homomorphism $f_*\colon \pi _1(X)\rightarrow \pi _1(Y)$ is surjective. Define {\mathcal{R} }^f\big (\pi _1(X), G\big )&= \lbrace \rho \in \operatorname{Hom}\big (\pi _1(X), G\big ) \mid A\circ \rho \ \text{factors} \text{through} ~ f_*\rbrace \,,\\[6pt] {\mathcal{R} }^f\big (\pi _1(X), K\big )&= \lbrace \rho \in \operatorname{Hom}\big (\pi _1(X), K\big ) \mid A\circ \rho \ \text{factors} \text{through} ~ f_*\rbrace \,, where $A\colon G\rightarrow \operatorname{GL}(\operatorname{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal{R} }^f(\pi _1(X, x_0),\, G)/\!\!/G$ admits a deformation retraction to ${\mathcal{R} }^f(\pi _1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.
AMSclassification: primary 14J60.
Keywords: Higgs bundle, flat connection, representation space, deformation retraction.
DOI: 10.5817/AM2015-4-191