Higgs Bundles and Representation Spaces Associated to Morphisms

Let G be a connected reductive affine algebraic group defined over the complex numbers, and K ⊂ G be a maximal compact subgroup. Let X, Y be irreducible smooth complex projective varieties and f : X → Y an algebraic morphism, such that π1(Y ) is virtually nilpotent and the homomorphism f∗ : π1(X)→ π1(Y ) is surjective. Define R ( π1(X), G ) = {ρ ∈ Hom ( π1(X), G ) | A ◦ ρ factors through f∗} , R ( π1(X),K ) = {ρ ∈ Hom ( π1(X),K ) | A ◦ ρ factors through f∗} , where A : G→ GL(Lie(G)) is the adjoint action. We prove that the geometric invariant theoretic quotient Rf (π1(X,x0), G)//G admits a deformation retraction to Rf (π1(X,x0), 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.