Higgs bundles and representation spaces associated to morphisms

TL;DR

This paper studies the topology of certain representation spaces associated with morphisms between complex projective varieties, showing they deformation retract to spaces related to maximal compact subgroups, thus revealing their topological structure.

Contribution

It proves that specific representation spaces related to morphisms and almost commuting elements deformation retract to their compact subgroup counterparts, extending understanding of their topological properties.

Findings

Representation spaces deformation retract to compact subgroup spaces.

Topological structure of representation spaces clarified.

Deformation retraction results apply to spaces of almost commuting elements.

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 $f: X \rightarrow Y$ an algebraic morphism, such that $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_* : \pi_1(X) \rightarrow\pi_1(Y)$ is surjective. Define $$ {\mathcal R }^f(\pi_1(X),\, G)\,=\, \{\rho\, \in\, \text{Hom}(\pi_1(X),\, G)\, \mid\, A\circ\rho \ \text{ factors through }~ f_*\}\, , $$ $$ {\mathcal R }^f(\pi_1(X),\, K)\,=\, \{\rho\, \in\, \text{Hom}(\pi_1(X),\, K)\, \mid\, A\circ\rho \ \text{ factors through }~ f_*\}\, , $$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ 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 $K$.