Higgs bundles and representation spaces associated to morphisms

Biswas, Indranil, and Florentino, Carlos. "Higgs bundles and representation spaces associated to morphisms." Archivum Mathematicum 051.4 (2015): 191-199. <http://eudml.org/doc/276268>.

@article{Biswas2015,
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 \begin\{align*\} \{\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 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 through \}~ f\_*\rbrace \,, \end\{align*\} 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 $K$.},
author = {Biswas, Indranil, Florentino, Carlos},
journal = {Archivum Mathematicum},
keywords = {Higgs bundle; flat connection; representation space; deformation retraction},
language = {eng},
number = {4},
pages = {191-199},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Higgs bundles and representation spaces associated to morphisms},
url = {http://eudml.org/doc/276268},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Biswas, Indranil
AU - Florentino, Carlos
TI - Higgs bundles and representation spaces associated to morphisms
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 4
SP - 191
EP - 199
AB - 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 \begin{align*} {\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 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 through }~ f_*\rbrace \,, \end{align*} 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 $K$.
LA - eng
KW - Higgs bundle; flat connection; representation space; deformation retraction
UR - http://eudml.org/doc/276268
ER -