| Title:
|
Higgs bundles and representation spaces associated to morphisms (English) |
| Author:
|
Biswas, Indranil |
| Author:
|
Florentino, Carlos |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
191-199 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $f\colon X\to Y$ an algebraic morphism, such that $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_*\colon \pi_1(X)\to\pi_1(Y)$ is surjective. Define \begin{align*} {\mathcal R }^f\big(\pi_1(X), G\big)&= \{\rho \in \operatorname{Hom}\big(\pi_1(X), G\big) \mid A\circ\rho \ \text{ factors through }~ f_*\}\,,\\[6pt] {\mathcal R }^f\big(\pi_1(X), K\big)&= \{\rho \in \operatorname{Hom}\big(\pi_1(X), K\big) \mid A\circ\rho \ \text{ factors through }~ f_*\}\,, \end{align*} where $A\colon G\to \operatorname{GL}(\operatorname{Lie}(G))$ 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 $K$. (English) |
| Keyword:
|
Higgs bundle |
| Keyword:
|
flat connection |
| Keyword:
|
representation space |
| Keyword:
|
deformation retraction |
| MSC:
|
14J60 |
| idZBL:
|
Zbl 06537724 |
| idMR:
|
MR3434602 |
| DOI:
|
10.5817/AM2015-4-191 |
| . |
| Date available:
|
2015-11-30T09:56:39Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144478 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| . |
