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RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

Published online by Cambridge University Press:  15 February 2018

Stefan Papadima
Affiliation:
Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania (Stefan.Papadima@imar.ro)
Alexander I. Suciu
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115, USA (a.suciu@northeastern.edu) URL: web.northeastern.edu/suciu/

Abstract

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

Type
Research Article
Copyright
© Cambridge University Press 2018

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Footnotes

The first author’s work was partially supported by the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III. The second author was partially supported by the Simons Foundation collaboration grant for mathematicians 354156.

Deceased 10 January 2018.

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