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Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds. (English) Zbl 1440.14119

For a path-connected space \(X\), having the homotopy type of a finite CW-complex and a complex, linear algebraic group \(G\), the representation variety \(\mathrm{Hom}(\pi_{1}(X),G)\) is the parameter space for locally constant sheaves on \(X\) with monodromies, factoring through the group \(G\).
The jump loci are the characteristic varieties of \(X\) w.r.t. a certain rational representation \(i\colon G\to \mathrm{GL}(V)\): \[ \mathcal V_{r}^{k}=\{\rho\in\mathrm{Hom}(\pi_{1}(X),G)\,|\,\dim H^{k}(X,V_{i\rho})\geq r\}, \] where degree \(k\geq 0\) and depth \(r\geq 0\). They describe variation of twisted cohomology inside the parameter space and carry important information about the topology of the space \(X\) and that of its covering spaces.
A basic problem of the deformation theory with homological constraints is the study of analytic germs of embedded cohomology jump loci. This paper is devoted to studying the germs at the origin of \(G\)-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. The authors relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on Morgan-Dupont models, associated to convenient compactifications of such manifolds.
In the case when \(G\) is either \(\mathrm{SL}_{2}(\mathbb{C})\) or its standard Borel subgroup and the depth of the jump locus is 1, this approach allows the authors to describe explicit irreducible decompositions for the germs of the embedded jump loci. However, when either \(G=\mathrm{SL}_{n}(\mathbb{C})\) for \(n\geq 3\), or the depth is \(>1\), certain natural inclusions of germs are strict.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
55N25 Homology with local coefficients, equivariant cohomology
20C15 Ordinary representations and characters
55P62 Rational homotopy theory
22E46 Semisimple Lie groups and their representations
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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References:

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