Rank Two Topological and Infinitesimal Embedded Jump Loci of Quasi-Projective Manifolds

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 SL 2 pCq 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 " SL n pCq for some n ě 3, or the depth is greater than 1, then certain natural inclusions of germs are strict.

The Topology of Compact Lie Group Actions Through the Lens of Finite Models

“…We single out in Examples 8.5 and 8.6 several more classes of quasi-projective manifolds for which the conclusions of the theorem hold. Whether those conclusions hold in complete generality remains an open question, which is investigated further in [43,44].…”

“…In view of a recent result from [44], a positive answer to the global Question 8.4 would imply a positive answer to the local Question 8.3. Equalities (31) and (32) are known to hold for several interesting classes of quasi-projective manifolds M. Let W ‚ denote the Deligne weight filtration on H ‚ pMq.…”

“…Furthermore, according to Theorem 1.3 from [5], Question 8.4 has a positive answer also for partial configuration spaces of smooth projective curves. Question 8.4 is analyzed in much detail in follow-up work, [43,44]. In particular, in [44, Theorem 1.2], we reinterpret this question in terms of the local structure of the representation variety Hompπ 1 pMq, Gq and of the embedded cohomology jump loci V 1 1 pπ 1 pMq, ιq.…”

Naturality properties and comparison results for topological and infinitesimal embedded jump loci

“…Further applications of the techniques that go into proving the above results can be found in our recent preprint [29]. In particular, in the context of Theorem 1.3, parts (1)-( 2), but for an arbitrary complex linear algebraic group G, it is shown in [29, Theorem 1.1 (2)] that the germs f !…”