Algebraic & Geometric Topology

On the geometry and topology of partial configuration spaces of Riemann surfaces

Barbu Berceanu, Daniela Anca Măcinic, Ştefan Papadima, and Clement Popescu

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We examine complements (inside products of a smooth projective complex curve of arbitrary genus) of unions of diagonals indexed by the edges of an arbitrary simple graph. We use Orlik–Solomon models associated to these quasiprojective manifolds to compute pairs of analytic germs at the origin, both for rank-1 and rank-2 representation varieties of their fundamental groups, and for degree-1 topological Green–Lazarsfeld loci. As a corollary, we describe all regular surjections with connected generic fiber, defined on the above complements onto smooth complex curves of negative Euler characteristic. We show that the nontrivial part at the origin, for both rank-2 representation varieties and their degree-1 jump loci, comes from curves of general type via the above regular maps. We compute explicit finite presentations for the Malcev Lie algebras of the fundamental groups, and we analyze their formality properties.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 2 (2017), 1163-1188.

Dates
Received: 22 April 2016
Accepted: 7 July 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1508431458

Digital Object Identifier
doi:10.2140/agt.2017.17.1163

Mathematical Reviews number (MathSciNet)
MR3623686

Zentralblatt MATH identifier
1379.55015

Subjects
Primary: 55N25: Homology with local coefficients, equivariant cohomology 55R80: Discriminantal varieties, configuration spaces
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 20F38: Other groups related to topology or analysis

Keywords
partial configuration space smooth projective curve Gysin model admissible maps onto curves representation variety cohomology jump loci Malcev completion

Citation

Berceanu, Barbu; Măcinic, Daniela Anca; Papadima, Ştefan; Popescu, Clement. On the geometry and topology of partial configuration spaces of Riemann surfaces. Algebr. Geom. Topol. 17 (2017), no. 2, 1163--1188. doi:10.2140/agt.2017.17.1163. https://projecteuclid.org/euclid.agt/1508431458


Export citation

References

  • J,F Adams, On the cobar construction, from “Colloque de topologie algébrique”, Georges Thone, Liège (1957) 81–87
  • D Arapura, Geometry of cohomology support loci for local systems, I, J. Algebraic Geom. 6 (1997) 563–597
  • S Ashraf, H Azam, B Berceanu, Representation theory for the Križ model, Algebr. Geom. Topol. 14 (2014) 57–90
  • B Berceanu, Ş Papadima, Universal representations of braid and braid-permutation groups, J. Knot Theory Ramifications 18 (2009) 999–1019
  • R Bezrukavnikov, Koszul DG-algebras arising from configuration spaces, Geom. Funct. Anal. 4 (1994) 119–135
  • C Bibby, J Hilburn, Quadratic-linear duality and rational homotopy theory of chordal arrangements, Algebr. Geom. Topol. 16 (2016) 2637–2661
  • K,T Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831–879
  • P Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1971) 5–57
  • P Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974) 5–77
  • A Dimca, Characteristic varieties and logarithmic differential $1$–forms, Compos. Math. 146 (2010) 129–144
  • A Dimca, Ş Papadima, Non-abelian cohomology jump loci from an analytic viewpoint, Commun. Contemp. Math. 16 (2014) 1350025, 47
  • A Dimca, Ş Papadima, A,I Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009) 405–457
  • C Dupont, The Orlik–Solomon model for hypersurface arrangements, Ann. Inst. Fourier $($Grenoble$)$ 65 (2015) 2507–2545
  • M Green, R Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987) 389–407
  • R,M Hain, The de Rham homotopy theory of complex algebraic varieties, I, $K$–Theory 1 (1987) 271–324
  • R Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597–651
  • A,D Măcinic, Cohomology rings and formality properties of nilpotent groups, J. Pure Appl. Algebra 214 (2010) 1818–1826
  • D,A Măcinic, Ş Papadima, C,R Popescu, A,I Suciu, Flat connections and resonance varieties: from rank one to higher ranks, Trans. Amer. Math. Soc. 369 (2017) 1309–1343
  • J,W Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978) 137–204
  • P Orlik, H Terao, Arrangements of hyperplanes, Grundl. Math. Wissen. 300, Springer, New York (1992)
  • D Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205–295
  • H,K Schenck, A,I Suciu, Resonance, linear syzygies, Chen groups, and the Bernstein–Gelfand–Gelfand correspondence, Trans. Amer. Math. Soc. 358 (2006) 2269–2289
  • D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 269–331