Alex Suciu Northeastern University Conference on Non-Isolated - PowerPoint PPT Presentation

G ERMS OF REPRESENTATION VARIETIES AND COHOMOLOGY JUMP LOCI Alex Suciu Northeastern University Conference on Non-Isolated Singularities and Derived Geometry A celebration of the 60th birthday of David Massey UNAM-Cuernavaca, Mexico August 1, 2019 A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 1 / 25

R EFERENCES R EFERENCES [PS1] S. Papadima and A. Suciu, The topology of compact Lie group actions through the lens of finite models , electronically published in Int. Math. Res. Not. IMRN (2018). [PS2] S. Papadima and A. Suciu, Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds , electronically published in J. Inst. Math. Jussieu (2018). [PS3] S. Papadima and A. Suciu, Naturality properties and comparison results for topological and infinitesimal embedded jump loci , Adv. in Math. 350 (2019), 256–303. [PS4] S. Papadima and A. Suciu, Infinitesimal finiteness obstructions , J. London Math. Soc. 99 (2019), no. 1, 173–193. [MPPS] A. M˘ acinic, S. Papadima, R. Popescu, and A. Suciu, Flat connections and resonance varieties: from rank one to higher ranks , Trans. Amer. Math. Soc. 369 (2017), no. 2, 1309–1343. A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 2 / 25

O VERVIEW O VERVIEW § The study of analytic germs of representation varieties and cohomology jump loci is a basic problem in deformation theory with homological constraints. § Building on work of Goldman–Millson [1988], it was shown by Dimca–Papadima [2014] that the germs at the origin of those loci are isomorphic to the germs at the origin of infinitesimal jump loci of a CDGA that is a finite model for the space in question. § Budur and Wang [2015] have extended this result away from the origin, by developing a theory of differential graded Lie algebra modules which control the corresponding deformation problem. A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 3 / 25

O VERVIEW § Work of Papadima–S [2017] reveals a surprising connection between SL 2 p C q representation varieties of arrangement groups and the monodromy action on the homology of Milnor fibers of hyperplane arrangements. § On the other hand, the universality theorem of Kapovich and Millson [1998] shows that SL 2 p C q -representation varieties of Artin groups may have arbitrarily bad singularities away from 1. § This lead us to focus on germs at the origin of the representation varieties Hom p π, G q , and look for explicit descriptions via infinitesimal CDGA methods. § This approach works very well when G “ SL p 2 , C q and π is a Kähler group, an arrangement group, or a right-angled Artin group. A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 4 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS R EPRESENTATION VARIETIES R EPRESENTATION VARIETIES § Let π be a finitely generated group. § G be a k -linear algebraic group. § The set Hom p π, G q has a natural structure of an affine variety, called the G-representation variety of π . § Every homomorphism ϕ : π Ñ π 1 induces an algebraic morphism, ϕ ˚ : Hom p π 1 , G q Ñ Hom p π, G q , which is an isomorphism onto a closed subvariety. § Example: Hom p F n , G q “ G n . § Hom p Z 2 , GL k p C qq is irreducible, but not much else is known about the varieties of commuting matrices, Hom p Z n , GL k p C qq . § The varieties Hom p π 1 p Σ g q , G q are connected if G “ SL k p C q , and irreducible if G “ GL k p C q . A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 5 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS C OHOMOLOGY JUMP LOCI C OHOMOLOGY JUMP LOCI § Let p X , x q be a pointed, path-connected space, and assume π “ π 1 p X , x q is finitely generated. § The variety Hom p π, G q is the parameter space for locally constant sheaves on X whose monodromies factor through G . § Given a rep τ : π Ñ GL p V q , we let V τ be the local system on X associated to τ , i.e., the left π -module V defined by g ¨ v “ τ p g q v . § We also let H . p X , V τ q be the twisted cohomology of X with coefficients in this local system. A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 6 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS C OHOMOLOGY JUMP LOCI § The characteristic varieties of X with respect to a representation ι : G Ñ GL p V q are the sets V i r p X , ι q “ t ρ P Hom p π, G q | dim C H i p X , V ι ˝ ρ q ě r u . ` Hom p π, G q , V i ˘ § The pairs r p X , ι q depend only on the homotopy type of X and on the representation ι . § If X is a finite-type CW-complex, and ι is a rational representation, the sets V i r p X , ι q are closed subvarieties of Hom p π, G q . § For G “ C ˚ , the variety Hom p π, C ˚ q “ H 1 p X , C ˚ q is the character group of π —a disjoint union of algebraic tori p C ˚ q b 1 p X q , indexed by Tors p H 1 p X , Z qq . » § For ι : C ˚ Ý Ñ GL 1 p C q and V “ C , we get the usual characteristic varieties, V i r p X q . A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 7 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS F LAT CONNECTIONS F LAT CONNECTIONS § The infinitesimal analogue of the G -representation variety is F p A , g q , the set of g -valued flat connections on a commutative, differential graded C -algebra p A . , d q , where g is a Lie algebra. § This set consists of all elements ω P A 1 b g which satisfy the Maurer–Cartan equation, d ω ` 1 2 r ω, ω s “ 0 . § If A 1 and g are finite dimensional, then F p A , g q is a Zariski-closed subset of the affine space A 1 b g . A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 8 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS I NFINITESIMAL COHOMOLOGY JUMP LOCI I NFINITESIMAL COHOMOLOGY JUMP LOCI § For each ω P F p A , g q , we turn A b V into a cochain complex, p A b V , d ω q : A 0 b V d ω � A 1 b V d ω � A 2 b V d ω � ¨ ¨ ¨ , using as differential the covariant derivative d ω “ d b id V ` ad ω . (The flatness condition on ω insures that d 2 ω “ 0.) § The resonance varieties of the CDGA p A . , d q with respect to a representation θ : g Ñ gl p V q are the sets R i r p A , θ q “ t ω P F p A , g q | dim C H i p A b V , d ω q ě r u . § If A , g , and V are all finite-dimensional, the sets R i r p A , θ q are closed subvarieties of F p A , g q . § For g “ C , we have F p A , g q – H 1 p A q . Also, for θ “ id C , we get the usual resonance varieties R i r p A q . A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 9 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS I NFINITESIMAL COHOMOLOGY JUMP LOCI § Let F 1 p A , g q “ t η b g P F p A , g q | d η “ 0 u . § Let Π p A , θ q “ t η b g P F 1 p A , g q | det p θ p g qq “ 0 u . § In the rank 1 case, F 1 p A , C q “ F p A , C q and Π p A , θ q “ t 0 u . T HEOREM ( MPPS) Let ω “ η b g P F 1 p A , g q . Then ω belongs to R i 1 p A , θ q if and only if there is an eigenvalue λ of θ p g q such that λη belongs to R i 1 p A q . Moreover, č R i Π p A , θ q Ď 1 p A , θ q . i : H i p A q‰ 0 A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 10 / 25

R EPRESENTATION VARIETIES AND FLAT CONNECTIONS L INEAR RESONANCE L INEAR RESONANCE § Suppose R 1 1 p A q “ Ť C P C C , a finite union of linear subspaces. § Let A C denote the sub- CDGA of the truncation A ď 2 defined by A 1 C “ C and A 2 C “ A 2 . T HEOREM ( MPPS) For any Lie algebra g , ď F p A , g q Ě F 1 p A , g q Y F p A C , g q , ( ˛ ) 0 ‰ C P C where each F p A C , g q is Zariski-closed in F p A , g q . Moreover, if A has zero differential, and g “ sl 2 , then ( ˛ ) holds as an equality, and R 1 ď 1 p A , θ q “ Π p A , θ q Y F p A C , g q . 0 ‰ C P C (For g “ sl 2 : if g , g 1 P g , then r g , g 1 s “ 0 if and only if rank t g , g 1 u ď 1.) A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 11 / 25

A LGEBRAIC MODELS AND GERMS OF JUMP LOCI A LGEBRAIC MODELS FOR SPACES A LGEBRAIC MODELS FOR SPACES § From now on, X will be a connected space having the homotopy type of a finite CW-complex. § Let A PL p X q be the Sullivan CDGA of piecewise polynomial C -forms on X . Then H . p A PL p X qq – H . p X , C q . § A CDGA p A , d q is a model for X if it may be connected by a zig-zag of quasi-isomorphisms to A PL p X q . § A is a finite model if dim C A ă 8 and A is connected. § X is formal if p H . p X , C q , d “ 0 q is a (finite) model. § E.g.: Compact Kähler manifolds, complements of hyperplane arrangements, etc, are all formal. § The converse is not true: all nilmanifolds, solvmanifolds, Sasakian manifolds, smooth quasi-projective varieties, etc, admit finite models, but many are non-formal. A LEX S UCIU (N ORTHEASTERN ) R EPRESENTATION VARIETIES & JUMP LOCI UNAM-C UERNAVACA 8/1/19 12 / 25