Naturality properties and comparison results for topological and infinitesimal embedded jump loci
NNATURALITY PROPERTIES AND COMPARISON RESULTS FORTOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI
STEFAN PAPADIMA : AND ALEXANDER I. SUCIU A bstract . We use augmented commutative di ff erential graded algebra ( acdga ) modelsto study G -representation varieties of fundamental groups π “ π p M q and their embeddedcohomology jump loci, around the trivial representation 1. When the space M admits afinite family of maps, uniformly modeled by acdga morphisms, and certain finitenessand connectivity assumptions are satisfied, the germs at 1 of Hom p π, G q and of the em-bedded jump loci can be described in terms of their infinitesimal counterparts, naturallywith respect to the given families. This approach leads to fairly explicit answers when M is either a compact K¨ahler manifold, the complement of a central complex hyperplanearrangement, or the total space of a principal bundle with formal base space, provided theLie algebra of the linear algebraic group G is a non-abelian subalgebra of sl p C q . C ontents
1. Introduction and statement of results 22. Artin approximation 73. Algebraic models of spaces and maps 84. Deformation theory of representation varieties 135. Cohomology jump loci and naturality properties 176. A natural comparison between embedded jump loci 237. K¨ahler manifolds 288. Principal bundles 339. Quasi-projective manifolds 37Acknowledgement 43References 43
Mathematics Subject Classification.
Primary 14B12, 14F35, 55N25, 55P62. Secondary 20C15,57S15.
Key words and phrases.
Representation variety, flat connection, cohomology jump loci, filtered di ff er-ential graded algebra, relative minimal model, mixed Hodge structure, analytic local ring, Artinian localring, di ff erential graded Lie algebra, deformation theory, formal spaces and maps, quasi-compact K¨ahlermanifold, hyperplane arrangement, principal bundle. Work partially supported by the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI,grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III. : Deceased January 10, 2018. Partially supported by the Simons Foundation collaboration grant for mathematicians 354156. a r X i v : . [ m a t h . AG ] A p r STEFAN PAPADIMA AND ALEXANDER I. SUCIU
1. I ntroduction and statement of results
Representation varieties and jump loci.
Sheaf cohomology is ubiquitous in ge-ometry and topology. The parameter space for rank n locally constant sheaves on a path-connected, pointed CW-complex X with finitely many 1-cells may be identified with theGL n -representation variety of the fundamental group of X . Twisted cohomology on X isencoded by the filtrations of these varieties by the (embedded) jump loci. While GL -representation varieties are finite unions of a ffi ne tori, the picture changes dramatically inhigher rank. For instance, the universality theorem of Kapovich and Millson [18] statesthat PSL -representation varieties may have arbitrarily bad singularities, away from theorigin 1 (the trivial representation). This is the reason why we focus here on analyticgerms at the origin of the embedded cohomology jump loci. The general case is ana-lyzed by Budur and Wang in [4], but it seems that explicit computations away from 1 areintractable in full generality.By the main result from [9], Theorem B, the germs at 1 of the embedded jump lociof X are isomorphic to the germs at 0 of the infinitesimal embedded jump loci of a com-mutative, di ff erential graded algebra A , provided this cdga models X , and certain mildfiniteness assumptions are satisfied. Furthermore, in the abelian case, this identificationis natural. One of our main goals here is to extend this natural comparison to the non-abelian setting, by studying the behavior of jump loci under suitable continuous mapsbetween spaces and cdga maps between their models.By construction, both representation varieties and their infinitesimal analogues are(bi)functorial. The naturality properties of both types of jump loci are summarized inCorollaries 5.8 and 5.10. As we point out in Example 5.9, naturality at this level requiresconnectivity assumptions for maps, defined in § case, even for compact Riemann surfaces.To avoid this major di ffi culty, we construct local analytic isomorphisms between thetwo types of jump loci by means of Artin approximation. That is, we replace the respectivelocal analytic rings by their completions (or by functors of Artin rings), and deduce localanalytic naturality from naturality at the level of completions. This we do in Proposition2.3, which is a general result about simultaneous Artin approximation. We need this‘simultaneous’ framework in view of the applications to be derived later on, which involvefamilies of maps between spaces.The condition that makes our approach to a natural comparison between embeddedjump loci work is based on the q -equivalence relation for morphisms between augmentedcommutative di ff erential graded algebras ( acdga s), denoted by » q , and detailed in Def-inition 3.2. The primary examples we have in mind are the acdga morphisms Ω p f q between Sullivan–de Rham models induced by pointed, continuous maps between topo-logical spaces. OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 3
Natural comparison with respect to finite families of maps.
In more detail, theembedded jump loci we consider in this paper are as follows. First let X be a pointed, path-connected space with fundamental group π , and let ι : G Ñ GL p V q be a representation.For each i , r ě
0, the embedded jump locus of X with respect to ι is the pair(1.1) ` Hom p π, G q , V ir p X , ι q ˘ , where V ir p X , ι q is the set of homomorphisms ρ for which the i -th cohomology group of X with coe ffi cients in the local system V ι ˝ ρ has dimension at least r .Next, let A be a cdga , and let θ : g Ñ gl p V q be a Lie algebra representation. Theinfinitesimal analog of the representation variety is the set F p A , g q Ď A b g of g -valuedflat connections on A . For each i , r ě
0, the infinitesimal embedded jump locus of A withrespect to θ is the pair(1.2) ` F p A , g q , R ir p A , θ q ˘ , where R ir p A , θ q is the set of flat connections ω for which the i -th cohomology group of thecochain complex p A b V , d ω q defined in (5.2) has dimension at least r .Assume now that ι is a rational representation of linear algebraic groups, and both g and V are finite-dimensional. Under mild q -finiteness conditions on X and A (explainedin § p π, G q and F p A , g q are a ffi ne varieties, and their jump loci are closedsubvarieties, for all i ď q and r ě t f : X Ñ X f u f P E be a finite family ofcontinuous maps between pointed, path-connected spaces. For each f P E , we denoteby f : π Ñ π f the induced homomorphism on fundamental groups. Let also t Φ f : A f Ñ A u f P E be a family of acdga morphisms. Consider a rational representation of linearalgebraic groups, ι : G Ñ GL p V q , over k “ R or C , with tangential representation θ : g Ñ gl p V q . For an a ffi ne k -variety X , we denote by X p x q the k -analytic germ of X at a point x P X . Theorem 1.1.
Fix an integer q ě , and suppose the following conditions hold:(1) All the above spaces and cdga s are q-finite.(2) Both f and Φ f are p q ´ q -connected maps, for all f P E.(3) Ω p f q » q Φ f in ACDGA , uniformly with respect to f P E.Under these assumptions, we may find local analytic isomorphisms a : F p A , g q p q » ÝÑ Hom p π, G q p q and a f : F p A f , g q p q » ÝÑ Hom p π f , G q p q for all f P E with the property thatthe following diagram commutes, for all f P E: F p A , g q p q a (cid:47) (cid:47) Hom p π, G q p q F p A f , g q p q Φ f b id (cid:79) (cid:79) a f (cid:47) (cid:47) Hom p π f , G q p q . f ! (cid:79) (cid:79) STEFAN PAPADIMA AND ALEXANDER I. SUCIU
Moreover, this construction induces the following commuting diagram of (local, reduced)embedded jump loci, for all f P E, i ď q, and r ě : p F p A , g q , R ir p A , θ qq p q a (cid:47) (cid:47) p Hom p π, G q , V ir p X , ι qq p q p F p A f , g q , R ir p A f , θ qq p q Φ f b id (cid:79) (cid:79) a f (cid:47) (cid:47) p Hom p π f , G q , V ir p X f , ι qq p q , f ! (cid:79) (cid:79) where both horizontal arrows are isomorphisms of analytic pairs. The meaning of the q -equivalence relation between continuous pointed maps and acdga maps, uniformly with respect to finite families, is explained in Definition 6.3. For a one-element family t f u , this condition simply means that Ω p f q » q Φ f in ACDGA . For a certaintype of two-element family, the uniformity condition is verified in the next theorem.Let f : X Ñ Y be a continuous, pointed map between path-connected spaces. Let π bethe fundamental group of X , let abf : π (cid:16) π abf be the projection onto its maximal torsion-free abelian quotient, and let f : X Ñ K p π abf , q be a classifying map for this projection.Set A . “ p Ź . H p X q , d “ q . Theorem 1.2.
Suppose that X and Y are q-finite, for some q ě , and Ω p f q » q Φ in ACDGA , where Φ : A Y Ñ A X is a morphism between q-finite acdga s. There is then an acdga map Φ : A Ñ A X inducing an isomorphism on H , and such that Ω p f q » q Φ in ACDGA , uniformly with respect to the families t f , f u and t Φ , Φ u . Moreover, if f and Φ are -connected maps, then all hypotheses from Theorem 1.1 are satisfied for q “ . A general framework for applications.
Let π be the fundamental group of a 1-finite manifold M . We aim at finding structural results for (non-abelian) embedded jumploci of M near the origin, in low degrees. To start with, we want to extract from thegeometry of M a finite family of group epimorphisms, t f : π (cid:16) π f u f P E p M q , induced onfundamental groups by maps f : M Ñ M f onto manifolds of smaller dimension. Next, weset E p M q “ E p M q Y t f u , where f is a classifying map for the projection abf : π (cid:16) π abf .Let ι : G Ñ GL p V q be a rational representation of C -linear algebraic groups. For agroup homomorphism h : π Ñ π , we let h ! : Hom p π , G q Ñ Hom p π, G q denote the in-duced morphism between the corresponding representation varieties. As explained inRemark 7.8, the abelian part of Hom p π, G q near the trivial representation coincides withthe germ abf ! p Hom p π abf , G qq p q . By naturality of jump loci, the natural inclusion,(1.3) Hom p π, G q Ě ď f P E p M q f ! Hom p π f , G q , induces inclusions(1.4) V ir p M , ι q Ě ď f P E p M q f ! V ir p M f , ι q OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 5 for all i ď r ě
1. Finally, we ask whether the inclusions (1.3) and (1.4) becomeequalities near 1.We focus on the rank 2 case, when the Lie algebra g of G is a non-abelian subalgebra of sl p C q . Our techniques allow us to treat simultaneously two interesting classes of exam-ples: (1) quasi-compact K¨ahler manifolds (in particular, quasi-projective manifolds), and(2) closed, smooth manifolds endowed with a free action of a compact, connected, realLie group K . In the first case, the family E p M q consists of equivalence classes of ‘admis-sible’ maps (in the sense of Arapura [2]) from M to smooth complex curves of negativeEuler characteristic. In the second case, E p M q has only one element, namely the bundleprojection M Ñ M { K .When the group G is SL p C q or PSL p C q and M is a quasi-projective manifold, equalityin (1.3) is related to deep results of Corlette–Simpson [6] and Loray–Pereira–Touzet [20],which give a rather intricate classification for the G -representations of π p M q , also validaway from 1. When M is a quasi-compact K¨ahler manifold, ι “ id C ˆ , and i “ r “ V p M , ι q from [2], againalso valid away from 1. Thus, our results below may be viewed as a more precise versionof the aforementioned work, in a broader context, albeit only near the origin. Theorem 1.3.
Let G be a C -linear algebraic group with non-abelian Lie algebra g Ď sl p C q , and let ι : G Ñ GL p V q be a rational representation. For i “ r “ and for i “ ,r ě , both (1.3) and (1.4) become equalities near the origin , provided π “ π p M q andeither(1) M is a compact, connected K¨ahler manifold;(2) M is the complement of a (central) complex hyperplane arrangement;(3) M is a closed, connected, di ff erentiable manifold supporting a free action by acompact, connected real Lie group K, and the orbit space M { K is formal in thesense of Sullivan [32] .Here E p M q “ E p M q Y t f u , where f realizes abf : π (cid:16) π abf and the set E p M q consistsof all admissible maps in the first two cases, and the projection M Ñ M { K in the third.
The common strategy of proof is to choose appropriate uniform acdga models for thefamily E p M q and apply Theorem 1.1 to replace topological by infinitesimal equalities. Inturn, the latter equalities are verified using results from [22] for parts (1)–(2) and from[28] for part (3).Compact K¨ahler manifolds and complements of complex hyperplane arrangements pro-vide highly non-trivial examples of uniform formality (over k “ R or C ) with respect tofinite families of maps. In Proposition 7.4, we reinterpret the main result from [7] in thefollowing form: Ω k p f q » H . p f , k q in ACDGA , uniformly with respect to an arbitrary finitefamily of holomorphic maps between compact K¨ahler manifolds. Similarly, we recast inProposition 9.3 the main result of [12], as follows: Ω k p f q » H . p f , k q in ACDGA , uniformlywith respect to the family E p M A q , for any central complex hyperplane arrangement A in STEFAN PAPADIMA AND ALEXANDER I. SUCIU C with complement M A . In this way, we are able to apply Theorem 1.1 in order to proveparts (1)–(2) of Theorem 1.3.For part (3), let N “ M { K be the orbit space of the free K -action on M , and let f : M Ñ N be the projection map of the resulting principal K -bundle. Assuming that N has a finitemodel A N over a field k of characteristic 0, we construct in Proposition 8.2 a finite model A M for M and an acdga map Φ f : A N Ñ A M such that Ω k p f q » Φ f in ACDGA . In the casewhen N is formal, we may take A N “ p H . p N , k q , d “ q . Applying now Theorems 1.1and 1.2 completes the proof of Theorem 1.3(3).Further applications of the techniques that go into proving the above results can befound 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, Theorem1.1(2)] that the germs f ! Hom p π f , G q p q and g ! Hom p π g , G q p q from decomposition (1.3)intersect only at the origin, provided the maps f , g P E p M q are distinct. This transversalityproperty is a substantial non-abelian extension of the corresponding rank 1 result, provedin [10] in the case when ι is the standard isomorphism C ˆ » ÝÑ GL p C q .1.4. Formal maps and regular maps.
The uniformity property for one-element familiesof maps may be verified in two further classes of examples: formal maps between formalspaces, and regular maps between quasi-projective manifolds.By definition, a continuous map f : X Ñ Y is formal over k if it is modeled in CDGA bythe morphism H . p f , k q : H . p Y , k q Ñ H . p X , k q , cf. [32, 34]. In Proposition 3.4, we provethe following: If f is formal over k and H p f , k q is injective, then Ω k p f q » H . p f , k q in k - ACDGA . Applying Theorem 1.1 yields relevant information (summarized in Proposition6.8) on the map induced by f between the corresponding embedded jump loci.To state the quasi-projective analogue of the formality property, we need to recall from[23, 5] some relevant facts. Every quasi-projective manifold M is of the form M z D , where M is a smooth, projective variety and D is a normal-crossing divisor in M . A regular mapbetween two such manifolds, f : M Ñ M , is induced by a regular map ¯ f : M Ñ M withthe property that ¯ f ´ p D q Ď D . The manifold M admits as a finite cdga model over C Morgan’s Gysin model MG p M , D q . Furthermore, the regular map f is modeled in C - CDGA by a certain map Φ p ¯ f q : MG p M , D q Ñ MG p M , D q .In Proposition 9.1, we use relative Sullivan models for cdga maps to improve on theseknown facts, as follows. Let f : M Ñ M be a regular map between quasi-projectivemanifolds, and let ¯ f : p M , D q Ñ p M , D q be an extension as above. If H p f q is injective,then Ω C p f q » Φ p ¯ f q in C - ACDGA . We indicate in Remark 9.2 some possible applicationsof this result.1.5.
Conventions.
All spaces are assumed to be path-connected. The default coe ffi cientring is a field k of characteristic 0. (When speaking about analytic germs and analyticalgebras, k will be either R or C .) Graded k -vector spaces are non-negatively graded. OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 7
2. A rtin approximation
Our approach to naturality properties of cohomology jump loci is based on (simultane-ous) Artin approximation, using the book Tougeron [33] as a basic reference. We start byproving a general result of this type.Given a local ring p R , m q , we denote by gr . p R q the associated graded ring with respectto the m -adic filtration. The m -adic completion of R will be denoted by p R . If I Ă R is anideal, p I will stand for the extended ideal p R ¨ I of p R . Morphisms between local rings areassumed to be local.We will use M. Artin’s theorem on approximating formal power series solutions ofanalytic equations by convergent power series (see [33, III.4]) in the following form. Theorem 2.1.
Let R and R be two analytic algebras, and let t I k u k P F and t I k u k P F be twofinite families of proper ideals in these algebras. Suppose α : p R Ñ p R is a morphismsending p I k to p I k for all k. There is then a morphism a : R Ñ R such that a p I k q Ă I k for allk and gr p a q “ gr p α q . The next lemma will also be useful in the sequel.
Lemma 2.2.
Let R and R be two analytic algebras, and let I Ă R and I Ă R be twoproper ideals. Suppose a : R Ñ R is a morphism that sends I to I, and α : p R Ñ p R is anisomorphism such that α p p I q “ p I and gr p a q “ gr p α q . Then a is an isomorphism anda p I q “ I.Proof.
It follows from [33, III.5] that a morphism of analytic algebras with isomorphiccompletions must be an isomorphism, provided the given morphism induces a surjectionon gr . Consequently, both a : R Ñ R and the induced morphism, a : R { I Ñ R { I , areisomorphisms, and the claim follows. (cid:3) We are now ready to describe the setup for our approximation result. Let t φ i : R (cid:16) R i u i P E and t ¯ φ i : R (cid:16) R i u i P E be two families of epimorphisms between analytic algebras,indexed by the same finite set E . Furthermore, let t I j Ď R u j P F , t I j Ď R u j P F , t I ji Ď R i u j P F ,and t I ji Ď R i u j P F be families of ideals in the respective analytic algebras, indexed by thesame finite set F . Finally, let α : p R » ÝÑ p R and t α i : p R i » ÝÑ p R i u i P E be isomorphisms betweenthe respective completions. Proposition 2.3.
In the above setup, assume the following conditions hold, for all i P Eand j P F (as the case may be):(1) I j ‰ R ô I j ‰ R , I ji ‰ R i ô I ji ‰ R i , I j “ R ñ I ji “ R i , I j “ R ñ I ji “ R i ;(2) α p p I j q “ p I j ;(3) α i p p I ji q “ p I ji ; STEFAN PAPADIMA AND ALEXANDER I. SUCIU (4) ˆ¯ φ i ˝ α “ α i ˝ ˆ φ i .There exist then isomorphisms a : R » ÝÑ R and t a i : R i » ÝÑ R i u i P E such that(i) a p I j q “ I j for all j P F;(ii) a i p I ji q “ I ji for all i P E and j P F;(iii) ¯ φ i ˝ a “ a i ˝ φ i for all i P E.Proof.
Without loss of generality, we may assume all ideals in sight are proper, replacingif need be the set F by subsets F Ď F and F i Ď F for i P E . For each i P E , put K i “ ker p φ i q and K i “ ker p ¯ φ i q , and pick proper ideals, t J ji Ă R u j P F i and t J ji Ă R u j P F i ,such that φ i p J ji q “ I ji and ¯ φ i p J ji q “ I ji . We claim that it is enough to find a morphism a : R Ñ R such that(a) gr p a q “ gr p α q ;(b) a p K i q Ď K i , for all i P E ;(c) a p I j q Ď I j , for all j P F ;(d) a p J ji q Ď J ji ` K i , for all p i , j q P E ˆ F i .Indeed, by (b), the morphism a : R Ñ R induces morphisms a i : R i Ñ R i for all i P E .In view of (a), we may apply Lemma 2.2 and deduce that the map a is an isomorphism.By construction, property (iii) is satisfied. By assumption, equality (4) holds, and so α p p K i q “ p K i . Again by Lemma 2.2, the maps a i must be isomorphisms, for all i P E . Inview of (c), we may also apply Lemma 2.2 to each of the ideals I j Ă R and I j Ă R for j P F , and deduce that a p I j q “ I j , thereby verifying property (i).Finally, to verify property (ii), we apply Lemma 2.2 to the ideals I ji “ p J ji ` K i q{ K i Ă R { K i “ R i and I ji “ p J ji ` K i q{ K i Ă R { K i “ R i for p i , j q P E ˆ F i . We know from (b) and(d) that the morphisms a i : R i Ñ R i preserve these ideals. The fact that the isomorphisms α i identify the completions of these ideals follows from their construction, together withassumption (3). Moreover, since a i is induced by a and α i is induced by α , property(a) implies that gr p a i q “ gr p α i q . Thus, Lemma 2.2 applies once again to show that a i p I ji q “ I ji . This completes the verification of our claim.Assumptions (2)–(4) insure that the map α : p R » ÝÑ p R is a formal series solution of theanalytic system (b)–(d). Applying now Theorem 2.1 completes the proof. (cid:3)
3. A lgebraic models of spaces and maps
The rational homotopy theory of Quillen [30], as reinterpreted by Sullivan in [32],provides a very useful mechanism for studying topological properties of spaces and con-tinuous maps by considering commutative di ff erential graded algebra (for short, cdga )models for them. In this section, we review the basics of this theory, and draw someconsequences in the formal case. OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 9 q -connectivity and q -equivalences. We start with some basic terminology, relatedto connectivity properties of spaces and cdga s. Fix 0 ď q ď 8 , usually to be omittedfrom notation when q “ 8 . Let ψ : C . Ñ C . be a morphism of graded vector spaces. Wesay ψ is q-connected if it is an isomorphism in degrees up to q and a monomorphism indegree q ` ψ is a q -connected morphism of cochain complexes, it is straight-forward to check that the induced morphism in cohomology, H . p ψ q : H . p C q Ñ H . p C q ,is again q -connected. Cochain maps inducing a q -connected map in cohomology will becalled q-equivalences . For q “ 8 , these maps are also called quasi-isomorphisms in theliterature.We say that a commutative graded algebra (for short, a cga ) A . is connected if A isthe k -span of the unit 1 (and thus A “ k ). If A is a connected cdga , then clearly itscohomology algebra, H . p A q , is again connected.This terminology is inspired by algebraic topology, where a continuous map f : X Ñ Y between two topological spaces is said to be q -connected if it induces isomorphisms onhomotopy groups up to degree q and an epimorphism in degree q `
1. By Hurewicz’stheorem, the induced map in cohomology, H . p f q : H . p Y q Ñ H . p X q , is also q -connected.Note that the map X Ñ t pt u is q -connected if and only if the space X is q -connected.Finally, let C be a subcategory of the category of cochain complexes, for instance, CDGA or the category of di ff erential graded Lie algebras, DGL . Two objects in this category, C and C , have the same q-type (denoted C » q C ) if they can be connected in C by a zig-zagof q -equivalences.3.2. CDGA models for spaces.
We now review the construction and some basic prop-erties of cdga models of spaces, following [32, 23, 17, 19, 13, 9]. We will denote by Ω . k p X q Sullivan’s de Rham algebra of a topological space X , constructed by using di ff er-ential forms with k -polynomial coe ffi cients on standard simplices, see [32]. The resultingfunctor Top Ñ k - CDGA has, among other things, the property that H . p Ω . k p X qq – H . p X , k q ,as graded k -algebras.To define monodromy representations of flat connections (over k “ R or C ), we willalso need the similar cdga Ω . p X , k q , constructed from usual smooth k -forms. It is knownthat Ω . p X , k q has the same -type as the sub- cdga Ω . k p X q , in a natural way.Let A be a cdga . For 0 ď q ď 8 , we say that A is a q-model for the space X if Ω k p X q » q A . We also say that X is q-finite if it has the homotopy type of a connectedCW-complex with finite q -skeleton. Similarly, we say that A is q -finite if it is connectedand dim À i ď q A i ă 8 . Once again, we shall omit q from the notation when q “ 8 .The category ACDGA of augmented , commutative di ff erential graded algebras has ob-jects p A , ε q , where the augmentation map ε : A Ñ k is a morphism of cdga s, while themorphisms in this category are the cdga maps commuting with augmentations. When X is a pointed space, both Ω . p X , k q and Ω . k p X q become ACDGA s, again in a natural way.
A connected cdga A has a unique augmentation map, sending A ` to 0, and the unit to1. Moreover, for every augmented cdga A , we have that(3.1) Hom ACDGA p A , A q “ Hom
CDGA p A , A q . Hirsch extensions and relative minimal models.
Let U . “ À i ě U i be a positivelygraded k -vector space. The free commutative graded algebra on U , denoted by Ź U . ,is the tensor product of the symmetric graded algebra on U even and the exterior gradedalgebra on U odd . We say that a cdga is free if the underlying cga has this property. Since Ź U . is connected, it has a unique augmentation, denoted by ε U .Let A “ p A . , d A q be a cdga , and denote by Z . p A q the graded vector space of cocycles.Given a finite-dimensional graded vector space U . , and a degree 1 linear map, τ : U . Ñ Z . ` p A q , we denote by p A b τ Ź U , d q the corresponding Hirsch extension . By definition,this is the cdga whose underlying cga is A . b Ź U . , and whose di ff erential restricts to d A on A and to τ on U . If A is an acdga with augmentation ε A , then A b τ Ź U is alsoan acdga , with augmentation ε A b ε U . The Hirsch extension depends only on the map r τ s : U . Ñ H . ` p A q , in the following sense: if r τ s “ r τ s , then A b τ Ź U – A b τ Ź U in CDGA , via an isomorphism extending id A . When dim U “
1, we speak of an ‘elementary’Hirsch extension.A relative Sullivan algebra with base B is a direct limit of elementary Hirsch extensions,starting from the cdga B . When the base is k , concentrated in degree 0, we simplyspeak of a Sullivan algebra. Such a cdga is necessarily of the form M “ p Ź U . , d q .If im p d q Ď Ź ě U , we say M is a minimal Sullivan algebra. If, moreover, all Hirschextensions have degree at most q , the cdga M is said to be q-minimal .A q-minimal model map for a cdga A is a q -equivalence ρ : M q Ñ A , with M q a q -minimal Sullivan algebra. Any cdga A whose cohomology algebra is connected admits a q -minimal model map. If ρ : M q Ñ A is another q -minimal model map for A , then M q and M q are isomorphic in CDGA . Consequently, if A and A are two cdga s with connectedhomology, then A » q A in CDGA if and only if there is a q -minimal cdga M q , and a shortzig-zag of q -equivalences in CDGA of the form(3.2) A M q ρ (cid:111) (cid:111) ¯ ρ (cid:47) (cid:47) A . Recall that a relative Sullivan algebra with base B is a cdga of the form A “ p B b Ź U , d q . When B is an augmented algebra, with augmentation ideal r B : “ ker p ε B q , thequotient cdga , A {p r B b Ź U q “ p Ź U , ¯ d q , is called the fiber of A . Following [17, 13], wesay that A is a minimal Sullivan algebra in the relative sense if the fiber is minimal. Al-lowing also degree 0 Hirsch extensions, we may speak of weak relative Sullivan (minimal)algebras.Let Φ : B Ñ C be a cdga map, and assume B is augmented and H . p B q is connected.A relative minimal model map for Φ is an -equivalence of cdga s, h : M Ñ C , where OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 11 M “ p B b Ź U , d q is a relative minimal Sullivan algebra and h | B “ Φ . If Φ is a 0-equivalence, then Φ admits a relative minimal model map; moreover, any two such maps, h and h , have isomorphic fibers, see [13, § H p Φ q is an isomorphism, see [17, Ch. 6]. Inparticular, if Φ : B Ñ C is a 0-equivalence and h : M Ñ C is any relative minimal modelmap for Φ in the weak sense, then necessarily the fiber of M is connected. Hence, if Φ isan acdga map, then M is canonically augmented and both h and the inclusion B ã Ñ M preserve augmentations.3.4. Homotopies and equivalences.
Let Ź p t , dt q be the free, contractible cdga gener-ated by t in degree 0 and dt in degree 1. For each s P k , let ev s : Ź p t , dt q Ñ k be the cdga map sending t to s and dt to 0. This map induces another cdga map,(3.3) Ev s “ id b ev s : A b Ź p t , dt q Ñ A b k “ A . Definition 3.1.
Two cdga maps ψ , ψ : A Ñ A are said to be homotopic (in CDGA ) ifthere is a cdga map Ψ : A Ñ A b Ź p t , dt q such that Ev s ˝ Ψ “ ψ s for s “ ,
1. Likewise,two acdga maps ψ and ψ as above are homotopic (in ACDGA ) if the homotopy Ψ alsosatisfies Ev ˝ Ψ “ ε , where Ev denotes the cdga map ε b id : A b Ź p t , dt q Ñ Ź p t , dt q .Plainly, equality of maps implies homotopy, in both categories. Note that augmentedhomotopy is strictly stronger than homotopy. Another useful remark is that homotopicmaps in CDGA induce the same map in cohomology.Denote by
ACDGA the full subcategory of ACDGA whose objects have connected coho-mology. Fix an integer q ě Definition 3.2. An elementary q-equivalence in ACDGA between two ACDGA -morphisms Φ : A Ñ A and Φ : A Ñ A consists of two acdga maps, ψ : A Ñ A and ψ : A Ñ A , both of which are q -equivalences, and such that ψ ˝ Φ is homotopic to Φ ˝ ψ in ACDGA .We denote by » q the associated equivalence relation between morphisms in ACDGA .In other words, if Φ : A Ñ A and ¯ Φ : B Ñ B are two such morphisms, we say that Φ » q ¯ Φ in ACDGA if there are two zig-zags, Z and Z , of q -equivalences in ACDGA , and acdga maps Φ , . . . , Φ (cid:96) ´ such that the following diagram commutes, up to augmentedhomotopy:(3.4) Z : A A ψ (cid:111) (cid:111) ψ (cid:47) (cid:47) ¨ ¨ ¨ A (cid:96) ´ (cid:111) (cid:111) ψ (cid:96) ´ (cid:47) (cid:47) BZ : A Φ (cid:79) (cid:79) A Φ (cid:79) (cid:79) ψ (cid:111) (cid:111) ψ (cid:47) (cid:47) ¨ ¨ ¨ A (cid:96) ´ Φ (cid:96) ´ (cid:79) (cid:79) (cid:111) (cid:111) ψ (cid:96) ´ (cid:47) (cid:47) B . ¯ Φ (cid:79) (cid:79) Forgetting augmentations in Definition 3.2, we obtain the equivalence relation » q be-tween maps in CDGA . When q “ 8 , we will simply write this as Φ » ¯ Φ . Lemma 3.3.
Let A and A be two acdga s. Assume H . p A q is connected, and let ρ : M q Ñ A be a q-minimal cdga model map as above. Then A » q A in
CDGA if and only if there isa short zig-zag of q-equivalences in
ACDGA as in (3.2) .Proof.
By the discussion from § A » q A in CDGA if and only if A and A share the same q -minimal model M q . The fact that M q is connected takes care of theaugmentations. (cid:3) Formal spaces and maps.
We conclude this section with some formality notions,for both spaces and maps, as well as models thereof. To start with, we say that a cdga A is q-formal if p A . , d q » q p H . p A q , d “ q in CDGA . Clearly, q -formality implies p -formality,for all p ď q . By definition, a space X is q -formal over k if Ω k p X q has this property. A q -finite, q -formal space X has the q -finite q -model p H . p X q , d “ q . As before, we willmostly omit q from notation when q “ 8 . Compact K¨ahler manifolds are well-known tobe formal, by the main result from [7].Following [32, 7, 34], we say that a morphism Φ : A Ñ A in CDGA is formal if there isa diagram consisting of two elementary equivalences in CDGA ,(3.5) A M ψ (cid:111) (cid:111) ¯ ψ (cid:47) (cid:47) p H . p A q , d “ q A Φ (cid:79) (cid:79) M ψ (cid:111) (cid:111) p Φ (cid:79) (cid:79) ¯ ψ (cid:47) (cid:47) p H . p A q , d “ q , H . p Φ q (cid:79) (cid:79) such that both M and M are minimal Sullivan algebras. Furthermore, we say that acontinuous map f : X Ñ X is formal (over k ) if the induced morphism between Sullivande Rham models, Ω k p f q : Ω k p X q Ñ Ω k p X q , has this property. Proposition 3.4.
Let f : X Ñ X be a continuous map between pointed spaces. Assumethat f is formal over k , and H p f q is injective. Then Ω p f q » H . p f q in k - ACDGA .Proof. In [34, II.3], Vigu´e-Poirrier uses the formality of the map f to construct a com-muting diagram in CDGA of the form(3.6) Ω p X q Ź U b Ź U ψ (cid:111) (cid:111) ¯ ψ (cid:47) (cid:47) p H . p X q , d “ q Ω p X q Ω p f q (cid:79) (cid:79) Ź U ψ (cid:111) (cid:111) (cid:63)(cid:31) j (cid:79) (cid:79) ¯ ψ (cid:47) (cid:47) p H . p X q , d “ q H . p f q (cid:79) (cid:79) where all horizontal arrows are -equivalences and j is the canonical inclusion. More-over, ψ and ¯ ψ are minimal model maps (in particular, Ź U is connected), and Ź U b Ź U is a relative minimal Sullivan algebra in the weak sense, with base Ź U .The injectivity assumption on H p f q implies that H . p f q ˝ ¯ ψ is a 0-equivalence. Fromthe discussion in § U “
0. This shows that Ź U b Ź U is alsoconnected. Hence, (3.6) is a commuting diagram in ACDGA , and our claim follows. (cid:3) OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 13
4. D eformation theory of representation varieties
Following Goldman–Millson [15] and Manetti [21], we recall, in a convenient form,two basic properties of the deformation functor associated to a di ff erential graded Liealgebra ( dgl for short). We then apply these techniques to the representation varieties ofdiscrete groups.4.1. Deformation functors.
We denote by
Art the category of Artinian local k -algebras.Given such an algebra p A , m A q and a di ff erential graded Lie algebra L , we consider the(nilpotent) dgl L b m A , and the set of solutions (called flat connections ) to the Maurer–Cartan equation,(4.1) F p L b m A q “ t ω P L b m A | d ω ` r ω, ω s “ u , with basepoint 0 P F p L b m A q . Clearly, this construction is bifunctorial. Let(4.2) G p L b m A q “ exp p L b m A q be the (Campbell–Hausdor ff ) gauge group of the nilpotent Lie algebra L b m A , withunderlying set L b m A . This group acts bifunctorially on F p L b m A q by(4.3) exp p α q ¨ ω “ ω ` ÿ n “ ad p α q n p n ` q ! pr α, ω s ´ d α q . The deformation functor , Def L : Art Ñ Set , is defined by(4.4) Def L p A q “ F p L b m A q{ G p L b m A q . It is readily seen that every dgl -morphism ψ : L Ñ L induces a natural transformation,Def ψ : Def L Ñ Def L .We now may state the Deligne–Schlesinger–Stashe ff theorem, as recorded and provedin [15, Thm. 2.4]. Theorem 4.1.
If L » L in DGL , then the deformation functors
Def L and Def L are natu-rally isomorphic. Homotopy invariance.
The homotopy relation in
DGL takes the following form.Given a dgl L , let us form the dgl L b Ź p t , dt q , endowed with the canonical tensor productstructure. For each s P k , we defined in § cdga map, ev s : Ź p t , dt q Ñ k ,which sends t ÞÑ s and dt ÞÑ
0. Proceeding as before, we extend this map to a dgl map,Ev s “ id b ev s : L b Ź p t , dt q Ñ L b k “ L .Two dgl maps ψ , ψ : L Ñ L are said to be homotopic if there is a dgl map Ψ : L Ñ L b Ź p t , dt q such that Ev s ˝ Ψ “ ψ s for s “ ,
1. The notion of homotopy between dgl maps is related to deformation functors via the following basic result of Manetti [21,Thm. 5.5].
Theorem 4.2.
Let L be a dgl , and let A be a local Artin algebra. Two flat connections β , β P F p L b m A q are equal in Def L p A q if and only if there is a flat connection ω P F p L b Ź p t , dt q b m A q such that p Ev s b id q ω “ β s for s “ , . This theorem has an immediate corollary, which will be useful in the sequel.
Corollary 4.3. If ψ , ψ : L Ñ L are homotopic in DGL , then
Def ψ “ Def ψ . Deformation theory of cdga s. We consider now the bifunctor
CDGA ˆ Lie Ñ DGL which associates to a cdga A and a Lie algebra g the dgl (4.5) L “ A b g , endowed with the canonical tensor product structure r a b g , a b g s “ aa b r g , g s anddi ff erential Bp a b g q “ da b g .We will also need an augmented version of this construction. Given an augmented cdga p A , ε q and a Lie algebra g , we denote by r A “ ker p ε q the augmentation di ff erentialideal, and we consider the sub- dgl r L “ r A b g of the dgl L “ A b g . This construction isagain bifunctorial. Remark 4.4.
Given a q -equivalence ψ P Hom
ACDGA p A , A q , it is easy to check that theinduced maps, ˜ ψ : r A Ñ r A and ˜ ψ b id : r L Ñ r L , are again q -equivalences, provided thatboth H . p A q and H . p A q are connected. Consequently, if A » q A in ACDGA then r L » q r L in DGL .Let g be a Lie algebra. The proof of the next lemma is straightforward. Lemma 4.5. A CDGA homotopy, Ψ : A Ñ A b Ź p t , dt q , between two maps, ψ and ψ ,induces a DGL homotopy, Ψ b id : A b g Ñ A b g b Ź p t , dt q , between the maps ψ b id and ψ b id . Moreover, if Ψ is an augmented homotopy, then Ψ induces a DGL homotopy, r Ψ b id : r A b g Ñ r A b g b Ź p t , dt q , between the maps ˜ ψ b id and ˜ ψ b id . Deformation theory of augmented cdga s. Our next goal is to relate q -types of cdga s to the deformation theory of acdga s. Fix q ě
1, and let Z be a zig-zag of q -equivalences in ACDGA ,(4.6) A A ψ (cid:111) (cid:111) ψ (cid:47) (cid:47) ¨ ¨ ¨ A (cid:96) ´ (cid:111) (cid:111) ψ (cid:96) ´ (cid:47) (cid:47) A (cid:96) , where H p A q “ k ¨
1. By Remark 4.4 and Theorem 4.1, the zig-zag Z induces a naturalbijection(4.7) β Z : F p r A (cid:96) b g b m A q{ G p r A (cid:96) b g b m A q » (cid:47) (cid:47) F p r A b g b m A q{ G p r A b g b m A q for all local Artin algebras A . It is important to note that, if Z and Z are two di ff erentzig-zags of q -equivalences connecting A to A (cid:96) , then the bijections β Z and β Z may also bedi ff erent. OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 15
Proposition 4.6.
Let ρ : M q Ñ A be a q-minimal model map. There is then a shortzig-zag of q-equivalences in ACDGA ,S : A M q ρ (cid:111) (cid:111) ¯ ρ (cid:47) (cid:47) A (cid:96) , such that β Z “ β S .Proof. We will construct, by induction on 0 ď i ď (cid:96) , a collection of q -minimal modelmaps ρ i : M q Ñ A i , which form, together with the maps ψ i from (4.6), homotopy-com-mutative triangles in ACDGA , starting with ρ “ ρ . Once this done, we set ¯ ρ “ ρ (cid:96) . Theequality β Z “ β S then follows from Lemma 4.5 and Corollary 4.3.For the induction step, we first assume that ψ i : A i Ñ A i ` . Then we take ρ i ` “ ψ i ˝ ρ i .Finally, suppose that ψ i : A i ` Ñ A i . The lifting property up to homotopy for cdga mapsalso holds for acdga maps, and implies that we may find a cdga map ρ i ` : M q Ñ A i ` such that ψ i ˝ ρ i ` is homotopic to ρ i in ACDGA . The fact that ρ i ` must be a q -equivalenceis easily checked, thereby completing the proof. (cid:3) Representation varieties.
Let π be a discrete group, and let G be a k -linear algebraicgroup. The set Hom p π, G q of group homomorphisms from π to G has a natural structureof an a ffi ne scheme. This set depends bi-functorially on π and G , and has a natural basepoint, the trivial representation, 1. Furthermore, G acts by conjugation on Hom p π, G q .Now suppose π is a finitely generated group. (Note that the fundamental group π “ π p X , x q of a pointed CW-space is finitely generated if and only if X is 1-finite.) In thiscase, the set Hom p π, G q has a natural structure of a ffi ne variety, called the G-representationvariety of π . Moreover, every homomorphism ϕ : π Ñ π induces an algebraic morphismbetween the corresponding representation varieties, ϕ ! : Hom p π , G q Ñ Hom p π, G q . Wewill come back to this point in Lemma 5.3.Clearly, the G -representation variety of the free group F n is equal to the n -fold directproduct G n . Much is known about the varieties of commuting matrices, for instance,that Hom p Z , GL n p C qq is irreducible. Nevertheless, many open questions remain aboutthe precise structure of the varieties Hom p Z n , G q , see for instance [3, 1] and referencestherein. Perhaps the most-studied family of representation varieties is that of fundamentalgroups of closed orientable surfaces Σ g . For instance, it is known that Hom p π p Σ g q , G q is connected if G “ SL n p C q , and an absolutely irreducible and Q -rational variety if G “ GL n p C q , see [14, 31].4.6. Flat connections.
The infinitesimal counterpart to the representation varieties is pro-vided by the space of flat connections. Given a cdga A and a Lie algebra g , we will denoteby F p A , g q the set of flat connections on the dgl A b g . This set behaves bi-functorially,and has a natural basepoint, the trivial flat connection 0. For a local Artin k -algebra A , thegauge group(4.8) G p A b g b m A q “ exp p A b g b m A q acts naturally on F p A b g b m A q . If A is an augmented cdga , we have that F p r A b g b m A q “ F p A b g b m A q and G p r A b g b m A q Ď G p A b g b m A q , with the augmented gauge groupacting by restriction. In the particular case when A is connected, the augmented gaugegroup is trivial, and we obtain a natural identification,(4.9) F p A , g b m A q “ Def r A b g p A q . If both A and g are finite-dimensional, then the set F p A , g q has a natural structure ofa ffi ne variety, which we shall call the g -variety of flat connections on the cdga A .Now let p X , x q be a pointed space with fundamental group π , and let G be a linearalgebraic group over k “ R or C , with Lie algebra g . The monodromy construction from[9, § F p Ω p X , k q , g q (cid:47) (cid:47) Hom p π, G q which extends the classical monodromy map for smooth manifolds, and has nice naturalityproperties. Furthermore, for each local Artin k -algebra A , we have a natural monodromymap(4.11) mon : F p Ω p X , k q , g b m A q (cid:47) (cid:47) Hom p π, exp p g b m A qq . The equivariance property of the monodromy map for smooth manifolds described in[15, (5-8)] can be extended to arbitrary topological spaces, as follows.
Lemma 4.7.
For any gauge equivalence a P G p Ω p X , k q b g b m A q , we have a commutingdiagram, F p Ω p X , k q , g b m A q mon (cid:47) (cid:47) a (cid:15) (cid:15) Hom p π, exp p g b m A qq c a (cid:15) (cid:15) F p Ω p X , k q , g b m A q mon (cid:47) (cid:47) Hom p π, exp p g b m A qq , where c a stands for the conjugation action by ´p ε b id qp a q and ε b id : Ω p X , k q b g b m A Ñ g b m A is given by the augmentation ε of Ω p X , k q corresponding to the basepointx. Consequently, the monodromy map factors through the action of the augmented gaugegroup. We will repeatedly work under the assumptions of Theorem B from [9]. Namely, wefix an integer q ě
1, and we let X be a pointed, q -finite space with fundamental group π .Next, we assume there is a q -finite cdga A such that Ω p X , k q » q A in CDGA . Finally, welet G be a linear algebraic group over k “ R or C , with Lie algebra g .Now let ρ : N Ñ Ω p X , k q be a ‘ π -adapted’ 1-minimal model map, as in [9, § cdga s, we may extend ρ to a q -minimal model map, ρ q : M q Ñ Ω p X , k q . By Lemma 3.3, we may find a zig-zag of q -equivalences in ACDGA of the form(4.12) S : Ω p X , k q M q ρ q (cid:111) (cid:111) ¯ ρ q (cid:47) (cid:47) A OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 17 which fits into the basic setup from [9, § special .The next result is a topological analog of Theorem 6.8 from [15], proved only forsmooth manifolds. Theorem 4.8.
Let X be a -finite space. Then the natural map mon : F p Ω p X , k q b g b m A q{ G p r Ω p X , k q b g b m A q (cid:47) (cid:47) Hom p π, exp p g b m A qq from Lemma 4.7 is a bijection, for all local Artin k -algebras A .Proof. Let ρ : N Ñ Ω p X , k q be a π -adapted 1-minimal model map. By [9, Prop. 6.16],the composite F p N b g b m A q ρ b id (cid:47) (cid:47) F p Ω p X , k q b g b m A q mon (cid:47) (cid:47) Hom p π, exp p g b m A qq is a bijection. Since N is connected, formula (4.9) allows us to replace F p N b g b m A q by Def Ă N b g p A q . Using now Lemma 4.7, we see that the above bijection is equal to thecomposite Def Ă N b g p A q ˜ ρ b id (cid:47) (cid:47) Def r Ω p X , k qb g p A q mon (cid:47) (cid:47) Hom p π, exp p g b m A qq . Finally, it follows from Theorem 4.1 and Remark 4.4 that the map ˜ ρ b id is also a bijection,and this completes the proof. (cid:3) Assume again that the hypotheses of Theorem B from [9] are satisfied. Let Z be a zig-zag of q -equivalences in ACDGA as in (4.6), connecting A “ Ω p X , k q to A (cid:96) “ A . UsingTheorem 4.8 and formula (4.9), we may then define a natural bijection(4.13) α Z : “ mon ˝ β Z : F p A , g b m A q » (cid:47) (cid:47) Hom p π, exp p g b m A qq . Corollary 4.9.
For any zig-zag Z as above, there is a special zig-zag S such that α Z “ α S .Proof. Let ρ q : M q Ñ Ω p X , k q be a q -minimal model map extending a π -adapted 1-minimal model map ρ : N Ñ Ω p X , k q . By Proposition 4.6, there is a special zig-zag S as in diagram (4.12) such that β Z “ β S . The claim follows. (cid:3)
5. C ohomology jump loci and naturality properties
We now define two types of cohomology jump loci (one for spaces and the other for cdga s), and study some of the naturality properties these algebraic varieties enjoy.
Embedded cohomology jump loci.
Let p X , x q be a pointed, path-connected space.Set π “ π p X , x q . For a k -linear algebraic group G , the set Hom p π, G q is a parameterspace for finite-dimensional local systems on X of type G . When the space X is 1-finite(or, equivalently, when the group π is finitely generated), this parameter space is an a ffi ne k -variety. When k “ R or C , we let Hom p π, G q p q be the analytic germ at 1 of this variety,and we denote by R “ R p π, G q the analytic local algebra of this germ.Given a cdga A . and a Lie algebra g , let F p A , g q be the set of g -valued flat connectionson A . When both A and g are finite-dimensional, this set is an a ffi ne variety. We shalldenote by R “ R p A , g q the analytic local algebra of the germ F p A , g q p q . Assume now thatboth X and A are 1-finite, and that Ω k p X q » A as cdga s. Letting g be the Lie algebra of G , it then follows from [9, Prop. 7.6] that the local algebras R and R are isomorphic.Given a representation τ : π Ñ GL p V q , we let V τ denote the local system on X as-sociated to τ , that is, the left π -module V defined by g ¨ v “ τ p g q v . Furthermore, welet H . p X , V τ q be the twisted cohomology of X with coe ffi cients in this local system, seee.g. [35]. Definition 5.1.
The characteristic varieties of the space X in degree i ě r ě ι : G Ñ GL p V q are the sets V ir p X , ι q “ t ρ P Hom p π, G q | dim k H i p X , V ι ˝ ρ q ě r u . For each i ě
0, the sequence t V ir p X , ι qu r ě is a descending filtration of Hom p π, G q “ V i p X , ι q . In the rank 1 case, i.e., when ι is the canonical identification k ˆ Ñ GL p k q , wewill drop the map ι from the notation, and simply write V ir p X q . When X “ K p π, q is aclassifying space for the group π , we will denote the corresponding characteristic varietiesby V ir p π, ι q .We will refer to the pairs(5.1) ` Hom p π, G q , V ir p X , ι q ˘ as the (global) embedded jump loci of X with respect to ι . Clearly, such pairs depend onlyon the homotopy type of X and on the representation ι . If ι is a rational representation and X is a q -finite space for some q ě
1, then the sets V ir p X , ι q are closed subvarieties of therepresentation variety Hom p π, G q , for all i ď q and r ě
0; see [9, 4].5.2.
Infinitesimal cohomology jump loci.
To define the infinitesimal counterpart of theseloci, we start with a cdga A . , a Lie algebra g , and a representation θ : g Ñ gl p V q . For eachflat connection ω P F p A , g q , we turn the tensor product A b V into a cochain complex,(5.2) p A b V , d ω q : A b V d ω (cid:47) (cid:47) A b V d ω (cid:47) (cid:47) A b V d ω (cid:47) (cid:47) ¨ ¨ ¨ , using as di ff erential the covariant derivative d ω “ d b id V ` ad ω . Here, if ω “ ř i a i b g i ,with a i P A and g i P g , then ad ω p a b v q “ ř i a i a b θ p g i qp v q , for all a P A and v P V . It isreadily checked that the flatness condition on ω insures that d ω “
0, see [9].
OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 19
Definition 5.2.
The resonance varieties of the cdga A . in degree i ě r ě θ : g Ñ gl p V q are the sets(5.3) R ir p A , θ q “ t ω P F p A , g q | dim k H i p A b V , d ω q ě r u . For each i ě
0, the sequence t R ir p A , θ qu r ě is a descending filtration of F p A , g q “ R i p A , θ q . In the rank one case, i.e., the case when θ is the canonical identification k Ñ gl p k q , we will simply write R ir p A q for the corresponding sets.We will refer to the pairs(5.4) ` F p A , g q , R ir p A , θ q ˘ as the (global) infinitesimal embedded jump loci of A with respect to θ . If A is q -finitefor some q ě
1, and both g and V are finite-dimensional, the sets R ir p A , θ q are closedsubvarieties of F p A , g q , for all i ď q and r ě
0; see [9, 4].Assume now that both the space X and the cdga A . are q -finite, for some q ě Ω k p X q » q A as cdga s. Let ι : G Ñ GL p V q be a rational representation, andlet θ : g Ñ gl p V q be its tangential representation. As shown in [9, Thm. B], there isthen an analytic isomorphism F p A , g q p q » ÝÑ Hom p π, G q p q restricting to isomorphisms R ir p A , θ q p q » ÝÑ V ir p X , ι q p q between the reduced analytic germs of the corresponding jumploci, for all i ď q and r ě q -models. We start witha preliminary observation, which follows directly from the definitions. Namely, for all i ď q and r ě P V ir p X , ι q ô P R ir p A , θ q ô b i ¨ dim p V q ě r , where b i “ b i p X q “ b i p A q denotes the i -th (untwisted) Betti number.5.3. Naturality properties of representation varieties.
As mentioned previously, bothambient spaces for jump loci, Hom p π, G q and F p A , g q , are bifunctorial. On the otherhand, for continuous and cdga maps, naturality of (global) jump loci requires certainconnectivity hypotheses. To begin, we only assume the minimally required connectivityand finiteness conditions. Lemma 5.3.
Let f : p X , x q Ñ p X , x q be a -connected, pointed map, and let f be theinduced homomorphism on fundamental groups. Assume that X is -finite. Then, for everylinear algebraic group G, the morphism induced by f on representation varieties, (5.6) f ! : Hom p π p X q , G q (cid:47) (cid:47) Hom p π p X q , G q , is an isomorphism onto a closed subvariety.Proof. Our 0-connectivity assumption on f means that the homomorphism f is surjective.Our 1-finiteness assumption on X , then, implies that both fundamental groups are finitely generated. Let us present π p X q as the quotient F m { R of a free group on m generators, andthen use the presentation for π p X q induced by f .By construction, the representation variety Hom p π p X q , G q is the closed subvariety of G m defined by the equations given by the relators in R . The variety Hom p π p X q , G q sitsalso in G m , with the same defining equations as Hom p π p X q , G q , plus the equations comingfrom the lifts to F m of the elements of ker p f q . The claim readily follows. (cid:3) Holonomy Lie algebras.
Before proceeding, let us recall from [22, §
4] the con-struction of the holonomy Lie algebra h p A q of a 1-finite cdga p A , d q . Set A i “ p A i q ˚ , andlet L p A q be the free Lie algebra on the dual vector space A . We then define(5.7) h p A q : “ L p A q{ ideal p im p d ˚ ` Y ˚ qq , where d ˚ : A Ñ A “ L p A q and Y ˚ : A Ñ A ^ A “ L p A q are the maps dual to thedi ff erential and the multiplication map in A , respectively. This construction is functorial:if ψ : A Ñ A is a morphism of 1-finite cdga s, then the linear map ψ “ p ψ q ˚ : A Ñ A extends to a Lie algebra morphism L p ψ q : L p A q Ñ L p A q , which in turn inducesa Lie algebra morphism h p ψ q : h p A q Ñ h p A q . Finally, as shown in [22, Prop. 4.5], thecanonical isomorphism A b g » ÝÑ Hom p A , g q restricts to an identification F p A , g q – Hom
Lie p h p A q , g q . Lemma 5.4.
Let ψ : A Ñ A be a -connected cdga map. Assume that A is -finite. Then,for every finite dimensional Lie algebra g , the morphism (5.8) ψ b id : F p A , g q (cid:47) (cid:47) F p A , g q is an isomorphism onto a closed subvariety.Proof. Our 0-connectivity assumption on ψ means that both A and A are connected cdga s, and that ψ is injective in degree 1. Our 1-finiteness assumption on A , then, im-plies that A is also 1-finite. Furthermore, the injectivity of ψ also implies that the map h p ψ q : h p A q Ñ h p A q is surjective.Using the above discussion, we may replace the a ffi ne map ψ b id : F p A , g q Ñ F p A , g q between spaces of flat connections by the induced map(5.9) h p ψ q ! : Hom Lie p h p A q , g q (cid:47) (cid:47) Hom
Lie p h p A q , g q between representation varieties of Lie algebras. The desired conclusion follows by thesame argument as in Lemma 5.3, with groups replaced by Lie algebras. (cid:3) As we saw in the above proof, the 0-connectivity of the cdga map ψ implies the sur-jectivity of the Lie algebra map h p ψ q . The next example shows that the latter property isstrictly weaker than the former. Nevertheless, we chose to state the lemma the way wedid, since higher connectivity properties for cdga maps will be needed later on. OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 21
Example 5.5.
Let A be the cohomology ring of S _ S , with trivial product and di ff er-ential. Plainly, the holonomy Lie algebra h “ h p A q is the free Lie algebra on 2 generators.Let h { Γ p h q be the third nilpotent quotient of h , and let A be the cochain cdga of thisnilpotent Lie algebra. Denote by ψ : A Ñ A the dual of the composite A Ñ h Ñ h { Γ h .It is not hard to check that ψ extends to a morphism ψ : A Ñ A between finite cdga s,with the property that h p ψ q is surjective. On the other hand, the map ψ is not Naturality properties of jump loci.
We now turn to the naturality properties ofembedded cohomology jump loci.
Lemma 5.6.
Let f : p X , x q Ñ p X , x q be a q-connected, pointed map, and let f be theinduced homomorphism on fundamental groups. Let ι : G Ñ GL p V q be a representation.Then the natural map (5.10) H . p f q : H . p X , V ι ˝ ρ q (cid:47) (cid:47) H . p X , V ι ˝ ρ q , where ρ “ f ! p ρ q for ρ P Hom p π p X q , G q , is q-connected.Proof. Without loss of generality, we may assume that G “ GL p V q and ι is the identitymap. Using standard CW-approximation results from homotopy theory, as recounted forinstance in [35, Ch. V], we may replace f , up to homotopy, by the inclusion of a CW-subcomplex X into a CW-complex X . Since f is assumed to be q -connected, X may beobtained by attaching cells of dimension at least q ` X .Using the long exact sequence in cohomology for the pair p X , X q , we see that our claimis equivalent to the vanishing of the twisted cohomology groups H i p X , X ; V q for i ď q ` V on X . Denote by t X n u the relative skeletal filtration of X .It is well-known that H . p X , X ; V q can be computed as the cohomology of the cellulartwisted cochain complex, whose degree n term is H n p X n , X n ´ ; V q , see e.g. [35, Ch. VI].On the other hand, X n “ X n ´ “ X for n ď q `
1, and this completes the proof. (cid:3)
Lemma 5.7.
Let ψ : A Ñ A be a q-connected map in
CDGA , and let θ : g Ñ gl p V q be aLie algebra representation. Then the natural map (5.11) H . p ψ q : H . p A b V , d ω q (cid:47) (cid:47) H . p A b V , d ω q , where ω “ p ψ b id qp ω q for ω P F p A , g q is q-connected.Proof. Without loss of generality, we may assume that g “ gl p V q and θ is the identitymap. Since ψ is q -connected, the cochain map ψ b id : p A b V , d ω q Ñ p A b V , d ω q is again q -connected. The claim follows from Lemma 2.6 in [22] and its proof. (cid:3) Corollary 5.8.
Let f : X Ñ X be a p q ´ q -connected map between q-finite pointedspaces, for some q ě , and let ι : G Ñ GL p V q be a rational representation. Then theinduced morphism (5.12) f ! : Hom p π p X q , G q (cid:47) (cid:47) Hom p π p X q , G q is a closed embedding which induces isomorphisms V ir p X , ι q Ñ V ir p X , ι qX Hom p π p X q , G q for all i ă q and r ě , and embeddings V qr p X , ι q Ñ V qr p X , ι q for all r ě .Proof. The fact that f ! is a closed embedding follows from Lemma 5.3. The other asser-tions are immediate consequences of Lemma 5.6. (cid:3) If f : π (cid:16) π is an epimorphism between finitely generated groups, the case q “ f ! : Hom p π , k ˆ q Ñ Hom p π, k ˆ q sends V p π q into V p π q . Without the 0-connectivity (i.e., surjectivity) assumption on f , the conclusionmay fail, as illustrated in the following simple example. Example 5.9.
Let f : Z Ñ F “ x x , y y be the inclusion sending 1 to x . Then f ! : p k ˆ q Ñ k ˆ is the projection onto the first factor. On the other hand, V p F q “ p k ˆ q , whereas V p Z q “ t u . Corollary 5.10.
Let ψ : A Ñ A be a p q ´ q -connected map between q-finite cdga sfor some q ě , and let θ : g Ñ gl p V q be a Lie algebra representation, with g and Vfinite-dimensional. Then the natural morphism (5.13) ψ b id : F p A , g q (cid:47) (cid:47) F p A , g q is a closed embedding which induces isomorphisms R ir p A , θ q Ñ R ir p A , θ q X F p A , g q forall i ă q and r ě , and embeddings R qr p A , θ q Ñ R qr p A , θ q for all r ě .Proof. The fact that ψ b id is a closed embedding follows from Lemma 5.4. The otherassertions are immediate consequences of Lemma 5.7. (cid:3) Corollary 5.11.
Let X be a pointed space with fundamental group π , let f : X Ñ K : “ K p π, q be a classifying map, and let ι : G Ñ GL p V q be a representation. Then theinduced isomorphism f ! : Hom p π p K q , G q Ñ Hom p π p X q , G q restricts to isomorphisms V ir p π, ι q – V ir p X , ι q for i ď and r ě .Proof. The map f is 1-connected, and so the claim follows from Lemma 5.6. (cid:3) Finite families of epimorphisms.
We conclude this section with a setup that willoften recur in the sequel. Let π be a finitely generated group, and let t f : π (cid:16) π f u f P E be afinite family of epimorphisms. Let ι : G Ñ GL p V q be a rational representation of C -linearalgebraic groups. By Corollary 5.8, the natural inclusion(5.14) Hom p π, G q Ě ď f P E f ! Hom p π f , G q induces for each i ď r ě V ir p π, ι q Ě ď f P E f ! V ir p π f , ι q . OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 23
One of our main goals for the remainder of this paper is to delineate several large classesof groups endowed with the required finite families of epimorphisms, for which the abovetwo inclusions hold as equalities near 1.6. A natural comparison between embedded jump loci
This section is devoted to proving our main naturality result.6.1.
Functors of Artin rings.
Returning now to the setup from §§ X be apointed, 1-finite space, and assume there is a 1-finite cdga such that Ω k p X q » A in CDGA . Let π “ π p X q , and let G be a linear algebraic group, with Lie algebra g . Wewish to compare the analytic germs Hom p π, G q p q and F p A , g q p q . Let R and R be therespective coordinate local algebras. By Artin approximation, we may start by looking atthe completions of these rings, p R and p R . Alternatively, we may analyze the correspondingfunctors of Artin rings,(6.1) h p R p A q “ Hom p p R , A q and h p R p A q “ Hom p p R , A q , for A a local Artin algebra, where Hom stands for morphisms of local algebras. We recallthat h p R p A q “ Hom p π, exp p g b m A qq , see [15], and h p R p A q “ F p A , g b m A q , see [9].Let Z be a zig-zag of 1-equivalences in ACDGA connecting Ω k p X q to A , as in (4.6). Itfollows from Theorem 4.8 that the natural bijection α Z defined in (4.13) yields an isomor-phism(6.2) α Z : p R (cid:47) (cid:47) p R . Assume now that X and A are q -finite, for some q ě
1, and that Ω k p X q » q A in CDGA .As usual, let ι : G Ñ GL p V q be a rational representation, with tangential representation θ : g Ñ gl p V q . For each i ď q and r ě
0, we denote by I ir Ď R the radical of the definingideal of the germ V ir p X , ι q p q inside Hom p π, G q p q . Similarly, we will let I ir Ď R stand forthe radical of the defining ideal of the germ R ir p A , θ q p q inside F p A , g q p q . Lemma 6.1.
For any zig-zag Z as above, the isomorphism α Z : p R Ñ p R from (6.2) identifies p I ir with p I ir , for all i ď q and r ě .Proof. In view of Corollary 4.9, we may replace the zig-zag Z by a special zig-zag S . Theclaim for S follows from [9, Lem. 9.9]. (cid:3) Now let f : X Ñ X be a pointed map, let Φ : A Ñ A be a cdga map, and assumeboth spaces and cdga s are q -finite, for some q ě
1. Let φ : R Ñ R be the morphism oflocal rings induced by the map f ! : Hom p π , G q Ñ Hom p π, G q , and let ¯ φ : R Ñ R be themorphism of local rings induced by the map Φ b id : F p A , g q Ñ F p A , g q . Suppose there is a q -equivalence in ACDGA between the maps Ω k p f q : Ω k p X q Ñ Ω k p X q and Φ : A Ñ A . We then obtain zig-zags Z from Ω k p X q to A and Z from Ω k p X q to A ; let α : p R Ñ p R and α : p R Ñ p R be the corresponding isomorphisms, given by (6.2). Lemma 6.2.
With the above setup, we have that α ˝ ˆ φ “ ˆ¯ φ ˝ α .Proof. In terms of functors of Artin rings, we have that h α “ mon ˝ β Z and h α “ mon ˝ β Z .First we show that the following diagram commutes, for every local Artin algebra A .(6.3) F p A , g b m A q F p r A b g b m A q G p r A b g b m A q β Z (cid:47) (cid:47) F p r Ω p X q b g b m A q G p r Ω p X q b g b m A q F p A , g b m A q Φ b id (cid:79) (cid:79) F p r A b g b m A q G p r A b g b m A q β Z (cid:47) (cid:47) Def r Φ b id p A q (cid:79) (cid:79) F p r Ω p X q b g b m A q G p r Ω p X q b g b m A q Def r Ω p f qb id p A q (cid:79) (cid:79) Plainly, it is enough to verify the commutativity of this diagram for an elementary q -equivalence in ACDGA . In this case, β Z “ Def r ψ b id p A q and β Z “ Def r ψ b id p A q , by construc-tion. The claim now follows from Lemma 4.5 and Corollary 4.3.Next, we show that the diagram(6.4) F p r Ω p X q b g b m A q G p r Ω p X q b g b m A q mon (cid:47) (cid:47) Hom p π p X q , exp p g b m A qq F p r Ω p X q b g b m A q G p r Ω p X q b g b m A q Def r Ω p f qb id p A q (cid:79) (cid:79) mon (cid:47) (cid:47) Hom p π p X q , exp p g b m A qq f ! (cid:79) (cid:79) commutes, where the horizontal arrows are as in Theorem 4.8. In fact, commutativityholds even before taking quotients by the gauge actions, due to the naturality propertiesof the monodromy construction, as detailed in [9, § φ takes the value Φ b id on A , whereas for ˆ φ we obtain the value f ! . The commutativity of the above two diagrams now verifies the claim. (cid:3) A natural comparison between embedded jump loci.
We now consider a familyof maps between pointed spaces, t f : X Ñ X f u f P E , indexed by a finite set E , and we let t f : π Ñ π f u be the family of induced homomorphisms on fundamental groups. We alsoconsider a family of ACDGA maps, t Φ f : A f Ñ A u f P E , indexed by the same set, and wewill assume that A and A f are connected cdga s. Fix an integer q ě OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 25
Suppose that Ω p f q » q Φ f in ACDGA . We then have a commuting diagram as in (3.4),(6.5) Z f : Ω p X q A ψ (cid:111) (cid:111) ψ (cid:47) (cid:47) ¨ ¨ ¨ A (cid:96) ´ (cid:111) (cid:111) ψ (cid:96) ´ (cid:47) (cid:47) AZ f : Ω p X f q Ω p f q (cid:79) (cid:79) A Φ (cid:79) (cid:79) ψ (cid:111) (cid:111) ψ (cid:47) (cid:47) ¨ ¨ ¨ A (cid:96) ´ Φ (cid:96) ´ (cid:79) (cid:79) (cid:111) (cid:111) ψ (cid:96) ´ (cid:47) (cid:47) A f . Φ f (cid:79) (cid:79) Let β Z f and β Z f be the associated natural bijections, defined as in display (4.7). Definition 6.3.
We will say that Ω p f q » q Φ f in ACDGA , uniformly with respect to f P E if the bijection β Z f is independent of f .We are now in a position to state and prove our main naturality result. As usual, G isa k -linear algebraic group (where k “ R or C ), and g is its Lie algebra. Furthermore, weconsider a rational representation ι : G Ñ GL p V q over k , and we let θ : g Ñ gl p V q be thetangential representation. Theorem 6.4.
Suppose the following conditions hold:(1) All the above spaces and cdga s are q-finite.(2) Both f and Φ f are p q ´ q -connected maps, for all f P E.(3) Ω p f q » q Φ f in ACDGA , uniformly with respect to f P E.Then we may find local analytic isomorphisms a : F p A , g q p q » ÝÑ Hom p π, G q p q anda f : F p A f , g q p q » ÝÑ Hom p π f , G q p q for all f P E such that the following diagram com-mutes, for all f P E, F p A , g q p q a (cid:47) (cid:47) Hom p π, G q p q F p A f , g q p q Φ f b id (cid:79) (cid:79) a f (cid:47) (cid:47) Hom p π f , G q p q . f ! (cid:79) (cid:79) Moreover, for all f P E, i ď q, and r ě , this construction induces a commutingdiagram of (local, reduced) embedded jump loci, p F p A , g q , R ir p A , θ qq p q a (cid:47) (cid:47) p Hom p π, G q , V ir p X , ι qq p q p F p A f , g q , R ir p A f , θ qq p q Φ f b id (cid:79) (cid:79) a f (cid:47) (cid:47) p Hom p π f , G q , V ir p X f , ι qq p q , f ! (cid:79) (cid:79) where both horizontal arrows are isomorphisms of analytic pairs.Proof. Due to our connectivity assumptions, Corollaries 5.8 and 5.10 apply, thereby show-ing that both f ! and Φ f b id respect the corresponding (global) jump loci. We will deduceall other claims from Proposition 2.3.To begin with, we denote by φ f : R Ñ R f and ¯ φ f : R Ñ R f the morphisms of analyticalgebras corresponding to the local analytic maps f ! : Hom p π f , G q p q Ñ Hom p π, G q p q and Φ f b id : F p A f , g q p q Ñ F p A , g q p q , respectively. To verify that both φ f and ¯ φ f areepimorphisms, we may use a standard, equivalent property, namely, the injectivity of theassociated natural transformation between Hom-functors, see for instance [33, III.4]. Inturn, this property readily follows from the injectivity on A -points of the correspondingmorphisms between a ffi ne coordinate rings, for an arbitrary commutative algebra A .For representation varieties, the map on A -points is given by f ! : Hom p π f , G p A qq Ñ Hom p π, G p A qq . Clearly, this map is injective, since by assumption, f is 0-connected, i.e., f is surjective. Likewise, for varieties of flat connections, the map on A -points is given by Φ f b id : F p A f b g b A q Ñ F p A b g b A q . Again, this map is injective, since by assumption Φ f is 0-connected, i.e., injective. This shows that the first preliminary hypotheses fromProposition 2.3 are satisfied.For i ď q and r ě
0, let I ir Ď R and I ir Ď R be defining radical ideals for the reducedanalytic germs V ir p X , ι q p q and R ir p A , θ q p q , as in Lemma 6.1. Similarly, for f P E , let I ir p f q Ď R f and I ir p f q Ď R f be defining radical ideals for V ir p X f , ι q p q , and R ir p A f , θ q p q .We deduce from display (5.5) that I ir ‰ R if and only if I ir ‰ R , which happens preciselywhen r ď b i ¨ dim p V q , where recall b i “ b i p X q “ b i p A q . Similarly, I ir p f q is a proper idealif and only I ir p f q is a proper ideal.Note that I ir “ R is equivalent to V ir p X , ι q p q “ ∅ , and similarly for I ir p f q . By Corollary5.8, if V ir p X , ι q p q is empty, then V ir p X f , ι q p q is also empty. Consequently, if I ir is non-proper, then I ir p f q is also non-proper. Likewise, Corollary 5.10 implies the following: if I ir is non-proper, then I ir p f q is also non-proper.The pairs p i , r q with 0 ď i ď q and 0 ď r ď b i ¨ dim p V q form a finite set, which we willdenote by F . Plainly, we need to verify the second claim of the theorem only for the pairs p i , r q P F and the maps f P E for which the ideal I ir p f q is proper.By assumption (3), Ω p f q » q Φ f in ACDGA , uniformly with respect to f P E . Inparticular, we have a zig-zag Z f of q -equivalences from Ω p X q to A and a zig-zag Z f from Ω p X f q to A f for each f P E , as in diagram (6.5). Let(6.6) α f : “ mon ˝ β Z f : p R f » ÝÑ p R f be the isomorphism from (6.2). By our uniformity assumption, the isomorphisms mon ˝ β Z f coincide with a fixed isomorphism, α : p R » ÝÑ p R .It follows from Lemma 6.1 that the isomorphism α f identifies the ideal p I ir p f q with p I ir p f q ,for all i ď q and r ě
0, and for all f P E . Likewise, the isomorphism α identifies the ideal p I ir with p I ir , for all i ď q and r ě
0. Finally, assumption (4) from Proposition 2.3 followsfrom Lemma 6.2.The desired conclusions follow from Proposition 2.3, applied to the above ideals. (cid:3)
OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 27
Naturality with respect to a single map.
For a one-element family E “ t f u , theuniform equivalence property from Definition 6.3 reduces to Ω p f q » q Φ f in ACDGA . Wethus have the following immediate corollary to Theorem 6.4. Corollary 6.5.
Let f : X Ñ X be a continuous, p q ´ q -connected map between q-finite,pointed spaces, for some q ě . Suppose Φ : A Ñ A is a p q ´ q -connected cdga mapbetween q-finite cdga s such that Ω p f q » q Φ in ACDGA . We then may find local analyticisomorphisms a and a which fit into the diagram F p A , g q p q a (cid:47) (cid:47) Hom p π p X q , G q p q F p A , g q p q Φ b id (cid:79) (cid:79) a (cid:47) (cid:47) Hom p π p X q , G q p q . f ! (cid:79) (cid:79) Furthermore, for all i ď q, and r ě , this construction induces a commuting diagram of(local, reduced) embedded jump loci, p F p A , g q , R ir p A , θ qq p q a (cid:47) (cid:47) p Hom p π p X q , G q , V ir p X , ι qq p q p F p A , g q , R ir p A , θ qq p q Φ b id (cid:79) (cid:79) a (cid:47) (cid:47) p Hom p π p X q , G q , V ir p X , ι qq p q , f ! (cid:79) (cid:79) where both horizontal arrows are isomorphisms of analytic pairs. Here is a situation where this type of property holds.
Lemma 6.6.
Let f : X Ñ X be a continuous map between pointed spaces, and assumeH . p f q is q-connected, for some q ě . Let A be a connected cdga , and suppose A is aq-model for X. Then Ω p f q » q id A in ACDGA .Proof. Consider the following commuting diagram in
ACDGA ,(6.7) Ω p X q M q ρ (cid:104) (cid:104) ρ (cid:118) (cid:118) ¯ ρ (cid:47) (cid:47) A , Ω p X q Ω p f q (cid:79) (cid:79) where ρ and ¯ ρ are q -minimal model maps provided by the assumption that Ω p X q » q Ω p X q » q A . Clearly, the map ρ “ Ω p f q ˝ ρ is a q -equivalence, since both Ω p f q and ρ are. Hence, Ω p f q » q id A in ACDGA , and the claim follows. (cid:3) Corollary 6.7.
Fix q ě . Let f : X Ñ X be a p q ´ q -connected map between q-finite,pointed spaces, such that H . p f q is q-connected. Let A be a q-finite cdga , and suppose Ais a q-model for X. Then the conclusions of Corollary 6.5 hold for A “ A and Φ “ id A . We conclude with one more class of spaces and maps where Corollary 6.5 applies.
Proposition 6.8.
Let f : X Ñ X be a p q ´ q -connected map between q-finite, pointedspaces, for some q ě . Assume that f is formal over k . Then the conclusions of Corollary6.5 hold for Φ “ H . p f q : H . p X , k q Ñ H . p X , k q .Proof. Our connectivity hypothesis implies that H p f q is injective. The claim follows atonce from Proposition 3.4. (cid:3)
7. K¨ ahler manifolds
In this section we show that pointed holomorphic maps between compact K¨ahler mani-folds can be uniformly modeled by the homomorphisms induced in (real) cohomology. Asan application, we derive a structural result on the germs at the origin of rank 2 embeddedjump loci of K¨ahler groups.7.1.
Essentially rank one flat connections.
We start with some preliminary lemmas.
Lemma 7.1.
A non-abelian Lie subalgebra g Ď sl p C q is either equal to sl p C q or isisomorphic to the standard Borel subalgebra, sol p C q .Proof. Easy exercise. (cid:3)
We now recall a few facts from [22]. Let A be a cdga , let g be a Lie algebra, and let F p A , g q Ă A b g be the set of g -valued flat connections on A . Let us define F p A , g q tobe the subset of A b g consisting of all tensors of the form η b g with d η “
0. We also fixa finite-dimensional representation θ : g Ñ gl p V q , and define Π p A , θ q to be the subset of F p A , g q consisting of all tensors as above which also satisfy det p θ p g qq “
0. When A is1-finite and g is finite-dimensional, both F p A , g q and Π p A , θ q are closed, homogeneoussubvarieties of F p A , g q . Moreover, if H i p A q ‰
0, then Π p A , θ q Ď R i p A , θ q . Lemma 7.2.
Under the above finiteness assumptions, every cdga map Φ : A Ñ A in-duces algebraic maps, Φ b id : F p A , g q Ñ F p A , g q and Φ b id : Π p A , θ q Ñ Π p A , θ q .Moreover, if H p Φ q is an isomorphism, then both these algebraic maps are isomorphisms.Proof. Follows directly from the definitions. (cid:3)
Lemma 7.3.
Let Φ : A Ñ A be a cdga map, where A “ p Ź . U , d “ q with ă dim U ă 8 and A is -finite, and assume H p Φ q is an isomorphism. Also let g Ď sl p C q be a Lie subalgebra, and let θ : g Ñ gl p V q be a finite-dimensional representation. Thenthe following hold:(1) Φ b id induces an isomorphism between F p A , g q and F p A , g q .(2) Φ b id induces an isomorphism between R p A , θ q and Π p A , θ q .Proof. By construction, A is the Chevalley–Eilenberg cochain cdga of the abelian Liealgebra U . Hence, by [22, Lem. 4.14], we have that F p A , g q “ F p A , g q . The firstclaim follows at once from Lemma 7.2. OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 29
It is easily checked that R p A q “ t u . Since F p A , g q “ F p A , g q , we may apply[22, Cor. 3.8] to conclude that R p A , θ q “ Π p A , θ q . The second claim now follows fromLemma 7.2. (cid:3) The uniformity property.
Let t f : M Ñ M f u f P E be a finite family of pointed, holo-morphic maps between compact K¨ahler manifolds. Each map f P E induces a homo-morphism H . p f q : H . p M f q Ñ H . p M q between the respective cohomology algebras withcoe ffi cients in k “ R or C . In fact, these homomorphisms may be viewed as ACDGA maps, by setting the di ff erentials to be zero, and taking the augmentations given by thebasepoints. Proposition 7.4.
In the above setup, Ω p f q » H . p f q in ACDGA , uniformly with respect tof P E.Proof.
To prove the claim, it is enough to show there is a functorial zig-zag of quasi-isomorphisms in
ACDGA connecting Ω p M q to H . p M q , for any pointed, compact K¨ahlermanifold M . In order to construct such a zig-zag, we proceed in two steps, following [7, §
6] and [13, § k “ R .Let p Ω dR p M q , d q be the de Rham cdga of the underlying di ff erentiable manifold, with d the exterior di ff erential. Set d c “ J ´ dJ , where J is the complex structure on the tangentbundle to M . This gives another cdga , p Ω dR p M q , d c q . By the first proof of the Main The-orem from [7], there is a zig-zag of quasi-isomorphisms in CDGA connecting p Ω dR p M q , d q to p H . p Ω dR p M q , d c q , d “ q , natural with respect to holomorphic maps. Taking homomor-phisms induced in cohomology, we obtain a natural zig-zag of cdga quasi-isomorphismsconnecting p H . dR p M q , d “ q to p H . p Ω dR p M q , d c q , d “ q . In fact, both zig-zags are in ACDGA , since all their terms are equal to R when M is a point. Combining these twozig-zags, we obtain a functorial zig-zag of quasi-isomorphisms in ACDGA from Ω dR p M q to p H . dR p M q , d “ q .On the other hand, the proof of the de Rham theorem from [13] provides a zig-zagof quasi-isomorphisms in CDGA connecting Ω dR p M q to Ω R p M q , which is natural with re-spect to di ff erentiable maps. An argument as above shows that this zig-zag is in ACDGA .Putting things together, and using the classical de Rham theorem, we arrive at the desiredconclusion. (cid:3)
Admissible maps and rank jump loci. A connected, complex manifold M issaid to be a quasi-compact K ¨ahler manifold if there is compact K¨ahler manifold M anda normal crossing divisor D Ă M such that M “ M z D . Of course, all compact K¨ahlermanifolds belong to this class. Furthermore, if M is an irreducible smooth, complex quasi-projective variety, or, for short, a quasi-projective manifold , then M is also of this type, byresolution of singularities.Given a quasi-compact K¨ahler manifold M , there is a certain finite family of pointedholomorphic maps, t f : M Ñ M f u f P E p M q , with each M f a quasi-projective manifold,which is intimately related to the structure near 1 of the characteristic variety V p M q . More precisely, a holomorphic map onto a smooth complex curve, f : M Ñ M f , issaid to be admissible if it extends to a holomorphic surjection with connected fibers, f : M Ñ M f , where M (respectively M f ) is a K¨ahler compactification of M (respectively M f ) obtained by adding a normal crossing divisor. It is known that, up to reparametriza-tion at the target, there is a finite family E p M q of such maps with the property that χ p M f q ă
0. For each f P E p M q , let us write π “ π p M q and π f “ π p M f q . It is readilyseen that the induced homomorphism on fundamental groups, f : π Ñ π f , is surjective.Work of Arapura [2] shows that the correspondence(7.1) f (cid:123) f ! Hom p π f , C ˆ q establishes a bijection between the set E p M q and the set of positive-dimensional, irre-ducible components of the characteristic variety V p M q passing through 1.For a pointed CW-space M with fundamental group π , we denote by f : M Ñ K p π abf , q the classifying map determined up to homotopy by the property that p f q “ abf : π (cid:16) π abf , where abf is the canonical projection of the group π onto its maximal torsion-freeabelian quotient. When M is a quasi-compact K¨ahler manifold, we set(7.2) E p M q “ E p M q Y t f u . In the rank one case, i.e., when ι “ id C ˆ , both inclusions, (5.14) and (5.15) for i “ r “
1, become equalities near the origin 1, for the family t f | f P E p M qu . Theorem 7.5.
Let M be a quasi-compact K¨ahler manifold, and let π “ π p M q . Then, Hom p π, C ˆ q p q “ ď f P E p M q f ! Hom p π f , C ˆ q p q , (7.3) V p π q p q “ ď f P E p M q f ! V p π f q p q . (7.4) Proof.
The first claim is easily verified. Indeed, the abelianization map, π (cid:16) π ab , inducesan isomorphism of character groups, while the map induced by the natural projection π ab (cid:16) π abf identifies Hom p π abf , C ˆ q with the identity component of Hom p π ab , C ˆ q . Itfollows that p f q “ π abf induces an isomorphism between germs at 1 of C ˆ -representationvarieties, and so (7.3) holds.The second claim is much more subtle. Since χ p M f q ă
0, it is easily seen that V p π f q “ V p M f q “ Hom p π f , C ˆ q , for f P E p M q . If b p M q “
0, we know from(5.5) that V p π q p q “ ∅ , and we are done. If b p M q ą
0, then either V p π q p q “ t u , orall irreducible components of V p π q passing through 1 are positive-dimensional. In thefirst case we are done, since 1 P p f q ! V p π abf q . In the second case, equality (7.4) followsfrom the aforementioned deep results of Arapura. This completes the proof. (cid:3) Rank embedded jump loci of K¨ahler manifolds. Let M be a compact K¨ahlermanifold with fundamental group π . A map f : M Ñ M f is admissible in the sensefrom § M f is a compact Riemann surface and f is a holomorphic surjection with OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 31 connected fibers. The Albanese map, f : M Ñ Alb p M q , is a holomorphic map betweencompact K¨ahler manifolds which classifies the canonical projection, π (cid:16) π abf .Our next goal is to extend the rank 1 results (7.3)–(7.4) to the rank 2 case. We start witha lemma. Lemma 7.6.
Let M be a compact K ¨ahler manifold with b p M q ą . Let g be a non-abelianLie subalgebra of sl p C q , and let θ : g Ñ gl p V q be a finite-dimensional representation.Then the following equalities hold: F p H . p M q , g q “ F p H . p M q , g q Y ď f P E p M q f ! p F p H . p M f q , g qq , (7.5) R p H . p M q , θ q “ Π p H . p M q , θ q Y ď f P E p M q f ! p F p H . p M f q , g qq , (7.6) where all cdga s are endowed with zero di ff erential.Proof. This is proved in [22, Cor. 7.2] for a 1-formal, quasi-projective manifold M . Byresults from [10], the proof also works for 1-formal, quasi-compact K¨ahler manifolds, inparticular, for a compact K¨ahler manifold M . (cid:3) Theorem 7.7.
Let M be a compact K¨ahler manifold with fundamental group π , and setE p M q “ E p M q Y t f u as in (7.2) . Let G be C -linear algebraic group with non-abelianLie algebra g Ď sl p C q , and let ι : G Ñ GL p V q be a rational representation. Then, Hom p π, G q p q “ ď f P E p M q f ! Hom p π f , G q p q , (7.7) and, for i “ r “ or i “ and r ě , V ir p π, ι q p q “ ď f P E p M q f ! V ir p π f , ι q p q . (7.8) Proof.
We wish to apply Theorem 6.4 with q “ t f : M Ñ M f u f P E p M q and cdga maps Φ f “ t H . p f q : H . p M f q Ñ H . p M qu f P E p M q , where all the dif-ferentials are set equal to 0. For that, we need to verify that the three hypotheses of thetheorem hold.First, all spaces and cdga s in question are 1-finite (in fact, -finite). Second, each map f is surjective, hence each f P E p M q is 0-connected. Thus, H . p f q is also 0-connected.Finally, by Proposition 7.4, Ω p f q » H . p f q in ACDGA , uniformly with respect to f P E p M q .In the case when i “
0, equality (7.7) clearly implies equality (7.8), by Corollary 5.8.Thus, we may assume i “ r “ b p M q “
0. By (5.5), we have that V p π, ι q p q “ ∅ . Therefore,equality (7.8) follows trivially. Moreover, the natural map Ω p K p , qq Ñ Ω p K p π, qq is a π has the same 1-minimal model as the trivial group. It then followsfrom [9, Thm. A] that Hom p π, G q p q “ t u . Therefore, equality (7.7) holds trivially.Thus, we may also assume that b p M q ą
0. We deduce from formula (7.5) and Lemma7.3, part (1) that(7.9) F p H . p M q , g q “ f !0 F p H . p M q , g q Y ď f P E p M q f ! p F p H . p M f q , g qq , where M denotes the Albanese variety Alb p M q » K p π abf , q , and H . p M q “ Ź . H p M q .By taking germs at the origin and using the naturality properties from Theorem 6.4, for-mula (7.9) implies that equality (7.7) holds.Similarly, formula (7.6) and Lemma 7.3 part (2) together imply that(7.10) R p H . p M q , θ q “ f !0 R p H . p M q , θ q Y ď f P E p M q f ! p F p H . p M f q , g qq . For each f P E p M q , note that M f is a 2-dimensional CW-complex with χ p M f q ă
0. Aneasy Euler characteristic argument then shows that V p π f , ι q “ V p M f , ι q “ Hom p π f , G q .Again by Theorem 6.4, formula (7.10) now implies that equality (7.8) holds. This com-pletes the proof. (cid:3) Remark 7.8.
In [20, Cor. B], Loray, Pereira, and Touzet prove the following result, whichrefines earlier results of Corlette and Simpson [6]. Let X be a quasi-projective mani-fold, and let ρ P Hom p π p X q , SL p C qq be a representation which is not virtually abelian.Then there is an orbifold morphism, f : X Ñ Y , such that the associated representation,˜ ρ P Hom p π p X q , PSL p C qq , belongs to f ! Hom p π orb1 p Y q , PSL p C qq , where Y is either a1-dimensional complex orbifold, or a polydisk Shimura modular orbifold.For a finitely generated group π and a linear algebraic group G , the abelian part ofthe representation variety Hom p π, G q coincides near 1 with abf ! Hom p π abf , G q p q . Indeed,[9, Thm. A] implies that the canonical projection π ab (cid:16) π abf induces an isomorphism ofgerms at the origin of the respective representation varieties.This remark shows that formula (7.7) from Theorem 7.7 may be viewed as a compactK¨ahler analogue near 1 of [20, Cor. B]. In this context, it provides a simpler classifica-tion: the representation ρ is either abelian, or it pulls back via an admissible map from acompact Riemann surface of genus g ą The main di ffi culty in the non-abelian case. The naturality property from [9, Thm.B(2)] is a consequence of the following fact, which holds in the abelian case. Let X be a1-finite space and A . a 1-finite cdga . The fact that Ω k p X q » A in CDGA means that thereis a zig-zag of 1-equivalences in
CDGA ,(7.11) Ω k p X q N ψ (cid:111) (cid:111) ¯ ψ (cid:47) (cid:47) A , where N “ p Ź . U , d q is a 1-minimal cdga , see Lemma 3.3. If g is an abelian Lie algebra,it follows from the definitions that F p B , g q “ H p B q b g , for any connected cdga B . OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 33
Applying this observation to the map ¯ ψ , we conclude that, in the abelian case, there is abijection(7.12) ¯ ψ b id : F p N , g q » (cid:47) (cid:47) F p A , g q . For a non-abelian Lie algebra g , though, this map is not necessarily surjective. Toillustrate this phenomenon, we first need a lemma. Lemma 7.9. If g “ sl p C q and ¯ ψ b id is surjective, then F p A , g q “ F p A , g q .Proof. The cdga N “ p Ź . U , d q comes endowed with the canonical filtration, N “ Ť n ě N n , of a 1-minimal cdga , where each cdga N n is of the form p Ź . U n , d q . Sincedim H p N q ă 8 , N n is the cochain algebra of a certain finite-dimensional, nilpotent Liealgebra. Since g “ sl p C q , it follows from [22, Lem. 4.14] that F p N n , g q “ F p N n , g q ,for each n ě
1. Hence, F p N , g q “ F p N , g q . Since ¯ ψ b id is surjective, we concludethat F p A , g q “ F p A , g q . (cid:3) Example 7.10.
Let Σ g be a compact Riemann surface of genus g ą
1. Since Σ g is a formalspace, the cdga A “ p H . p Σ g q , d “ q is a finite model for it. Let N be a 1-minimal modelfor A , and let ¯ ψ : N Ñ A be the corresponding map. It follows from [22, Lem. 7.3] that F p A , g q ‰ F p A , g q . By Lemma 7.9, then, the map ¯ ψ b id is not surjective.Thus, in the case when G “ SL p C q , we have no natural analytic map F p A , g q Ñ Hom p π p X q , G q . This is the reason why we have to construct a local analytic isomorphismbetween the germs at the origin of the two varieties, in a manner which is compatiblewith both continuous maps and cdga maps, using the simultaneous Artin approximationtechnique from Proposition 2.3. 8. P rincipal bundles In this section, we apply our theory to principal bundles. When the base manifold isformal, we obtain a structural result for the germs at the origin of rank 2 embedded jumploci of the total space.8.1.
Two-element families with the uniform property.
In the applications of Theorem6.4, we also need to take into account the projection of a group π onto its maximal torsion-free abelian quotient, abf : π (cid:16) π abf . Theorem 8.1.
Let f : M Ñ N be a continuous, pointed map. Denote by f : M Ñ K p π p M q abf , q the classifying map for the above projection. Suppose that M and N areq-finite, for some q ě , and that Ω p f q » q Φ in ACDGA , where Φ : A N Ñ A M is a cdga map between q-finite objects. Set A . “ p Ź . H p M q , d “ q . There is then a cdga map Φ : A Ñ A M inducing an isomorphism on H , and such that Ω p f q » q Φ in ACDGA ,uniformly with respect to the families t f , f u and t Φ , Φ u . Moreover, if f and Φ are -connected maps, then all the hypotheses from Theorem 6.4 are satisfied for q “ . Proof.
The assumption that Ω p f q » q Φ provides a zig-zag Z of q -equivalences in ACDGA connecting Ω k p M q to A M . Let ρ : N Ñ Ω k p M q be a π -adapted 1-minimal model map, asin [9, § q -minimal model map ρ : M Ñ Ω k p M q . ByProposition 4.6, there is a special zig-zag S of the form Ω k p M q M ρ (cid:111) (cid:111) ¯ ρ (cid:47) (cid:47) A M such that β Z “ β S .Now, as explained in [9, § cdga inclusion, j : A ã Ñ N ,inducing an isomorphism on H . It follows that the map Φ “ ¯ ρ ˝ j : A Ñ A M has thesame property. Putting things together, we obtain the following commuting diagram in ACDGA :(8.1) M ρ (cid:122) (cid:122) ¯ ρ (cid:35) (cid:35) Ω p M q N (cid:63)(cid:31) i (cid:79) (cid:79) ρ (cid:111) (cid:111) ¯ ρ (cid:47) (cid:47) A M Ω p π abf q Ω p f q (cid:79) (cid:79) A ρ abf (cid:111) (cid:111) (cid:63)(cid:31) j (cid:79) (cid:79) A . Φ (cid:79) (cid:79) Both upper-diagonal arrows are q -equivalences, and both lower-horizontal arrows are -equivalences. It follows that Ω p f q » q Φ in ACDGA , as claimed. The uniform propertyfollows from the equality β Z “ β S . It is obvious that the map f is 0-connected. Finally,since H p Φ q is injective, the map Φ is also 0-connected. (cid:3) Models for principal bundle projections.
Let K be a compact, connected real Liegroup acting freely on a closed, smooth manifold M . Let N “ M { K be the orbit space,and let f : M Ñ N be the projection map of the resulting principal K -bundle. Of course,both M and N have the homotopy type of a finite CW-complex. We will fix compatiblebasepoints for M and N . Note that f is 0-connected, by the exact homotopy sequence ofthe fibration K Ñ M f ÝÑ N and the connectivity of K .By a classical result of H. Hopf, the cohomology algebra of K (with coe ffi cients in afield k of characteristic 0) is of the form H . p K q “ Ź P . , where P . is a finite-dimensional,oddly graded k -vector space. Let r τ s : P . Ñ H . ` p N q be the transgression in the Serrespectral sequence of our fibration.Suppose A N is a cdga model for N , so that there is a zig-zag of quasi-isomorphismsconnecting Ω k p N q to A N . Such a zig-zag yields an isomorphism of H . p N q with H . p A N q .Let τ : P . Ñ Z . ` p A N q be a lift of r τ s . As noted in § A M : “ A N b τ Ź P is well-defined, up to a cdga isomorphism extending id A N . Proposition 8.2.
Let f : M Ñ N be the projection map of a principal K-bundle as above,and suppose N admits a finite model A N . Let Φ : A N ã Ñ A M be the canonical cdga inclusion. Then A M is a finite model for M, and both f and Φ are -connected maps.Moreover, Ω p f q » Φ in ACDGA , and thus the conclusions of Corollary 6.5 hold for q “ . OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 35
Proof.
Clearly, since A N is a finite cdga , then A M is also a finite cdga . Equally clearly,the map Φ is 0-connected. By the classical Hirsch Lemma (see [13, pp. 216–218]) thereis a commutative diagram in ACDGA ,(8.2) Ω p M q Ω p N q b τ Ź P h (cid:111) (cid:111) Ω p N q Ω p f q (cid:103) (cid:103) (cid:63)(cid:31) (cid:79) (cid:79) where h is an -equivalence. Since by assumption Ω p N q » A N , there is a minimal cdga N , connected by quasi-isomorphisms ψ : N Ñ Ω p N q and ¯ ψ : N Ñ A N . We then obtaina commuting diagram in CDGA ,(8.3) Ω p N q b τ Ź P Ω p N q b τ N Ź P » (cid:111) (cid:111) N b τ N Ź P ψ b id (cid:111) (cid:111) ¯ ψ b id (cid:47) (cid:47) A N b τ A Ź P » (cid:47) (cid:47) A M Ω p N q (cid:63)(cid:31) (cid:79) (cid:79) (cid:53) (cid:85) (cid:104) (cid:104) N (cid:63)(cid:31) (cid:79) (cid:79) ψ (cid:111) (cid:111) ¯ ψ (cid:47) (cid:47) A N (cid:63)(cid:31) (cid:79) (cid:79) (cid:44)(cid:12) Φ (cid:58) (cid:58) where the transgression r τ N s is identified with r τ N s using H . p ψ q , while r τ N s is identifiedwith r τ A s using H . p ¯ ψ q . Note that all cdga s in (8.3) are augmented, and all maps respectaugmentations. By [13, Lem. 14.2], the maps ψ b id and ¯ ψ b id are quasi-isomorphisms,since both ψ and ¯ ψ are. Splicing together diagrams (8.2) and (8.3) we reach the desiredconclusions. (cid:3) Embedded jump loci of principal bundles.
Before stating and proving the mainresult of this section, we need two more lemmas. According to the guiding philosophyof [22], the essentially rank 1 part of the higher-rank resonance varieties of a cdga isdetermined by rank 1 resonance. We begin with a version of this general principle, validfor families of cdga morphisms.Fix an integer q ě
1. Let t φ f : A f Ñ A u f P E be a finite family of p q ´ q -connectedmaps between connected C - cdga s. Also, let g be a Lie algebra, and let θ : g Ñ gl p V q be afinite-dimensional representation. For each i ď q such that H i p A q ‰
0, Corollary 3.8 andLemma 2.6 from [22] give an inclusion(8.4) R i p A , θ q Ě Π p A , θ q Y ď f P E p φ f b id q R i p A f , θ q . Lemma 8.3.
Assume (8.4) holds as an equality in the rank case. Then R i p A , θ q X F p A , g q Ď Π p A , θ q Y ď f P E p φ f b id q R i p A f , θ q . Proof.
Let ω “ η b g be a non-zero element in ` R i p A , θ q X F p A , g q ˘ z Π p A , θ q . From[22, Cor. 3.8], we know that η b g belongs to R i p A , θ q X F p A , g q if and only if there is aneigenvalue λ of θ p g q such that λη P R i p A q . By our assumption on the rank 1 resonance, λη “ φ f p η q , for some η P R i p A f q . Since λ ‰ ω “ p φ f b id qp η f b g q , forsome η f P A f such that d η f “ λη f P R i p A f q . Again by [22, Cor. 3.8], we concludethat η f b g belongs to R i p A f , θ q X F p A f , g q , and we are done. (cid:3) Lemma 8.4.
Let A be a -finite C - cdga with d “ , and let θ : g Ñ gl p V q be a finite-dimensional representation of a non-abelian Lie subalgebra of sl p C q . Then F p A , g q “ F p A , g q Y R p A , θ q .Proof. Let ω P F p A , g qz F p A , g q . It follows from the proof of [22, Prop. 5.3] that ω P U b g , where U Ď A is a linear subspace of dimension at least 2 which is isotropic withrespect to the multiplication map A ^ A Ñ A . Clearly, A U : “ C ¨ ‘ U is a finite sub- cdga of A , and χ p H . p A U qq ă
0. By [22, Prop. 2.4], we have that F p A U , g q “ R p A U , θ q .Therefore, ω P R p A U , θ q Ď R p A , θ q , and this completes the proof. (cid:3) Theorem 8.5.
Let f : M Ñ N be the projection map of a principal K-bundle, where bothM and N are smooth, closed manifolds, and K is a compact, connected real Lie group.Let G be a complex linear algebraic group, with non-abelian Lie algebra g Ď sl p C q .Let ι : G Ñ GL p V q be a rational representation. Let f : π p M q (cid:16) π p N q be the inducedhomomorphism on fundamental groups, and let abf : π p M q (cid:16) π p M q abf be the canonicalprojection. Suppose N is formal. Then, Hom p π p M q , G q p q “ abf ! Hom p π p M q abf , G q p q Y f ! Hom p π p N q , G q p q , (8.5) and, for i “ r “ or i “ and r ě , V ir p π p M q , ι q p q “ abf ! V ir p π p M q abf , ι q p q Y f ! V ir p π p N q , ι q p q . (8.6) Proof.
By Corollary 5.8, equality of germs at 1 in (5.14) implies equality at 1 in (5.15) for i “ r ě
1. Thus, in order to verify equality (8.6), it is enough to assume i “ r “ b p M q “ b p M q ą N is formal, we may take as a model for it the cdga A N “p H . p N q , d “ q . As usual, let r τ s : P . Ñ H . ` p N q be the transgression in the Serrespectral sequence of the fibration K Ñ M Ñ N . By Proposition 8.2, the Hirsch extension A M : “ A N b τ Ź P is a finite cdga model for M , and the canonical inclusion Φ : H . p N q ã Ñ A . M is a model for the map Ω p f q : Ω p N q Ñ Ω p M q .Now set A . “ p Ź . H p M q , d “ q , and let f : M Ñ K p π p M q abf , q be the canonicalmap defined by the homomorphism abf. By Theorem 8.1 (with q “ cdga map Φ : A Ñ A M such that Ω p f q » Φ and Ω p f q » Φ in ACDGA , uniformly withrespect to the families t f , f u and t Φ , Φ u . Since, as was mentioned in Proposition 8.2,both f and Φ are 0-connected, Theorem 6.4 applies, giving an analytic isomorphism of OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 37 embedded germs,(8.7) ` Hom p π p M q , G q , abf ! Hom p π p M q abf , G q Y f ! Hom p π p N q , G q ˘ p q –p F p A M , g q , p Φ b id q F p A , g q Y p Φ b id q F p A N , g qq p q . On the other hand, Proposition 5.3 from [28] guarantees the global equality(8.8) F p A M , g q “ F p A M , g q Y p Φ b id q F p A N , g q . We may also apply Lemma 7.3 to the map Φ to deduce the global equalities F p A M , g q “ p Φ b id q F p A , g q , (8.9) Π p A M , θ q “ p Φ b id q R p A , θ q . Using equations (8.7)–(8.9) as well as Theorem 6.4, we see that, in order to completethe proof, it is enough to show the following: if the inclusion F p A M , g q Ď F p A M , g q Y p Φ b id q F p A N , g q (8.10)holds, then the inclusion R p A M , θ q Ď Π p A M , θ q Y p Φ b id q R p A N , θ q (8.11)also holds. Pick ω P R p A M , θ q . There are two cases to consider.First suppose that ω P F p A M , g q . Using [28, Prop. 5.5] and induction on the dimensionof P , we see that R p A M q Ď t u Y Φ p R p A N qq . Hence, we may apply Lemma 8.3(for i “ q “
1) to the one-element family t Φ : A N ã Ñ A M u and conclude that ω P Π p A M , θ q Y p Φ b id q R p A N , θ q , as required.Finally, suppose that ω R F p A M , g q . Then ω “ Φ b id p ω q , for some ω P F p A N , g qz F p A N , g q , by assumption (8.10). By Lemma 8.4, we have that ω P R p A N , θ q . Thisverifies that (8.11) holds in this case, too, thereby completing the proof. (cid:3) Theorem 8.5 improves on Theorem 1.5(2) from [28], where an extra assumption (in-jectivity of the transgression in degree 1) was required. Our stronger result here is ofthe same flavor as the equality (7.7) from Theorem 7.7, in the context provided by Re-mark 7.8. Namely, if G is a C -linear algebraic group with Lie algebra g as above, and if ρ : π p M q Ñ G is a representation near the origin 1, then ρ is either abelian or pulls backvia f from a G -representation of π p N q .9. Q uasi - projective manifolds We conclude with another interesting class of examples where the uniform propertyholds for one-element families of maps, namely, regular maps between smooth, quasi-projective varieties. We also derive a non-compact analogue of Theorem 7.7 for a spe-cial class of quasi-projective manifolds, namely, complements of complex hyperplanearrangements.
Mixed Hodge diagrams.
Let M be an irreducible, smooth, complex quasi-projec-tive variety, or, for short, a quasi-projective manifold . Note that M is a finite space. Byresolution of singularities, we have that M “ M z D , where M is a smooth projectivevariety, and D Ă M is a normal crossing divisor. A map between such pairs, ¯ f : p M , D q Ñp M , D q , is called a regular morphism if the map ¯ f : M Ñ M is a regular map with theproperty that ¯ f ´ p D q Ď D . Clearly, the restriction f : M z D Ñ M z D is also a regularmap. Conversely, any regular map between quasi-projective manifolds is induced by aregular morphism between convenient compactifications with normal crossing divisors.We want to prove a quasi-projective analogue of Proposition 3.4. For that, we willneed the theory of relative minimal models for mixed Hodge diagrams (MHDs, for short),developed by Cirici and Guill´en in [5]. We start by recalling some pertinent definitionsand results from [5].The objects of the category FDGA are of the form p A . , W . q , where p A . , d q is a cdga defined over Q and W . is an increasing, multiplicative, regular, exhaustive filtration on p A . , d q , called a weight filtration. Such an object gives rise to a spectral sequence in thecategory of bigraded cdga s, t E r p A qu r ě , which converges to H . p A q . A morphism in FDGA is a cdga map which respects filtrations. Such a morphism ψ induces a map of spectralsequences, t E r p ψ qu r ě .The objects of the category MHD are strings of morphisms in
FDGA defined over C ,(9.1) H : A ψ (cid:47) (cid:47) A ¨ ¨ ¨ (cid:111) (cid:111) (cid:47) (cid:47) A (cid:96) ´ A (cid:96)ψ (cid:96) ´ (cid:111) (cid:111) , where p A , W q is defined over Q and all the induced maps E p ψ i q are isomorphisms. Thereare also additional data and axioms, related to the mixed Hodge structure (MHS) on A (cid:96) ,see [5, Def. 3.1]. A morphism of mixed Hodge diagrams, Φ : H Ñ H , is a tuple of fdga maps, p Φ , Φ , . . . , Φ (cid:96) q , commuting with the maps ψ and ψ , and such that Φ is definedover Q . There is also an extra condition on Φ (cid:96) pertaining to the MHS, see [5, Def. 3.5].9.2. The Gysin model of Morgan and Navarro.
Returning to our setup, let M be aquasi-projective manifold, and let M “ M Y D be a normal-crossing compactification.Given these data, Navarro constructs in [25] a mixed Hodge diagram H p M , D q , functorialwith respect to regular morphisms of pairs (see also Hain [16]). Furthermore, there isan equivalence A p M , D q » Ω C p M q in CDGA , natural with respect to the pair p M , D q .Moreover, E p A p M , D qq is isomorphic (as a bigraded cdga ) to MG p M , D q , the Gysinmodel of M “ M z D constructed by Morgan in [23, 24] (see also Dupont [11]). Note thatthis is a finite C -model, defined over Q , and that MG p M , ∅ q “ p H . p M q , d “ q .Suppose ¯ f : p M , D q Ñ p M , D q is a regular morphism, such that the restriction f : M Ñ M preserves basepoints. Naturality in the sense of Navarro yields an equivalence(9.2) Ω C p f q » Φ p ¯ f q OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 39 in C - ACDGA . Following Cirici and Guill´en [5], we define(9.3) Φ p f q “ E p Φ p f qq : MG p M , D q Ñ MG p M , D q , over C . Proposition 9.1.
Let f : M Ñ M be a pointed, regular map between quasi-projectivemanifolds, inducing an injection on H . Extend f to a regular morphism, ¯ f : p M , D q Ñp M , D q , by adding divisors with normal crossings in a suitable manner. Then Ω C p f q » Φ p f q in C - ACDGA .Proof. Looking at Q -components of MHDs and ignoring additional MHS data, we extractfrom [5, Theorems 3.17 & 3.19] the following commuting square in FDGA :(9.4) A p M , D q Ź U b Ź U ρ (cid:111) (cid:111) A p M , D q Φ p ¯ f q (cid:79) (cid:79) Ź U ρ (cid:111) (cid:111) (cid:63)(cid:31) j (cid:79) (cid:79) By [5, Lemma 3.4], the induced maps E p ρ q and E p ρ q are known to be isomorphisms.Hence, the maps ρ and ρ are quasi-isomorphisms. Furthermore, the CDGA diagram un-derlying (9.4) has the following properties: ρ is a minimal model map, and ρ is a relativeminimal model map for Φ p ¯ f q ˝ ρ , in the sense of § H p f q , together with the equivalence from (9.2), implythat the map Φ p ¯ f q ˝ ρ is a 0-equivalence. Using the discussion from § Ź U and Ź U b Ź U are connected cdga s. In particular, all maps from diagram(9.4) respect augmentations.It’s time now to take into account the available MHS data. We know from the work ofCirici and Guill´en that the map j is actually a morphism of mixed Hodge cdga s, in thesense of [5, Definition 3.14]. According to Deligne’s functorial splitting over C of mixedHodge structures, we have the following identifications in CDGA ,(9.5) E p Ź U q “ Ź U , E p Ź U b Ź U q “ Ź U b Ź U , E p j q “ j . This can be verified using the argument of Morgan from [23, Thm. 9.6]. See also [5,Lemma 3.20], where no extra finite-type assumptions are needed (over C ).Applying the E functor to diagram (9.4), we obtain the following commuting diagramin CDGA ,(9.6) E p Ź U b Ź U q E p ρ q (cid:47) (cid:47) E p A q MG p M , D q E p Ź U q E p j q (cid:79) (cid:79) E p ρ q (cid:47) (cid:47) E p A q E p Φ q (cid:79) (cid:79) MG p M , D q Φ p ¯ f q (cid:79) (cid:79) Here both horizontal arrows are quasi-isomorphisms, since E p ρ q and E p ρ q are iso-morphisms. Since all cdga s in sight are connected, (9.6) is a commuting diagram in ACDGA . The desired conclusion follows by putting together the information from displays(9.2) and (9.4)–(9.6). (cid:3)
Remark 9.2.
As mentioned previously, it is known that the Navarro model E p A p M , D qq is isomorphic in CDGA to Morgan’s Gysin model MG p M , D q . It is also known that thelatter is functorial with respect to regular morphisms of pairs; see [11] for a convenient,explicit description of the cdga map MG p ¯ f q : MG p M , D q Ñ MG p M , D q induced by¯ f : p M , D q Ñ p M , D q . But we do not know whether under this identification on objectsthe map MG p ¯ f q coincides with the map Φ p ¯ f q defined in (9.3). If that were the case, onecould use [9, Ex. 5.3] to infer that the map Φ p ¯ f q “ MG p ¯ f q is injective, whenever f : M Ñ M is a regular surjection onto a curve, with connected generic fiber. This observation,together with Proposition 9.1, would then imply that the conclusions of Corollary 6.5hold for regular admissible maps defined on quasi-projective manifolds, in the case when q “ Hyperplane arrangements.
Let A be an arrangement of hyperplanes, that is, a fi-nite, non-empty collection of complex a ffi ne hyperplanes in C (cid:96) , for some (cid:96) ą
0. Theunion of these hyperplanes is an a ffi ne hypersurface, V A , defined by an equation of theform Q A “
0, where Q A “ ś H P A α H and α H “ H . The complement of the arrangement, M A “ C (cid:96) z V A , is a connected, smooth,quasi-projective variety, which has the homotopy type of a finite CW-complex of dimen-sion at most (cid:96) .A nice feature of this class of quasi-projective manifolds is that formality over k “ R or C holds in the following strong sense. For each H P A , the logarithmic 1-form(9.7) ξ H “ π i d log α H P Ω dR p M A q is a closed form. Let e H P H p M A , k q be the cohomology class corresponding to r ξ H s P H p M A q under the de Rham isomorphism. It is known that t e H | H P A u forms a basisfor H p M A , k q . Thus, the k -linear map ξ A : H p M A , k q Ñ Ω p M A q sending each e H to ξ H yields an isomorphism r ξ A s : H p M A , k q » ÝÑ H p M A q .The celebrated Brieskorn–Orlik–Solomon theorem (see [26]) states that the cohomol-ogy ring H . p M A , Z q is the quotient of the exterior algebra Ź . H p M A , Z q by an ideal gen-erated in degrees at least 2 and depending only on the intersection lattice of A . Moreover,the extension of ξ A to a cdga map, ξ A : p Ź . H p M A , k q , d “ q Ñ Ω . dR p M A q , factorsthrough a quasi-isomorphism(9.8) ξ A : p H . p M A , k q , d “ q (cid:47) (cid:47) Ω . dR p M A q . We now suppose that the arrangement A is central , i.e., all hyperplanes H P A passthrough the origin 0 P C (cid:96) . For the purpose of studying the fundamental group π “ OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 41 π p M A q , we may assume that A is a central arrangement in C . This can be achievedby taking a generic 3-slice (if (cid:96) ą C ´ (cid:96) (if (cid:96) ă M A is a quasi-projective manifold. By the discussion from § E p M A q of admissible maps f : M A Ñ M f (up to reparametrization at thetarget), such that M f is a smooth curve with χ p M f q ă
0. It turns out that the mixed Hodgestructure on M A is pure of weight 2. Consequently, each curve M f must be of the form CP zt k points u , for some k ě E p M A q , well-suited for our purposes here (see also [8, §
5] and [27, § k ě
3, a k-multinet N on acentral arrangement A in C consists of a partition, A “ A \ ¨ ¨ ¨ \ A k , and a multi-plicity function, m : A Ñ N , satisfying several axioms, one of which being that the sum ř H P A i m H is independent of i P r k s .The multinet axioms imply that the polynomials Q i “ ś H P A i α m H H belong to a pencil ofcurves, that is, for each i ą a i and b i such that Q i “ a i Q ` b i Q .Consider the central line arrangement L “ t L , . . . , L k u in C , with L i “ t g i “ u ,where g “ z , g “ z , and g i “ a i z ` b i z for i ą
2. Let f N : M A Ñ M L be the regularmap with components p Q , Q q . Projectivizing, we obtain an admissible map,(9.9) f N : M A Ñ CP zt k points u . More generally, there is a complete set of representatives for E p M A q consisting ofadmissible maps f N : M A Ñ CP zt k points u obtained by restricting to M A the map f N : M B Ñ CP zt k points u , where N is a k -multinet on a sub-arrangement B Ď A ;see [27, Corollary 6.6].Now set n “ | A | , and identify π abf “ Z n . Let C be the Boolean arrangement in C n ,consisting of all coordinate hyperplanes. Clearly, M C “ p C ˆ q n . Consider the regularmap f : M A Ñ M C with components p α H q H P A . As noted for instance in [8, Lem. 5.1],the induced homomorphism, p f q : π p M A q Ñ π p M C q , coincides with the canonicalprojection, abf : π (cid:16) π abf . We let E p M A q “ E p M A q Y t f u , as in (7.2).By construction, all maps f P E p M A q are of the form f : M Ñ M f , where M “ M A and M f “ M A f , for some (a ffi ne) arrangement A f . Proposition 9.3.
Let A be a central hyperplane arrangement in C , and fix a basepointin M A . For k “ R or C , we have that Ω k p f q » H . p f , k q in k - ACDGA , uniformly withrespect to f P E p M A q . Proof.
In view of the Brieskorn–Orlik–Solomon isomorphism (9.8), it is enough to checkthe commutativity of the following diagram in
CDGA :(9.10) p H . p M A q , d “ q ξ A (cid:47) (cid:47) Ω . dR p M A qp H . p M A f q , d “ q ξ A f (cid:47) (cid:47) H . p f q (cid:79) (cid:79) Ω . dR p M A f q Ω . dR p f q (cid:79) (cid:79) Since the cohomology ring of an arrangement complement is generated in degree 1,we may assume that . “ ξ in degree 1, we can further reduce to showing that Ω dR p f qp d log α H q belongs to the Z -span of t d log α H | H P A u , for every H P A f .First assume f “ f . Then the claim follows from the formula Ω dR p f qp d log z H q “ d log α H , for every H P A , which in turn follows directly from the definition of f .Next assume f “ f N , for some multinet N on a sub-arrangement B Ď A . Clearly,we may assume that B “ A . The claim is now an easy consequence of the formula Ω dR p f N qp d log g i q “ ř H P A i m H d log α H , which is verified in [27, Lem. 6.3]. (cid:3) Theorem 9.4.
Let A be a central hyperplane arrangement with complement M “ M A .Write π “ π p M q , and, for each map f : M Ñ M f in E p M q , set π f “ π p M f q . Let G be a C -linear algebraic group with non-abelian Lie algebra g Ď sl p C q , and let ι : G Ñ GL p V q be a rational representation. Then, Hom p π, G q p q “ ď f P E p M q f ! Hom p π f , G q p q , (9.11) and, for i “ r “ or i “ and r ě , V ir p π, ι q p q “ ď f P E p M q f ! V ir p π f , ι q p q . (9.12) Proof.
As noted before, we may assume (cid:96) “
3. The argument we give is closely modeledon the proof of Theorem 7.7. To begin with, note that the conclusions of Lemma 7.6 holdfor the formal, quasi-projective manifold M “ M A , with the same proof. Next, considerthe map f : M A Ñ M C , and the induced cdga map H . p f q : H . p M C q Ñ H . p M A q ,where both di ff erentials are 0. Since M C “ p C ˆ q n , where n “ | A | , we may iden-tify H . p M C q with Ź . H p M C q . Furthermore, H p f q is an isomorphism, by construction.Hence, Lemma 7.3 may be applied to the map Φ “ H . p f q .By Proposition 9.3, we have that Ω p f q » H . p f q in ACDGA , uniformly with respect to f P E p M A q . We may now apply Theorem 6.4 for q “ t f : M A Ñ M A f u f P E p M A q and the family of cdga maps t H . p f q : H . p M A f q Ñ H . p M A qu f P E p M A q , where again all di ff erentials are 0. The rest of the argument goes ex-actly as in the proof of Theorem 7.7. (cid:3) OPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI 43 A cknowledgement Part of this work was done while the second author visited the Institute of Mathematicsof the Romanian Academy in June, 2016. He thanks IMAR for its hospitality, support,and excellent research atmosphere. R eferences [1] A. Adem, F.R. Cohen,
Commuting elements and spaces of homomorphisms , Math. Ann. (2007),no. 3, 587–626. MR2317932 4.5[2] D. Arapura,
Geometry of cohomology support loci for local systems
I, J. Algebraic Geom. (1997),no. 3, 563–597. MR1487227 1.3, 7.3[3] A. Borel, R. Friedman, J.W. Morgan, Almost commuting elements in compact Lie groups , Mem. Amer.Math. Soc. (2002), no. 747. MR1895253 4.5[4] N. Budur, B. Wang,
Cohomology jump loci of di ff erential graded Lie algebras , Compos. Math. (2015), no. 8, 1499–1528. MR3383165 1.1, 5.1, 5.2[5] J. Cirici, F. Guill´en, E -formality of complex algebraic varieties , Algebr. Geom. Topol. (2014),no. 5, 3049–3079. MR3276854 1.4, 9.1, 9.1, 9.2, 9.2, 9.2, 9.2[6] K. Corlette, C. Simpson, On the classification of rank-two representations of quasiprojective funda-mental groups , Compos. Math. (2008), no. 5, 1271–1331. MR2457528 1.3, 7.8[7] P. Deligne, P. Gri ffi ths, J.W. Morgan, D. Sullivan, Real homotopy theory of K¨ahler manifolds , Invent.Math. (1975), no. 3, 245–274. MR0382702 1.3, 3.5, 7.2[8] G. Denham, A.I. Suciu, Multinets, parallel connections, and Milnor fibrations of arrangements , Proc.London Math. Soc. (2014), no. 6, 1435–1470. MR3218315 9.3, 9.3[9] A. Dimca, S. Papadima,
Non-abelian cohomology jump loci from an analytic viewpoint , Commun.Contemp. Math. (2014), no. 4, 1350025, 47 pp. MR3231055 1.1, 3.2, 4.6, 4.6, 4.6, 4.6, 5.1, 5.1,5.2, 5.2, 6.1, 6.1, 6.1, 7.4, 7.8, 7.5, 8.1, 9.2[10] A. Dimca, S. Papadima, A.I. Suciu, Topology and geometry of cohomology jump loci , Duke Math.Journal (2009), no. 3, 405–457. MR2527322 1.3, 7.4[11] C. Dupont,
The Orlik–Solomon model for hypersurface arrangements , Ann. Inst. Fourier (Grenoble), (2015), no. 6, 2507–2545. MR3449588 9.2, 9.2[12] M. Falk, S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves , Compositio Math. (2007), no. 4, 1069–1088. MR2339840 1.3, 9.3[13] Y. F´elix, S. Halperin, J.-C. Thomas,
Rational homotopy theory , Grad. Texts in Math., vol. 205,Springer-Verlag, New York, 2001. MR1802847 3.2, 3.3, 7.2, 8.2, 8.2[14] W. Goldman,
Topological components of spaces of representations , Invent. Math. (1988), no. 3,557–607. MR0952283 4.5[15] W. Goldman, J. Millson, The deformation theory of representations of fundamental groups of compactK¨ahler manifolds , Inst. Hautes ´Etudes Sci. Publ. Math. (1988), 43–96. MR0972343 4, 4.1, 4.6, 4.6,6.1[16] R. Hain, The de Rham homotopy theory of complex algebraic varieties. I . K-Theory (1987), no. 3,271–324. MR0908993 9.2[17] S. Halperin, Lectures on minimal models , M´em. Soc. Math. France, S´er. 2, (1983), 1–261.MR0736299 3.2, 3.3[18] M. Kapovich, J. Millson,
On representation varieties of Artin groups, projective arrangements and thefundamental groups of smooth complex algebraic varieties , Inst. Hautes ´Etudes Sci. Publ. Math. (1998), 5–95. MR1733326 1.1 [19] D. Lehmann, Th´eorie homotopique des formes dif´erentielles (d’apr`es D. Sullivan) , Ast´erisque, no. 45,Soci´et´e Math´ematique de France, Paris, 1977. MR0488041 3.2[20] F. Loray, J.V. Pereira, F. Touzet,
Representations of quasiprojective groups, flat connections and trans-versely projective foliations , J. ´Ec. Polytech. Math. (2016), 263–308. MR3522824 1.3, 7.8[21] M. Manetti, Deformation theory via di ff erential graded Lie algebras , arxiv:math.AG/0507284v1 .4, 4.2[22] A. M˘acinic, S. Papadima, R. Popescu, A.I. Suciu, Flat connections and resonance varieties: from rankone to higher ranks , Trans. Amer. Math. Soc. (2017), no. 2, 1309–1343. MR3572275 1.3, 5.4, 5.4,5.5, 7.1, 7.1, 7.4, 7.5, 7.10, 8.3, 8.3, 8.3[23] J.W. Morgan,
The algebraic topology of smooth algebraic varieties , Inst. Hautes ´Etudes Sci. Publ.Math. (1978), 137–204. MR0516917 1.4, 3.2, 9.2, 9.2[24] J.W. Morgan, Correction to: “The algebraic topology of smooth algebraic varieties” , Inst. Hautes´Etudes Sci. Publ. Math. (1986), 185. MR0876163 9.2[25] V. Navarro Aznar, Sur la th´eorie de Hodge-Deligne , Invent. Math. (1987), no. 1, 11–76.MR0906579 9.2[26] P. Orlik, H. Terao, Arrangements of hyperplanes , Grundlehren Math. Wiss., vol. 300, Springer-Verlag,Berlin, 1992. MR1217488 9.3[27] S. Papadima, A.I. Suciu,
The Milnor fibration of a hyperplane arrangement: from modular resonanceto algebraic monodromy , Proc. London Math. Soc. (2017), no. 6, 961–1004. MR3661343 9.3, 9.3,9.3[28] S. Papadima, A.I. Suciu,
The topology of compact Lie group actions through the lens of finite models ,Int. Math. Res. Notices (2018), 1–47, doi:10.1093 / imrn / rnx294. 1.3, 8.3, 8.3[29] S. Papadima, A.I. Suciu, Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds , J. Inst. Math. Jussieu (2018), 1–35, doi:10.1017 / S1474748018000063. 1.3[30] D. Quillen,
Rational homotopy theory , Ann. of Math. (2) (1969), 205–295. MR0258031 3[31] A.S. Rapinchuk, V.V. Benyash-Krivetz, V.I. Chernousov, Representation varieties of the fundamentalgroups of compact orientable surfaces , Israel J. Math. (1996), no. 1, 29–71. MR1380633 4.5[32] D. Sullivan, Infinitesimal computations in topology , Inst. Hautes ´Etudes Sci. Publ. Math. (1977),no. 47, 269–331. MR0646078 3, 1.4, 3, 3.2, 3.5[33] J.-Cl. Tougeron,
Id´eaux de fonctions di ff ´erentiables , Ergeb. Math. Grenzgeb., vol. 71, Springer-Verlag,Berlin-New York, 1972. MR0440598 2, 2, 6.2[34] M. Vigu´e-Poirrier, R´ealisation de morphismes donn´es en cohomologie et suite spectrale d’Eilenberg–Moore , Trans. Amer. Math. Soc. (1981), no. 2, 447–484. MR0610959 1.4, 3.5, 3.5[35] G.W. Whitehead,
Elements of homotopy theory , Grad. Texts in Math., vol. 61, Springer-Verlag, NewYork-Berlin, 1978. MR0516508 5.1, 5.5S imion S toilow I nstitute of M athematics , P.O. B ox ucharest , R omania E-mail address : [email protected] D epartment of M athematics , N ortheastern U niversity , B oston , MA 02115, USA E-mail address : [email protected] URL ::