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Home , Cohomology, General position, Induced homomorphism

GEOMETRYANDTOPOLOGYOF COHOMOLOGYJUMPLOCI

LECTURE 1: CHARACTERISTICVARIETIES

Alex Suciu

Northeastern University

MIMS Summer School: New Trends in and

Mediterranean Institute for the Mathematical Sciences Tunis, Tunisia

July 9–12, 2018

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 1 / 25 OUTLINE

1 CASTOFCHARACTERS The The equivariant Characteristic varieties Degree1 characteristic varieties

2 EXAMPLES AND COMPUTATIONS Warm-up examples Toric complexes and RAAGs Quasi-projective

3 APPLICATIONS of finite abelian covers Dwyer–Fried sets and propagation

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 2 / 25 CASTOFCHARACTERS THECHARACTERGROUP

THECHARACTERGROUP

Throughout, X will be a connected CW-complex with finite q-skeleton, for some q ě 1. We may assume X has a single 0-cell, call it e0. 0 Let G = π1(X, e ) be the of X: a finitely 1 1 generated group, with generators x1 = [e1], ... , xm = [em]. The character group,

G = Hom(G, Cˆ) Ă (Cˆ)m

is a (commutative) algebraicp group, with multiplication ρ ¨ ρ1(g) = ρ(g)ρ1(g), and identity G Ñ Cˆ, g ÞÑ 1. 1 Let Gab = G/G – H1(X, Z) be the abelianization of G. The » projection ab: G Ñ Gab induces an Gab ÝÑ G.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMERp SCHOOLp2018 3 / 25 CASTOFCHARACTERS THECHARACTERGROUP

The identity component, G0, is isomorphic to a complex algebraic of n = rank Gab. p The other connected components are all isomorphic to G0 = (Cˆ)n, and are indexed by the finite Tors(Gab). p

Char(X ) = G is the moduli space of rank1 local systems on X:

ˆ p ρ : G Ñ C Cρ the complex C, viewed as a right over the group ZG via a ¨ g = ρ(g)a, for g P G and a P C.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 4 / 25 CASTOFCHARACTERS THEEQUIVARIANTCHAINCOMPLEX

THEEQUIVARIANTCHAINCOMPLEX

Let p : X Ñ X be the universal cover. The cell structure on X lifts to a cell structure on X. 0 ´1 0 0 Fixing ar lift e˜ P p (e ) identifies G = π1(X, e ) with the group of deck transformationsr of X. Thus, we may view the cellular chain complex of X as a chain complex of left ZG-modules,r r B˜i+1 B˜i ¨ ¨ ¨ / Ci+1(X, Z) / Ci (X, Z) / Ci´1(X, Z) / ¨ ¨ ¨ .

˜ 1 0 B1(e˜ i ) = (xi ´ 1)er˜ . r r ˜ 2 m φ 1 B2(e˜ ) = i=1 Br/Bxi ¨ e˜ i , where r P F = xx , , x y is the word traced by the attaching of e2; mř 1 ... m Br/Bxi P ZFm are the Fox derivatives of r; φ : ZFm Ñ ZG is the linear extension of the projection Fm  G.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 5 / 25 CASTOFCHARACTERS THEEQUIVARIANTCHAINCOMPLEX

H˚(X, Cρ) is the homology of the chain complex of C-vector spaces Cρ bZG C‚(X, Z):

rB˜i+1(ρ) B˜i (ρ) ¨ ¨ ¨ / Ci+1(X, C) / Ci (X, C) / Ci´1(X, C) / ¨ ¨ ¨ ,

where the evaluation of B˜i at ρ is obtained by applying the ring ZG Ñ C, g ÞÑ ρ(g) to each entry of B˜i . Alternatively, consider the universal abelian cover, X ab, and its ab equivariant chain complex, C‚(X , Z) = ZGab bZG C‚(X, Z), ab with differentials Bi = id b Bi . r Then H˚(X, Cρ) is computed from the resulting C-chain complex, ab ˜r with differentials Bi (ρ) = Bi (ρ).

The identity1 P Char(X ) yields the trivial , C1 = C, and H˚(X, C) is the usual homology of X with C-coefficients. Denote by bi (X ) = dimC Hi (X, C) the ith Betti number of X.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 6 / 25 CASTOFCHARACTERS CHARACTERISTICVARIETIES

CHARACTERISTICVARIETIES

DEFINITION The characteristic varieties of X are the sets

i Vk (X ) = tρ P Char(X ) | dimC Hi (X, Cρ) ě ku.

i i i For each i, get stratification Char(X ) = V0 Ě V1 Ě V2 Ě ¨ ¨ ¨ i 1 P Vk (X ) ðñ bi (X ) ě k. 0 0 V1 (X ) = t1u and Vk (X ) = H, for k ą 1.

i ˆ Define analogously Vk (X, k) Ă Hom(G, k ), for arbitrary field k. i i ˆ Then Vk (X, k) = Vk (X, K) X Hom(G, k ), for any k Ď K.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 7 / 25 CASTOFCHARACTERS CHARACTERISTICVARIETIES

LEMMA

i For each 0 ď i ď q andk ě 0, the set Vk (X ) is a Zariski closed subset of the algebraic group G = Char(X ).

PROOF (FOR i ă q). p

Let R = C[Gab] be the coordinate ring of G = Gab. By definition, a i character ρ belongs to Vk (X ) if and only if p p ab ab rank Bi+1(ρ) + rank Bi (ρ) ď ci ´ k, where ci = ci (X ) is the number of i-cells of X. Hence,

i ab ab Vk (X ) = tρ P G | rank Bi+1(ρ) ď r ´ 1 or rank Bi (ρ) ď s ´ 1u r+s=c ´k+1; r,sě0 i č  p ab ab  = V Ir (Bi ) ¨ Is(Bi+1) , r+s=c ´k+1; r,sě0 i ÿ where Ir (ϕ) = of r ˆ r minors of ϕ. ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 8 / 25 CASTOFCHARACTERS CHARACTERISTICVARIETIES

The characteristic varieties are -type invariants of a space:

LEMMA SupposeX » X 1. There is then an isomorphism G 1 – G, which i 1 i restricts to Vk (X ) – Vk (X ), for alli ď q andk ě 0. p p PROOF. Let f : X Ñ X 1 be a (cellular) homotopy equivalence. 0 1 10 The f7 : π1(X, e ) Ñ π1(X , e ), yields an 1 isomorphism of algebraic groups, fˆ7 : G Ñ G. Lifting f to a cellular homotopy equivalence, f˜ X X 1, defines x p : Ñ 1 1 isomorphisms Hi (X, Cρ˝f7 ) Ñ Hi (X , Cρ), for each ρ P G . r r ˆ i 1 i Hence, f7 restricts to isomorphisms Vk (X ) Ñ Vk (X ). p

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 9 / 25 CASTOFCHARACTERS DEGREE 1 CHARACTERISTICVARIETIES

DEGREE 1 CHARACTERISTICVARIETIES

1 2 Vk (X ) depends only on G = π1(X ) (in fact, only on G/G ), so we 1 may write these sets as Vk (G).

Suppose G = xx1, ... , xm | r1, ... , rpy is finitely presented

1 ab Away from1 P G, we have that Vk (G) = V (Ek (B1 )), the zero-set of the ideal of k minors of the Alexander matrix

p ab ab p m B1 = Bri /Bxj : ZGab Ñ ZGab.

If ϕ : G  Q is an epimorphism, then, for each k ě 1, the induced monomorphism between character groups, ϕ˚ : Q ãÑ G, restricts 1 1 to an embedding Vk (Q) ãÑ Vk (G). p p Given any subvariety W Ă (Cˆ)n defined over Z, there is a finitely n 1 presented group G such that Gab = Z and V1 (G) = W .

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 10 / 25 EXAMPLES AND COMPUTATIONS WARM-UPEXAMPLES

WARM-UPEXAMPLES

EXAMPLE (THECIRCLE) We have S1 = R. 1 ˘1 Identify π1(S , ˚) = Z = xty and ZZ = Z[t ]. Then: Ă 1 ˘1 t´1 ˘1 C‚(S ) : 0 / Z[t ] / Z[t ] / 0

For ρ P Hom(Z, CĂˆ) = Cˆ, we get

ρ´1 1 Cρ bZZ C‚(S ) : 0 / C / C / 0

1 1 which is exact, except for ρĂ= 1, when H0(S , C) = H1(S , C) = C. Hence: 0 1 1 1 V1 (S ) = V1 (S ) = t1u i 1 Vk (S ) = H, otherwise.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 11 / 25 EXAMPLES AND COMPUTATIONS WARM-UPEXAMPLES

EXAMPLE (THE n-TORUS) n n n ˆ ˆ n Identify π1(T ) = Z , and Hom(Z , C ) = (C ) . Using the Koszul n resolution C‚(T ) as above, we get

n Ă i n t1u if k ď ( i ), Vk (T ) = #H otherwise.

EXAMPLE (NILMANIFOLDS) More generally, let M be a nilmanifold. An inductive argument on the nilpotency class of π1(M), based on the Hochschild-Serre , yields

i t1u if k ď bi (M), Vk (M) = #H otherwise

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 12 / 25 EXAMPLES AND COMPUTATIONS WARM-UPEXAMPLES

EXAMPLE (WEDGEOFCIRCLES) n 1 ˆ ˆ n Identify π1( S ) = Fn, and Hom(Fn, C ) = (C ) . Then:

Ž ˆ n n (C ) if k ă n, 1 1 Vk S = $t1u if k = n, ’ ł &H if k ą n. %’ EXAMPLE (ORIENTABLE SURFACE OF g ą 1)

Write π1(Σg ) = xx1, ... , xg, y1, ... , yg | [x1, y1] ¨ ¨ ¨ [xg, yg ] = 1y, and ˆ ˆ 2g identify Hom(π1(Σg ), C ) = (C ) . Then:

(Cˆ)2g if i = 1, k ă 2g ´ 1, i Vk (Σg ) = $t1u if i = 1, k = 2g ´ 1, 2g; or i = 2, k = 1, &’H otherwise.

%’ ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 13 / 25 EXAMPLES AND COMPUTATIONS TORICCOMPLEXES

TORICCOMPLEXESAND RAAGS

Given L on n vertices, define the toric complex n TL as the subcomplex of T obtained by deleting the cells corresponding to the missing simplices of L:

σ σ n TL = T , where T = tx P T | xi = ˚ if i R σu. PL σď Let Γ = (V, E) be the graph with vertex set the0-cells of L, and edge set the1-cells of L. Then π1(TL) is the right-angled Artin group associated to Γ:

GΓ = xv P V | vw = wv if tv, wu P Ey. Properties: 1 2 Γ = K n ñ GΓ = Fn Γ = Γ Γ ñ GΓ = GΓ1 ˚ GΓ2 n 1 2 Γ = Kn ñ GΓ = Z Γ = Γ š˚ Γ ñ GΓ = GΓ1 ˆ GΓ2

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 14 / 25 EXAMPLES AND COMPUTATIONS TORICCOMPLEXES

ˆ Identify character group GΓ = Hom(GΓ, C ) with the algebraic torus (Cˆ)V := (Cˆ)n. For each subsetW Ď V,p let (Cˆ)W Ď (Cˆ)V be the corresponding coordinate subtorus; in particular, (Cˆ)H = t1u.

THEOREM (PAPADIMA–S. 2006/09)

i ˆ W Vk (TL) = (C ) , WĎV dimC Hi ď1 (lkL (σ),C)ěk σPLVzW ´ ´|σ| W ř r whereL W is the subcomplex induced byL on W, and lkK (σ) is the link of a simplex σ in a subcomplexK Ď L.

In particular: 1 ˆ W V1 (GΓ) = (C ) . WĎV ΓW disconnectedď

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 15 / 25 EXAMPLES AND COMPUTATIONS QUASI-PROJECTIVEMANIFOLDS

QUASI-PROJECTIVEMANIFOLDS

A space M is said to be a quasi-projective variety if M is a Zariski open subset of a projective variety M (i.e., a Zariski closed subset of some ). By resolution of singularities, a connected, smooth, complex quasi-projective variety M can realized as M = MzD, where M is a smooth, complex projective variety, and D is a normal crossing divisor. For short, we say M is a quasi-projective . When M = Σ is a smooth complex curve with χ(M) ă 0, we saw 1 that V1 (M) = Char(M).

THEOREM (GREEN–LAZARSFELD,...,ARAPURA,...,BUDUR–WANG) All the characteristic varieties of a quasi-projective manifoldM are finite unions of torsion-translates of subtori of Char(M), i.e., i nα Vk (M) = α ραTα, whereT α is an algebraic subtorus and ρα = 1.

ALEX SUCIU (NŤORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 16 / 25 EXAMPLES AND COMPUTATIONS QUASI-PROJECTIVEMANIFOLDS An algebraic map f : M Ñ Σ to a smooth complex curve Σ is admissible if f is a surjection and has connected generic fiber.

The homomorphism f7 : π1(M) Ñ π1(Σ) is surjective; thus, f7 : Char(Σ) Ñ Char(M) is injective, and im(f7) is a complex 1 subtorus of V1 (M). p p Up to reparametrization at the target, there is a finite set E(M) of admissible maps f : M Ñ Σ with χ(Σ) ă 0.

THEOREM (ARAPURA 1997)

The correspondencef f7 Char(Σ) defines a bijection between E(M) 1 and the set of positive-dimensional, irreducible components of V1 (M) passing through 1. p

THEOREM (DIMCA–PAPADIMA–S. (2008–09)) 1 1 1 If ρT and ρ T are two distinct irreducible components of V1 (M), then 1 1 1 eitherT = T orT X T = t1u. Hence, distinct components of V1 (M) meet only in a finite set of finite-order characters. ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 17 / 25 EXAMPLES AND COMPUTATIONS QUASI-PROJECTIVEMANIFOLDS

EXAMPLE (ORDERED CONFIGURATION SPACE OF n POINTSIN C) n Let Confn(Σ) = tz P Σ | zi ‰ zj u, and set Mn = Confn(C).

ˆ (n) Then π1(Mn) = Pn, and so Char(Mn) = (C ) 2 .

1 (D. Cohen–S. 1999) The set of irreducible components of V1 (Mn) n n n+1 passing through1 consists of the following (3) + (4) = ( 4 ) subtori of dimension2:

Tijk = tij tik tjk = 1 and trs = 1 if tr, su Ć ti, j, ku . ( Tijk` = tij = tjk , tjk = ti`, tik = tj`, tpq = 1, and trs = 1 if tr, su Ć ti, j, k, `u . 1ďpăqďn ź (

EXAMPLE (ORDERED CONFIGURATION SPACE OF E = Σ1) (Dimca 2010) The set of positive-dimensional components of 1 n ˆ n(n´1) V1 (Confn(E)) consists of (2) two-dimensional subtori of (C ) , of n ´1 the form Tij = im(fij 7), where fij : E Ñ Ezt1u is given by z ÞÑ zi zj .

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 18 / 25 x APPLICATIONS HOMOLOGY OF FINITE ABELIAN COVERS

HOMOLOGY OF FINITE ABELIAN COVERS The characteristic varieties can be used to compute the homology of finite, abelian, regular covers (work of A. Libgober, E. Hironaka, P. Sarnak–S. Adams, M. Sakuma, D. Matei–A. S. from the 1990s).

THEOREM LetY Ñ X be a regular cover, defined by an epimorphism ν from G = π1(X ) to a finite abelian groupA. Let k be an algebraically closed field of characteristic not dividing the order ofA. Then, for eachi ě 0,

i dimk Hi (Y , k) = im(ν) X Vk (X, k) . kě1 ˇ ˇ ÿ ˇ ˇ ˇ p ˇ PROOF (SKETCH). By Shapiro’s Lemma and Maschke’s Theorem,

Hi (Y , k) – Hi (X, k[A]) – Hi (X, kρ). ρPim(ν) ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCIà MIMSSUMMER SCHOOL 2018 19 / 25 p APPLICATIONS HOMOLOGY OF FINITE ABELIAN COVERS

EXAMPLE n 1 Let X = 1 S , and let Y Ñ X be the2-fold cover defined by ν : F Ñ Z , x ÞÑ 1. (Of course, Y = 2n´1 S1.) n Ž2 i 1 Inside Char(X ) = (Cˆ)n, we have thatŽim(ν) = t1, ´1u, and 1 1 ˆ n 1 V1 (X ) = ¨ ¨ ¨ = Vn´1(X ) = (C ) , while Vn (X ) = t1u. p Hence, b1(Y ) = n + (n ´ 1) = 2n ´ 1.

EXAMPLE

Let X = Σg with g ě 2, and let Y Ñ X be an n-fold regular abelian cover. (Of course, Y = Σh, where h = ng ´ n + 1.) Inside Char(X ) = (Cˆ)2g, we have 1 1 ˆ 2g 1 1 V1 (X ) = ¨ ¨ ¨ = V2g´2(X ) = (C ) and V2g´1(X ) = V2g (X ) = t1u. Hence, b1(Y ) = 2g + (n ´ 1)(2g ´ 2) = 2(ng ´ n + 1).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 20 / 25 APPLICATIONS DWYER–FRIEDSETS

DWYER–FRIEDSETS The characteristic varieties can also be used to determine the homological finiteness properties of free abelian, regular covers. For a fixed r P N, the regular Zr -covers of a space X are r classified by epimorphisms ν : π  Z . n Such covers are parameterized by the Grr (Q ), where n = b1(X ), via the correspondence regular Zr -covers of X ÐÑ r-planes in H1(X, Q) ν ˚ r 1 X Ñ X ÐÑ Pν :=(im(ν : Q Ñ H (X, Q)) ( The Dwyer–Fried invariants of X are the subsets i n ν Ωr (X ) = Pν P Grr (Q ) bj (X ) ă 8 for j ď i . For each r ą 0, we get a descendingˇ filtration, ( n 0 ˇ 1 2 Grr (Q ) = Ωr (X ) Ě Ωr (X ) Ě Ωr (X ) Ě ¨ ¨ ¨ . i i Ω1(X ) is open, but Ωr (X ) may be non-open for r ą 1. ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 21 / 25 APPLICATIONS DWYER–FRIEDSETS

THEOREM (DWYER–FRIED 1987, PAPADIMA–S. 2010)

r For an epimorphism ν : π1(X )  Z , the following are equivalent: i ν The vector space j=0 Hj (X , C) is finite-dimensional. r  The algebraic torusÀT ν = im νˆ : Z ãÑ π1(X ) intersects the i j variety W (X ) = jďi V1(X ) in only finitely many points. x { Ť Note that exp(Pν b C) = Tν. Thus:

COROLLARY i 1 i  Ωr (X ) = P P Grr (H (X, Q)) dim exp(P b C) X W (X ) = 0 ˇ ( COROLLARY ˇ i i n If W (X ) is finite, then Ωr (X ) = Grr (Q ), wheren = b1(X ). i q If W (X ) is infinite, then Ωn(X ) = H, for allq ě i.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 22 / 25 APPLICATIONS DUALITY AND PROPAGATION

DUALITYANDABELIANDUALITY

Let X be a connected, finite-type CW-complex, with G = π1(X ).

(Bieri–Eckmann 1978) X is a duality space of dimension n if Hi (X, ZG) = 0 for i ‰ n and Hn(X, ZG) ‰ 0 and torsion-free.

Let D = Hn(X, ZG) be the dualizing ZG-module. Given any i ZG-module A, we have: H (X, A) – Hn´i (X, D b A).

(Denham–S.–Yuzvinsky 2016/17) X is an abelian duality space of i n dimension n if H (X, ZGab) = 0 for i ‰ n and H (X, ZGab) ‰ 0 and torsion-free.

n Let B = H (X, ZGab) be the dualizing ZGab-module. Given any i ZGab-module A, we have: H (X, A) – Hn´i (X, B b A).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 23 / 25 APPLICATIONS DUALITY AND PROPAGATION

THEOREM (DSY) LetX be an abelian duality space of dimensionn. Then:

b1(X ) ě n ´ 1.

bi (X ) ‰ 0, for 0 ď i ď n andb i (X ) = 0 fori ą n. (´1)nχ(X ) ě 0. 1 n The characteristic varieties propagate, i.e., V1 (X ) Ď ¨ ¨ ¨ Ď V1 (X ).

THEOREM (DENHAM–S. 2018) LetM be a quasi-projective manifold of dimensionn. SupposeM has a smooth compactification M for which 1 Components of MzM form an arrangement of hypersurfaces A; 2 For each submanifoldX in the intersection posetL (A), the complement of the restriction of A toX is a Stein manifold. ThenM is both a duality space and an abelian duality space of dimensionn.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 24 / 25 APPLICATIONS DUALITY AND PROPAGATION

LINEAR, ELLIPTIC, ANDTORICARRANGEMENTS

THEOREM (DS18) Suppose that A is one of the following: n 1 An affine-linear arrangement in C , or a arrangement in CPn; n 2 A non-empty elliptic arrangement inE ; n 3 A toric arrangement in (Cˆ) . Then the complementM (A) is both a duality space and an abelian duality space of dimensionn ´ r,n + r, andn, respectively, wherer is the corank of the arrangement.

This theorem extends several previous results: 1 Davis, Januszkiewicz, Leary, and Okun (2011); 2 Levin and Varchenko (2012); 3 Davis and Settepanella (2013), Esterov and Takeuchi (2018).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 25 / 25 GEOMETRYANDTOPOLOGYOF COHOMOLOGYJUMPLOCI

LECTURE 2: RESONANCE VARIETIES

Alex Suciu

Northeastern University

MIMS Summer School: New Trends in Topology and Geometry

Mediterranean Institute for the Mathematical Sciences Tunis, Tunisia

July 9–12, 2018

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 1 / 24 OUTLINE

1 RESONANCE VARIETIES OF CDGAS Commutative differential graded algebras Resonance varieties Tangent cone inclusion 2 RESONANCE VARIETIES OF SPACES Algebraic models for spaces Germs of jump loci Tangent cones and exponential maps The tangent cone theorem Detecting non-formality 3 INFINITESIMAL FINITENESS OBSTRUCTIONS Spaces with finite models Associated graded Lie algebras Holonomy Lie algebras Malcev Lie algebras Finiteness obstructions for groups

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 2 / 24 RESONANCE VARIETIES OF CDGAS COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS

COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS

Let A “ pA‚, dq be a commutative, differential graded algebra over a field k of characteristic0. That is: i i A “ iě0 A , where A are k-vector spaces. i j i`j TheÀ multiplication ¨: A b A Ñ A is graded-commutative, i.e., ab “ p´1q|a||b|ba for all homogeneous a and b. The differential d: Ai Ñ Ai`1 satisfies the graded Leibnitz rule, i.e., dpabq “ dpaqb ` p´1q|a|a dpbq.

A CDGA A is of finite-type (or q-finite) if it is connected (i.e., A0 “ k ¨ 1); i dimk A is finite for i ď q.

Let Hi pAq “ kerpd: Ai Ñ Ai`1q{ impd: Ai´1 Ñ Ai q. Then H‚pAq inherits an algebra structure from A.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 3 / 24 RESONANCE VARIETIES OF CDGAS COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS

A cdga ϕ: A Ñ B is both an algebra map and a cochain map. Hence, it induces a morphism ϕ˚ : H‚pAq Ñ H‚pBq.

A map ϕ: A Ñ B is a quasi-isomorphism if ϕ˚ is an isomorphism. Likewise, ϕ is a q-quasi-isomorphism (for some q ě 1) if ϕ˚ is an isomorphism in degrees ď q and is injective in degree q ` 1.

Two cdgas, A and B, are (q-)equivalent (»q) if there is a zig-zag of (q-)quasi-isomorphisms connecting A to B.

A cdga A is formal (or just q-formal) if it is (q-)equivalent to pH‚pAq, d “ 0q.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 4 / 24 RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

RESONANCE VARIETIES

Since A is connected and dp1q “ 0, we have Z 1pAq “ H1pAq.

For each a P Z 1pAq, we construct a cochain complex,

δ0 δ1 δ2 ‚ 0 a 1 a 2 a pA , δaq: A / A / A / ¨ ¨ ¨ ,

i i with differentials δapuq “ a ¨ u ` d u, for all u P A .

The resonance varieties of A are the sets

i 1 i ‚ Rk pAq “ ta P H pAq | dim H pA , δaq ě ku.

i If A is q-finite, then Rk pAq are algebraic varieties for all i ď q.

If A is a CGA (so that d “ 0), these varieties are homogeneous subvarieties of H1pAq “ A1.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 5 / 24 RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

1 Fix a k- te1,..., er u for H pAq, and let tx1,..., xr u be the dual 1 ˚ basis for H1pAq “ pH pAqq .

Identify SympH1pAqq with S “ krx1,..., xr s, the coordinate ring of the affine space H1pAq. ‚ Define a cochain complex of free S-modules, LpAq :“ pA bk S, δq,

δi δi`1 ¨ ¨ ¨ / Ai b S / Ai`1 b S / Ai`2 b S / ¨ ¨ ¨ ,

i n where δ pu b f q “ j“1 ej u b f xj ` d u b f . 1 The specialization ofřpA bk S, δq at a P A coincides with pA, δaq. i Hence, Rk pAq is the zero-set of the ideal generated by all minors i`1 i of size bi pAq ´ k ` 1 of the block-matrix δ ‘ δ . 1 1 In particular, Rk pAq “ V pIr´k pδ qq, the zero-set of the ideal of codimension k minors of δ1.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 6 / 24 RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

EXAMPLE (EXTERIORALGEBRA) Let E “ V , where V “ kn, and S “ SympV q. Then LpEq is the on V . E.g., for n “ 3: Ź

x2 x3 0 x1 2 δ1 x δ “ ´x1 0 x3 “ 2 3 x3 ˜ 0 ´x1 ´x2 ¸ δ “p x3 ´x2 x1 q S ˆ ˙ / S3 / S3 / S .

Hence, n i t0u if k ď i , Rk pEq “ #H otherwise` ˘ .

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 7 / 24 RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

EXAMPLE (NON-ZERORESONANCE)

Let A “ pe1, e2, e3q{xe1e2y, and set S “ krx1, x2, x3s. Then

x1 1 x3 0 ´x1 Ź δ “ x2 δ2“ x3 0 x3 ´x2 LpAq : S ˆ ˙/ S3 ˆ ˙ / S2 .

tx3 “ 0u if k “ 1, R1pAq “ k $t0u if k “ 2 or 3, &H if k ą 3. % EXAMPLE (NON-LINEARRESONANCE)

Let A “ pe1,..., e4q{xe1e3, e2e4, e1e2 ` e3e4y. Then

x1 Ź x x4 00 ´x1 δ1“ 2 2 ¨ x3 ˛ δ “ 0 x3 ´x2 0 x4 4 ˜ ´x2 x1 x4 ´x3 ¸ 3 LpAq : S ˝ ‚/ S / S .

1 R1pAq “ tx1x2 ` x3x4 “ 0u

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 8 / 24 RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

EXAMPLE (NON-HOMOGENEOUSRESONANCE) Let A “ pa, bq with d a “ 0, d b “ b ¨ a.

H1pAq “ŹC, generated by a. Set S “ Crxs. Then:

δ1 0 2 “p x q δ “p x´10 q LpAq : S / S2 / S .

Hence, R1pAq “ t0, 1u, a non-homogeneous subvariety of C.

Let A1 be the sub-CDGA generated by a. The , A1 ãÑ A, induces an isomorphism in .

But R1pA1q “ t0u, and so the resonance varieties of A and A1 differ, although A and A1 are quasi-isomorphic.

PROPOSITION 1 i i 1 IfA »q A , then Rk pAqp0q – Rk pA qp0q, for alli ď q andk ě 0.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 9 / 24 RESONANCE VARIETIES OF CDGAS TANGENTCONEINCLUSION

TANGENTCONEINCLUSION

THEOREM (BUDUR–RUBIO,DENHAM–S. 2018) IfA is a connected k-CDGA A with locally finite cohomology, then

i i ‚ TC0pRk pAqq Ď Rk pH pAqq.

i i In general, we cannot replace TC0pRk pAqq by Rk pAq.

EXAMPLE Let A “ pa, bq with d a “ 0 and d b “ b ¨ a. Then H‚ A a , and so R1 A 0 . Źp q “ p q 1p q “ t u 1 1 Hence R1pAq “ t0, 1u is not contained in R1pAq, though 1 Ź TC0pR pAqq “ t0u is.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 10 / 24 RESONANCE VARIETIES OF CDGAS TANGENTCONEINCLUSION

i i ‚ In general, the inclusion TC0pRk pAqq Ď Rk pH pAqq is strict.

EXAMPLE Let A “ pa, b, cq with d a “ d b “ 0 and d c “ a ^ b.

Writing SŹ“ krx, ys, we have: y ´x 1 x δ2“ 00 ´x δ1“ y ¨ ˛ 00 ´y ˆ 0 ˙ 3 3 LpAq : S / S ˝ ‚/ S .

1 Hence R1pAq “ t0u. ‚ 1 ‚ 2 But H pAq “ pa, bq{pabq, and so R1pH pAqq “ k . Ź

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 11 / 24 RESONANCE VARIETIES OF SPACES ALGEBRAICMODELSFORSPACES

ALGEBRAICMODELSFORSPACES

Given any space X, there is an associated Sullivan Q-cdga, ‚ ‚ APLpXq, such that H pAPLpXqq “ H pX, Qq. We say X is q-finite if X has the homotopy type of a connected CW-complex with finite q-skeleton, for some q ě 1.

An algebraic (q-)model (over k) for X is a k-cgda pA, dq which is (q-) equivalent to APLpXq bQ k.

If M is a smooth manifold, then ΩdRpMq is a model for M (over R). Examples of spaces having finite-type models include: Formal spaces (such as compact Kähler manifolds, hyperplane arrangement complements, toric spaces, etc). Smooth quasi-projective varieties, compact solvmanifolds, Sasakian manifolds, etc.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 12 / 24 RESONANCE VARIETIES OF SPACES GERMSOFJUMPLOCI

GERMSOFJUMPLOCI

THEOREM (DIMCA–PAPADIMA 2014) LetX be aq-finite space, and supposeX admits aq-finite,q-modelA. Then the map exp: H1pX, Cq Ñ H1pX, C˚q induces a local analytic 1 isomorphismH pAqp0q Ñ CharpXqp1q, which identifies the germ at 0 of i i Rk pAq with the germ at 1 of Vk pXq, for alli ď q andk ě 0.

COROLLARY i i IfX is aq-formal space, then Vk pXqp1q – Rk pXqp0q, fori ď q andk ě 0.

A precursor to corollary can be found in work of Green, Lazarsfeld, and Ein on cohomology jump loci of compact Kähler manifolds.

The case when q “ 1 was first established in [DPS 2019].

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 13 / 24 RESONANCE VARIETIES OF SPACES TANGENTCONESANDEXPONENTIALMAPS

TANGENTCONESANDEXPONENTIALMAPS

n ˆ n z zn The map exp: C Ñ pC q , pz1,..., znq ÞÑ pe 1 ,..., e q is a homomorphism taking0 to1. For a Zariski-closed subset W “ V pIq inside pCˆqn, define: The tangent cone at1 to W as TC1pW q “ V pinpIqq. The exponential tangent cone at1 to W as n τ1pW q “ tz P C | exppλzq P W , @λ P Cu

These sets are homogeneous subvarieties of Cn, which depend only on the analytic germ of W at1. Both commute with finite unions and arbitrary intersections.

τ1pW q Ď TC1pW q. “ if all irred components of W are subtori. ‰ in general.

(DPS 2009) τ1pW q is a finite union of rationally defined subspaces.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 14 / 24 RESONANCE VARIETIES OF SPACES THE TANGENT CONE THEOREM

THE TANGENT CONE THEOREM

Let X be a connected CW-complex with finite q-skeleton.

THEOREM (LIBGOBER 2002, DPS 2009) For alli ď q andk ě 0,

i i i τ1pVk pXqq Ď TC1pVk pXqq Ď Rk pXq.

THEOREM (DPS-2009, DP-2014)

SupposeX is aq-formal space. Then, for alli ď q andk ě 0,

i i i τ1pVk pXqq “ TC1pVk pXqq “ Rk pXq.

i In particular, all irreducible components of Rk pXq are rationally defined linear subspaces ofH 1pX, Cq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 15 / 24 RESONANCE VARIETIES OF SPACES DETECTINGNON-FORMALITY

DETECTINGNON-FORMALITY

EXAMPLE 1 Let π “ xx1, x2 | rx1, rx1, x2ssy. Then V1 pπq “ tt1 “ 1u, and so

1 1 τ1pV1 pπqq “ TC1pV1 pπqq “ tx1 “ 0u.

1 2 On the other hand, R1pπq “ C , and so π is not1-formal.

EXAMPLE ´2 ´1 Let π “ xx1,..., x4 | rx1, x2s, rx1, x4srx2 , x3s, rx1 , x3srx2, x4sy. Then

1 4 2 2 R1pπq “ tz P C | z1 ´ 2z2 “ 0u.

This is a quadric hypersurface which splits into two linear subspaces over R, but is irreducible over Q. Thus, π is not1-formal.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 16 / 24 RESONANCE VARIETIES OF SPACES DETECTINGNON-FORMALITY

EXAMPLE 3 Let π be a finitely presented group with πab “ Z and

1 ˚ 3 V1 pπq “ pt1, t2, t3q P pC q | pt2 ´ 1q “ pt1 ` 1qpt3 ´ 1q ,

This is a complex, 2-dimensional torus passing through the origin,( but this torus does not embed as an algebraic subgroup in pC˚q3. Indeed,

1 τ1pV1 pπqq “ tx2 “ x3 “ 0u Y tx1 ´ x3 “ x2 ´ 2x3 “ 0u.

Hence, π is not1-formal.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 17 / 24 RESONANCE VARIETIES OF SPACES DETECTINGNON-FORMALITY

EXAMPLE

Let ConfnpEq be the configuration space of n labeled points of an elliptic curve E “ Σ1.

‚ Using the computation of H pConfnpΣgq, Cq by Totaro (1996), we 1 find that R1pConfnpEqq is equal to

n n n n xi “ yi “ 0, px, yq P C ˆ C i“1 i“1 xi yj ´ xj yi “ 0, for 1 ď i ă j ă n " ˇ ř ř * ˇ ˇ For n ě 3, this is an irreducible,ˇ non-linear variety (a rational normal scroll). Hence, ConfnpEq is not1-formal.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 18 / 24 INFINITESIMAL FINITENESS OBSTRUCTIONS SPACES WITH FINITE MODELS

SPACES WITH FINITE MODELS

THEOREM (EXPONENTIAL AX–LINDEMANNTHEOREM) LetV Ď Cn andW Ď pC˚qn be irreducible algebraic subvarieties. 1 Suppose dim V “ dim W and exppV q Ď W . ThenV is a translate of a linear subspace, andW is a translate of an algebraic subtorus. n n 2 Suppose the exponential map exp: C Ñ pC˚q induces a local analytic isomorphismV p0q Ñ Wp1q. ThenW p1q is the germ of an algebraic subtorus.

THEOREM (BUDUR–WANG 2017) IfX is aq-finite space which admits aq-finiteq-model, then, for all i i ď q andk ě 0, the irreducible components of Vk pXq passing through 1 are algebraic subtori of CharpXq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 19 / 24 INFINITESIMAL FINITENESS OBSTRUCTIONS SPACES WITH FINITE MODELS

EXAMPLE n Let G be a f.p. group with Gab “ Z and 1 ˆ n n V1 pGq “ t P pC q | i“1 ti “ n . Then G admits no1-finite1-model. ř ( THEOREM (PAPADIMA–S. 2017)

SupposeX is pq ` 1q finite, orX admits aq-finiteq-model. Let MqpXq be Sullivan’sq-minimal model ofX. Thenb i pMqpXqq ă 8, @i ď q ` 1.

COROLLARY LetG be a f.g. group. Assume that eitherG is finitely presented, orG has a 1-finite 1-model. Thenb 2pM1pGqq ă 8.

EXAMPLE 2 1 1 ˆ n Let G “ Fn { Fn with n ě 2. We have V1 pGq “ V1 pFnq “ pC q , and so G passes the Budur–Wang test. But b2pM1pGqq “ 8, and so G admits no1-finite1-model (and is not finitely presented).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 20 / 24 INFINITESIMAL FINITENESS OBSTRUCTIONS ASSOCIATED GRADED LIEALGEBRAS

ASSOCIATED GRADED LIEALGEBRAS

The lower central series of a group G is defined inductively by γ1G “ G and γk`1G “ rγk G, Gs.

This forms a filtration of G by characteristic subgroups. The LCS quotients, γk G{γk`1G, are abelian groups. The group commutator induces a graded Lie algebra structure on

grpG, q “ pγ G{γ Gq b . k kě1 k k`1 Z k à Assume G is finitely generated. Then grpGq is also finitely generated (in degree1) by gr1pGq “ H1pG, kq. n For instance, grpFnq is the free graded Lie algebra Ln :“ Liepk q.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 21 / 24 INFINITESIMAL FINITENESS OBSTRUCTIONS HOLONOMY LIEALGEBRAS

HOLONOMY LIEALGEBRAS

i ˚ i Let A be a1-finite cdga. Set Ai “ pA q “ HomkpA , kq.

˚ Let µ : A2 Ñ A1 ^ A1 be the dual to the multiplication map µ: A1 ^ A1 Ñ A2.

˚ 1 2 Let d : A2 Ñ A1 be the dual of the differential d : A Ñ A .

The holonomy Lie algebra of A is the quotient

˚ ˚ hpAq “ LiepA1q{ximpµ ` d qy.

For a f.g. group G, set hpGq :“ hpH‚pG, kqq. There is then a canonical surjection hpGq  grpGq, which is an isomorphism precisely when grpGq is quadratic.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 22 / 24 INFINITESIMAL FINITENESS OBSTRUCTIONS MALCEV LIEALGEBRAS

MALCEV LIEALGEBRAS

The group-algebra kG has a natural structure, with comultiplication ∆pgq “ g b g and counit ε: kG Ñ k. Let I “ ker ε.

(Quillen 1968) The I-adic completion of the group-algebra, G “ lim G{Ik , is a filtered, complete Hopf algebra. k ÐÝk k

Anx element x P kG is called primitive if ∆x “ xb1 ` 1bx. The set of all such elements, with bracket rx, ys “ xy ´ yx, and endowed with the inducedx filtration, is a complete,p filteredp Lie algebra.p

We then have mpGq – PrimpkGq and grpmpGqq – grpGq.

(Sullivan 1977) G is1-formal xðñ mpGq is quadratic, namely: mpGq “ hpH‚pG, kq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI{ MIMSSUMMER SCHOOL 2018 23 / 24 INFINITESIMAL FINITENESS OBSTRUCTIONS FINITENESSOBSTRUCTIONSFORGROUPS

FINITENESSOBSTRUCTIONSFORGROUPS

THEOREM (PS 2017) A f.g. groupG admits a 1-finite 1-modelA if and only if mpGq is the lcs completion of a finitely presented Lie algebra, namely,

mpGq – hpAq.

z THEOREM (PS 2017) LetG be a f.g. group which has a free, non-cyclic quotient. Then: G{G2 is not finitely presentable. G{G2 does not admit a 1-finite 1-model.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 24 / 24 GEOMETRYANDTOPOLOGYOF COHOMOLOGYJUMPLOCI

LECTURE 3: FUNDAMENTALGROUPSANDJUMPLOCI

Alex Suciu

Northeastern University

MIMS Summer School: New Trends in Topology and Geometry

Mediterranean Institute for the Mathematical Sciences Tunis, Tunisia

July 9–12, 2018

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 1 / 18 OUTLINE

1 FUNDAMENTALGROUPSINGEOMETRY Fundamental groups of manifolds Kähler groups Quasi-projective groups Complements of hypersurfaces arrangements Artin groups

2 COMPARINGCLASSESOFGROUPS Kähler groups vs other groups Quasi-projective groups vs other groups

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 2 / 18 FUNDAMENTALGROUPSINGEOMETRY FUNDAMENTALGROUPSOFMANIFOLDS

FUNDAMENTALGROUPSOFMANIFOLDS

Every finitely presented group π can be realized as π “ π1pMq, for some smooth, compact, connected manifold Mn of dim n ě 4.

Mn can be chosen to be orientable.

If n even, n ě 4, then Mn can be chosen to be symplectic (Gompf).

If n even, n ě 6, then Mn can be chosen to be complex (Taubes).

Requiring that n “ 3 puts severe restrictions on the (closed) 3 3-manifold group π “ π1pM q.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 3 / 18 FUNDAMENTALGROUPSINGEOMETRY KÄHLERGROUPS

KÄHLERGROUPS

A Kähler manifold is a compact, connected, complex manifold, with a Hermitian metric h such that ω “ imphq is a closed2-form.

Smooth, complex projective varieties are Kähler manifolds.

A group π is called a Kähler group if π “ π1pMq, for some Kähler manifold M.

The group π is a projective group if M can be chosen to be a projective manifold.

The classes of Kähler and projective groups are closed under finite direct products and passing to finite-index subgroups.

Every finite group is a projective group. [Serre „1955]

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 4 / 18 FUNDAMENTALGROUPSINGEOMETRY KÄHLERGROUPS

The Kähler condition puts strong restrictions on π, e.g.: π is finitely presented.

b1pπq is even. [by ]

π is1-formal [Deligne–Griffiths–Morgan–Sullivan 1975]

π cannot split non-trivially as a free product. [Gromov 1989]

Problem: Are all Kähler groups projective groups?

Problem [Serre]: Characterize the class of projective groups.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 5 / 18 FUNDAMENTALGROUPSINGEOMETRY QUASI-PROJECTIVEGROUPS

QUASI-PROJECTIVEGROUPS

A group π is said to be a quasi-Kähler group if π “ π1pMzDq, where M is a Kähler manifold and D is a divisor. The group π is a quasi-projective group if M can be chosen to be a projective manifold. qK/qp groups are finitely presented. The classes of qK/qp groups are closed under finite direct products and passing to finite-index subgroups. For a qp group π,

b1pπq can be arbitrary (e.g., the free groups Fn). π may be non-1-formal (e.g., the Heisenberg group).

π can split as a non-trivial free product (e.g., F2 “ Z ˚ Z).

Problem: Are all quasi-Kähler groups quasi-projective groups?

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 6 / 18 FUNDAMENTALGROUPSINGEOMETRY QUASI-PROJECTIVEGROUPS

RESONANCEOFQUASI-KÄHLERMANIFOLDS

THEOREM (DIMCA–PAPADIMA–S. 2009)

LetX be a quasi-Kähler manifold, andG “ π1pXq. Let tLαuα be the 1 non-zero irreducible components of R1pGq. IfG is 1-formal, then 1 EachL α is a linear subspace ofH pG, Cq.

EachL α isp-isotropic (i.e., restriction of YG toL α has rankp), with dim Lα ě 2p ` 2, for somep “ ppαq P t0, 1u.

If α ‰ β, thenL α X Lβ “ t0u. R1 G 0 L . k p q “ t u Y α:dim Lαąk`ppαq α Furthermore, Ť

IfX is compact, thenG is 1-formal, and eachL α is 1-isotropic. 1 IfW 1pH pX, Cqq “ 0, thenG is 1-formal, and eachL α is 0-isotropic.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 7 / 18 FUNDAMENTALGROUPSINGEOMETRY COMPLEMENTSOFHYPERSURFACES

COMPLEMENTSOFHYPERSURFACES A subclass of quasi-projective groups consists of fundamental n groups of complements of hypersurfaces in CP , n π “ π1pCP ztf “ 0uq, f P Crz0,..., zns homogeneous. All such groups are1-formal. [Kohno 1983] 2 By the Lefschetz hyperplane sections theorem, π “ π1pCP zCq, for some algebraic curve C. Zariski asked Van Kampen to find presentations for such groups. Using the Alexander polynomial, Zariski showed that π is not determined by the combinatorics of C (number and type of singularities), but also depends on the position of its singularities.

PROBLEM (ZARISKI) 2 Is π “ π1pCP zCq residually finite, i.e., is the map to the profinite completion, π Ñ πalg :“ lim π{G, injective? ÐÝGŸf.i.π ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 8 / 18 FUNDAMENTALGROUPSINGEOMETRY LINEARRANGEMENTS

HYPERPLANEARRANGEMENTS Even more special are the arrangement groups, i.e., the fundamental groups of complements of complex hyperplane arrangements (or, equivalently, complex line arrangements). 2 Let A be an arrangement of lines in CP , defined by a polynomial f “ LPA fL, with fL linear forms so that L “ PpkerpfLqq. Theś combinatorics of A is encoded in the intersection poset, LpAq, with L1pAq “ tlinesu and L2pAq “ tintersection pointsu.

P1 P2 P3 P4 L4 L3 P4 L2

P3

L1 P1 P2 L1 L2 L3 L4

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 9 / 18 FUNDAMENTALGROUPSINGEOMETRY LINEARRANGEMENTS

2 Let UpAq “ CP z LPA L. The group π “ π1pUpAqq has a finite presentation with Ť Meridional generators x1,..., xn, where n “ |A|, and xi “ 1. ´1 Commutator relators xi αj pxi q , where α1, . . . αs P Pn śĂ AutpFnq, and s “ |L2pAq|. 1 Let γ1pπq “ π, γ2pπq “ π “ rπ, πs, γk pπq “ rγk´1pπq, πs, be the lower central series of π. Then:

n´1 πab “ π{γ2 equals Z .

π{γ3 is determined by LpAq.

π{γ4 (and thus, π) is not determined by LpAq (G. Rybnikov).

PROBLEM (ORLIK) Is π torsion-free?

Answer is yes if UpAq is a K pπ, 1q. This happens if the cone on A is a simplicial arrangement (Deligne), or supersolvable (Terao).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 10 / 18 FUNDAMENTALGROUPSINGEOMETRY ARTINGROUPS

ARTINGROUPS

Let Γ “ pV , Eq be a finite, simple graph, and let `: E Ñ Zě2 be an edge-labeling. The associated Artin group:

AΓ,` “ xv P V | vwv ¨ ¨ ¨ “ wvw ¨ ¨ ¨, for e “ tv, wu P Ey. `peq `peq loomoon looomooon If pΓ, `q is Dynkin diagram of type An´1 with `pti, i ` 1uq “ 3 and `pti, juq “ 2 otherwise, then AΓ,` is the braid group Bn. If `peq “ 2, for all e P E, then

AΓ “ xv P V | vw “ wv if tv, wu P Ey. is the right-angled Artin group associated to Γ.

1 Γ – Γ ô AΓ – AΓ1 [Kim–Makar-Limanov–Neggers–Roush 80 / Droms 87]

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 11 / 18 FUNDAMENTALGROUPSINGEOMETRY ARTINGROUPS The corresponding Coxeter group, 2 WΓ,` “ AΓ,`{xv “ 1 | v P V y,

fits into exact sequence1 / PΓ,` / AΓ,` / WΓ,` / 1.

THEOREM (BRIESKORN 1971)

IfW Γ,` is finite, thenG Γ,` is quasi-projective.

Idea: let AΓ,` “ reflection arrangement of type WΓ,` (over C) X “ nz H, where n “ |A | Γ,` C HPAΓ,` Γ,` PΓ,` “ π1pXŤΓ,`q then: n AΓ,` “ π1pXΓ,`{WΓ,`q “ π1pC ztδΓ,` “ 0uq

THEOREM (KAPOVICH–MILLSON 1998)

There exist infinitely many pΓ, `q such thatA Γ,` is not quasi-projective.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 12 / 18 COMPARINGCLASSESOFGROUPS KÄHLERGROUPSVSOTHERGROUPS

KÄHLERGROUPSVSOTHERGROUPS

QUESTION (DONALDSON–GOLDMAN 1989) Which3-manifold groups are Kähler groups?

Reznikov gave a partial solution in 2002.

THEOREM (DIMCA–S. 2009) LetG be the fundamental group of a closed 3-manifold. ThenG is a Kähler group ðñ π is a finite subgroup of Op4q, acting freely onS 3.

Idea of our proof: compare the resonance varieties of3-manifolds to those of Kähler manifolds.

By passing to a suitable index-2 subgroup of G, we may assume that the closed3-manifold is orientable.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 13 / 18 COMPARINGCLASSESOFGROUPS KÄHLERGROUPSVSOTHERGROUPS

PROPOSITION LetM be a closed, orientable 3-manifold. Then: 1 1 H pM, Cq is not 1-isotropic.

2 1 1 Ifb 1pMq is even, then R1pMq “ H pM, Cq. On the other hand, it follows from a previous theorem that:

PROPOSITION

LetM be a compact Kähler manifold withb 1pMq ‰ 0. If 1 1 1 R1pMq “ H pM, Cq, thenH pM, Cq is 1-isotropic.

If G is a Kähler, then b1pGq even.

Thus, if G is both a3-mfd group and a Kähler group ñ b1pGq “ 0. Using work of Fujiwara (1999) and Reznikov (2002) on Kazhdan’s property (T), as well as Perelman (2003), it follows that G is a finite subgroup of Op4q.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGYJUMPLOCI MIMSSUMMER SCHOOL 2018 14 / 18 COMPARINGCLASSESOFGROUPS KÄHLERGROUPSVSOTHERGROUPS

Alternative proofs have later been given by Kotschick (2012) and Biswas, Mj, and Seshadri (2012).

THEOREM (FRIEDL–S. 2014)

LetN be a 3-manifold with non-empty, toroidal boundary. If π1pNq is a Kähler group, thenN – S1 ˆ S1 ˆ I.

Subsequent generalization by Kotschick (dropping the toroidal boundary assumption): If G is both an infinite3-manifold group and a Kähler group, then G is a surface group.

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THEOREM (DPS 2009)

Let Γ be a finite simple graph, and leA Γ be the corresponding RAAG. The following are equivalent:

1 AΓ is a Kähler group.

2 AΓ is a of even rank. 3 Γ is a complete graph on an even number of vertices.

THEOREM (S. 2011) 2 Let A be an arrangement of lines in CP , with group π “ π1pUpAqq. The following are equivalent: 1 π is a Kähler group. 2 π is a free abelian group of even rank. 3 A consists of an odd number of lines in .

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QUASI-PROJECTIVEGROUPSVSOTHERGROUPS

THEOREM (DIMCA–PAPADIMA–S. 2011) Let π be the fundamental group of a closed, orientable 3-manifold. Assume π is 1-formal. Then the following are equivalent:

1 mpπq – mpπ1pXqq, for some quasi-projective manifoldX. 3 n 1 2 1 2 mpπq – mpπ1pNqq, whereN is eitherS , # S ˆ S , orS ˆ Σg.

THEOREM (FRIEDL–S. 2014)

LetN be a 3-mfd with empty or toroidal boundary. If π1pNq is a quasi- projective group, then all prime components ofN are graph manifolds.

In particular, the fundamental group of a hyperbolic3-manifold with empty or toroidal boundary is never a qp-group.

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THEOREM (DPS 2009)

A right-angled Artin groupA Γ is a quasi-projective group if and only if Γ is a complete multipartite graphK n1,...,nr “ K n1 ˚ ¨ ¨ ¨ ˚ K nr , in which caseA Γ “ Fn1 ˆ ¨ ¨ ¨ ˆ Fnr .

THEOREM (S. 2011)

Let π “ π1pUpAqq be an arrangement group. The following are equivalent: 1 π is a RAAG. 2 π is a finite direct product of finitely generated free groups. 3 GpAq is a forest.

Here GpAq is the ‘multiplicity’ graph, with

vertices: points P P L2pAq with multiplicity at least3; edges: tP, Qu if P, Q P L, for some L P A.

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