Alex Suciu Northeastern University MIMS Summer School: New Trends - - PowerPoint PPT Presentation

GEOMETRY AND TOPOLOGY OF

COHOMOLOGY JUMP LOCI

LECTURE 1: CHARACTERISTIC 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 1 / 25

OUTLINE

1

CAST OF CHARACTERS

The character group The equivariant chain complex Characteristic varieties Degree 1 characteristic varieties

2

EXAMPLES AND COMPUTATIONS

Warm-up examples Toric complexes and RAAGs Quasi-projective manifolds

3

APPLICATIONS

Homology of finite abelian covers Dwyer–Fried sets Duality and propagation

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 2 / 25

CAST OF CHARACTERS THE CHARACTER GROUP

THE CHARACTER GROUP

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. Let G = π1(X, e0) be the fundamental group of X: a finitely generated group, with generators x1 = [e1

1], . . . , xm = [e1 m].

The character group, p G = Hom(G, Cˆ) Ă (Cˆ)m is a (commutative) algebraic group, with multiplication ρ ¨ ρ1(g) = ρ(g)ρ1(g), and identity G Ñ Cˆ, g ÞÑ 1. Let Gab = G/G1 – H1(X, Z) be the abelianization of G. The projection ab: G Ñ Gab induces an isomorphism p Gab

»

Ý Ñ p G.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 3 / 25

CAST OF CHARACTERS THE CHARACTER GROUP

The identity component, p G0, is isomorphic to a complex algebraic torus of dimension n = rank Gab. The other connected components are all isomorphic to p G0 = (Cˆ)n, and are indexed by the finite abelian group Tors(Gab). Char(X) = p G is the moduli space of rank 1 local systems on X: ρ: G Ñ Cˆ

• Cρ

the complex vector space C, viewed as a right module over the group ring ZG via a ¨ g = ρ(g)a, for g P G and a P C.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 4 / 25

CAST OF CHARACTERS THE EQUIVARIANT CHAIN COMPLEX

THE EQUIVARIANT CHAIN COMPLEX

Let p : r X Ñ X be the universal cover. The cell structure on X lifts to a cell structure on r X. Fixing a lift ˜ e0 P p´1(e0) identifies G = π1(X, e0) with the group of deck transformations of r X. Thus, we may view the cellular chain complex of r X as a chain complex of left ZG-modules, ¨ ¨ ¨

Ci+1(r

X, Z)

˜ Bi+1 Ci(r

X, Z)

˜ Bi

Ci´1(r

X, Z)

¨ ¨ ¨ .

˜ B1(˜ e1

i ) = (xi ´ 1)˜

e0. ˜ B2(˜ e2) = řm

i=1

• Br/Bxi

φ ¨ ˜ e1

i , where

r P Fm = xx1, . . . , xmy is the word traced by the attaching map of e2; 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 5 / 25

CAST OF CHARACTERS THE EQUIVARIANT CHAIN COMPLEX

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

Ci+1(X, C)

˜ Bi+1(ρ) Ci(X, C) ˜ Bi(ρ)

Ci´1(X, C) ¨ ¨ ¨ ,

where the evaluation of ˜ Bi at ρ is obtained by applying the ring homomorphism ZG Ñ C, g ÞÑ ρ(g) to each entry of ˜ Bi. Alternatively, consider the universal abelian cover, X ab, and its equivariant chain complex, C‚(X ab, Z) = ZGab bZG C‚(r X, Z), with differentials Bab

i

= id b r Bi. Then H˚(X, Cρ) is computed from the resulting C-chain complex, with differentials Bab

i (ρ) = ˜

Bi(ρ). The identity 1 P Char(X) yields the trivial local system, 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 6 / 25

CAST OF CHARACTERS CHARACTERISTIC VARIETIES

CHARACTERISTIC VARIETIES

DEFINITION The characteristic varieties of X are the sets Vi

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

For each i, get stratification Char(X) = Vi

0 Ě Vi 1 Ě Vi 2 Ě ¨ ¨ ¨

1 P Vi

k(X) ð

ñ bi(X) ě k. V0

1(X) = t1u and V0 k (X) = H, for k ą 1.

Define analogously Vi

k(X, k) Ă Hom(G, kˆ), for arbitrary field k.

Then Vi

k(X, k) = Vi k(X, K) X Hom(G, kˆ), for any k Ď K.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 7 / 25

CAST OF CHARACTERS CHARACTERISTIC VARIETIES

LEMMA For each 0 ď i ď q and k ě 0, the set Vi

k(X) is a Zariski closed subset

• f the algebraic group p

G = Char(X). PROOF (FOR i ă q). Let R = C[Gab] be the coordinate ring of p G = p

• Gab. By definition, a

character ρ belongs to Vi

k(X) if and only if

rank Bab

i+1(ρ) + rank Bab i (ρ) ď ci ´ k,

where ci = ci(X) is the number of i-cells of X. Hence,

Vi

k(X) =

č

r+s=ci´k+1; r,sě0

tρ P p G | rank Bab

i+1(ρ) ď r ´ 1 or rank Bab i (ρ) ď s ´ 1u

= V

• ÿ

r+s=ci´k+1; r,sě0

Ir(Bab

i ) ¨ Is(Bab i+1)

• ,

where Ir(ϕ) = ideal of r ˆ r minors of ϕ.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 8 / 25

CAST OF CHARACTERS CHARACTERISTIC VARIETIES

The characteristic varieties are homotopy-type invariants of a space: LEMMA Suppose X » X 1. There is then an isomorphism p G 1 – p G, which restricts to isomorphisms Vi

k(X 1) – Vi k(X), for all i ď q and k ě 0.

PROOF. Let f : X Ñ X 1 be a (cellular) homotopy equivalence. The induced homomorphism f7 : π1(X, e0) Ñ π1(X 1, e

10), yields an

isomorphism of algebraic groups, ˆ f7 : x G1 Ñ p G. Lifting f to a cellular homotopy equivalence, ˜ f : r X Ñ r X 1, defines isomorphisms Hi(X, Cρ˝f7) Ñ Hi(X 1, Cρ), for each ρ P p G 1. Hence, ˆ f7 restricts to isomorphisms Vi

k(X 1) Ñ Vi k(X).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 9 / 25

CAST OF CHARACTERS DEGREE 1 CHARACTERISTIC VARIETIES

DEGREE 1 CHARACTERISTIC VARIETIES

V1

k (X) depends only on G = π1(X) (in fact, only on G/G2), so we

may write these sets as V1

k (G).

Suppose G = xx1, . . . , xm | r1, . . . , rpy is finitely presented Away from 1 P p G, we have that V1

k (G) = V(Ek(Bab 1 )), the zero-set

• f the ideal of codimension k minors of the Alexander matrix

Bab

1 =

• Bri/Bxj

ab : ZGp

ab Ñ ZGm ab.

If ϕ: G ։ Q is an epimorphism, then, for each k ě 1, the induced monomorphism between character groups, ϕ˚ : p Q ã Ñ p G, restricts to an embedding V1

k (Q) ã

Ñ V1

k (G).

Given any subvariety W Ă (Cˆ)n defined over Z, there is a finitely presented group G such that Gab = Zn and V1

1(G) = W.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 10 / 25

EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES

WARM-UP EXAMPLES

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

Z[t˘1]

t´1 Z[t˘1]

For ρ P Hom(Z, Cˆ) = Cˆ, we get Cρ bZZ C‚(Ă S1) : 0

C

ρ´1 C

which is exact, except for ρ = 1, when H0(S1, C) = H1(S1, C) = C. Hence: V0

1(S1) = V1 1(S1) = t1u

Vi

k(S1) = H,

• therwise.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 11 / 25

EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES

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

k(T n) =

# t1u if k ď (n

i ),

H

• therwise.

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

k(M) =

# t1u if k ď bi(M), H

• therwise

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 12 / 25

EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES

EXAMPLE (WEDGE OF CIRCLES) Identify π1(Žn S1) = Fn, and Hom(Fn, Cˆ) = (Cˆ)n. Then: V1

k

• n

ł S1 = $ ’ & ’ % (Cˆ)n if k ă n, t1u if k = n, H if k ą n. EXAMPLE (ORIENTABLE SURFACE OF GENUS g ą 1) Write π1(Σg) = xx1, . . . , xg, y1, . . . , yg | [x1, y1] ¨ ¨ ¨ [xg, yg] = 1y, and identify Hom(π1(Σg), Cˆ) = (Cˆ)2g. Then: Vi

k(Σg) =

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

• therwise.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 13 / 25

EXAMPLES AND COMPUTATIONS TORIC COMPLEXES

TORIC COMPLEXES AND RAAGS

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

σPL

T σ, where T σ = tx P T n | xi = ˚ if i R σu. Let Γ = (V, E) be the graph with vertex set the 0-cells of L, and edge set the 1-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:

Γ = K n ñ GΓ = Fn Γ = Kn ñ GΓ = Zn Γ = Γ1 š Γ2 ñ GΓ = GΓ1 ˚ GΓ2 Γ = Γ1 ˚ Γ2 ñ GΓ = GΓ1 ˆ GΓ2

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 14 / 25

EXAMPLES AND COMPUTATIONS TORIC COMPLEXES

Identify character group p GΓ = Hom(GΓ, Cˆ) with the algebraic torus (Cˆ)V := (Cˆ)n. For each subset W Ď V, let (Cˆ)W Ď (Cˆ)V be the corresponding coordinate subtorus; in particular, (Cˆ)H = t1u. THEOREM (PAPADIMA–S. 2006/09) Vi

k(TL) =

ď

WĎV

ř

σPLVzW dimC r

Hi´1´|σ|(lkLW(σ),C)ěk

(Cˆ)W, where LW is the subcomplex induced by L on W, and lkK (σ) is the link

• f a simplex σ in a subcomplex K Ď L.

In particular: V1

1(GΓ) =

ď

WĎV

ΓW disconnected

(Cˆ)W.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 15 / 25

EXAMPLES AND COMPUTATIONS QUASI-PROJECTIVE MANIFOLDS

QUASI-PROJECTIVE MANIFOLDS

A space M is said to be a quasi-projective variety if M is a Zariski

• pen subset of a projective variety M (i.e., a Zariski closed subset
• f some projective space).

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 manifold.

When M = Σ is a smooth complex curve with χ(M) ă 0, we saw that V1

1(M) = Char(M).

THEOREM (GREEN–LAZARSFELD, . . . , ARAPURA, . . . , BUDUR–WANG) All the characteristic varieties of a quasi-projective manifold M are finite unions of torsion-translates of subtori of Char(M), i.e., Vi

k(M) = Ť α ραTα, where Tα is an algebraic subtorus and ρnα α = 1.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 16 / 25

EXAMPLES AND COMPUTATIONS QUASI-PROJECTIVE MANIFOLDS

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, p f7 : Char(Σ) Ñ Char(M) is injective, and im(p f7) is a complex subtorus of V1

1(M).

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 correspondence f p f7 Char(Σ) defines a bijection between E(M) and the set of positive-dimensional, irreducible components of V1

1(M)

passing through 1. THEOREM (DIMCA–PAPADIMA–S. (2008–09)) If ρT and ρ1T 1 are two distinct irreducible components of V1

1(M), then

either T = T 1 or T X T 1 = t1u. Hence, distinct components of V1

1(M)

meet only in a finite set of finite-order characters.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 17 / 25

EXAMPLES AND COMPUTATIONS QUASI-PROJECTIVE MANIFOLDS

EXAMPLE (ORDERED CONFIGURATION SPACE OF n POINTS IN C) Let Confn(Σ) = tz P Σn | zi ‰ zju, and set Mn = Confn(C). Then π1(Mn) = Pn, and so Char(Mn) = (Cˆ)(n

2).

(D. Cohen–S. 1999) The set of irreducible components of V1

1(Mn)

passing through 1 consists of the following (n

3) + (n 4) = (n+1 4 )

subtori of dimension 2: Tijk =

• tijtiktjk = 1 and trs = 1 if tr, su Ć ti, j, ku

( . Tijkℓ =

• tij = tjk, tjk = tiℓ, tik = tjℓ,

ź

1ďpăqďn

tpq = 1, and trs = 1 if tr, su Ć ti, j, k, ℓu

( . EXAMPLE (ORDERED CONFIGURATION SPACE OF E = Σ1) (Dimca 2010) The set of positive-dimensional components of V1

1(Confn(E)) consists of (n 2) two-dimensional subtori of (Cˆ)n(n´1), of

the form Tij = im(x fij7), where fij : En Ñ Ezt1u is given by z ÞÑ ziz´1

j

.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 18 / 25

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 Let Y Ñ X be a regular cover, defined by an epimorphism ν from G = π1(X) to a finite abelian group A. Let k be an algebraically closed field of characteristic not dividing the order of A. Then, for each i ě 0, dimk Hi(Y, k) = ÿ

kě1

ˇ ˇ ˇim(p ν) X Vi

k(X, k)

ˇ ˇ ˇ . PROOF (SKETCH). By Shapiro’s Lemma and Maschke’s Theorem, Hi(Y, k) – Hi(X, k[A]) – à

ρPim(p ν)

Hi(X, kρ).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 19 / 25

APPLICATIONS HOMOLOGY OF FINITE ABELIAN COVERS

EXAMPLE Let X = Žn

1 S1, and let Y Ñ X be the 2-fold cover defined by

ν: Fn Ñ Z2, xi ÞÑ 1. (Of course, Y = Ž2n´1

1

S1.) Inside Char(X) = (Cˆ)n, we have that im(p ν) = t1, ´1u, and V1

1(X) = ¨ ¨ ¨ = V1 n´1(X) = (Cˆ)n, while V1 n(X) = t1u.

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 V1

1(X) = ¨ ¨ ¨ = V1 2g´2(X) = (Cˆ)2g and V1 2g´1(X) = V1 2g(X) = t1u.

Hence, b1(Y) = 2g + (n ´ 1)(2g ´ 2) = 2(ng ´ n + 1).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 20 / 25

APPLICATIONS DWYER–FRIED SETS

DWYER–FRIED SETS

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 classified by epimorphisms ν: π ։ Zr. Such covers are parameterized by the Grassmannian Grr(Qn), where n = b1(X), via the correspondence

• regular Zr-covers of X

( Ð Ñ

• r-planes in H1(X, Q)

( X ν Ñ X Ð Ñ Pν := im(ν˚ : Qr Ñ H1(X, Q)) The Dwyer–Fried invariants of X are the subsets Ωi

r(X) =

• Pν P Grr(Qn)

ˇ ˇ bj(X ν) ă 8 for j ď i ( . For each r ą 0, we get a descending filtration, Grr(Qn) = Ω0

r (X) Ě Ω1 r (X) Ě Ω2 r (X) Ě ¨ ¨ ¨ .

Ωi

1(X) is open, but Ωi r(X) may be non-open for r ą 1.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 21 / 25

APPLICATIONS DWYER–FRIED SETS

THEOREM (DWYER–FRIED 1987, PAPADIMA–S. 2010) For an epimorphism ν: π1(X) ։ Zr, the following are equivalent: The vector space Ài

j=0 Hj(X ν, C) is finite-dimensional.

The algebraic torus Tν = im ˆ ν: x Zr ã Ñ { π1(X)

• intersects the

variety Wi(X) = Ť

jďi Vj 1(X) in only finitely many points.

Note that exp(Pν b C) = Tν. Thus: COROLLARY Ωi

r(X) =

• P P Grr(H1(X, Q))

ˇ ˇ dim

• exp(P b C) X Wi(X)

= 0 ( COROLLARY If Wi(X) is finite, then Ωi

r(X) = Grr(Qn), where n = b1(X).

If Wi(X) is infinite, then Ωq

n(X) = H, for all q ě i.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 22 / 25

APPLICATIONS DUALITY AND PROPAGATION

DUALITY AND ABELIAN DUALITY

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 ZG-module A, we have: Hi(X, A) – Hn´i(X, D b A). (Denham–S.–Yuzvinsky 2016/17) X is an abelian duality space of dimension n if Hi(X, ZGab) = 0 for i ‰ n and Hn(X, ZGab) ‰ 0 and torsion-free. Let B = Hn(X, ZGab) be the dualizing ZGab-module. Given any ZGab-module A, we have: Hi(X, A) – Hn´i(X, B b A).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 23 / 25

APPLICATIONS DUALITY AND PROPAGATION

THEOREM (DSY) Let X be an abelian duality space of dimension n. Then: b1(X) ě n ´ 1. bi(X) ‰ 0, for 0 ď i ď n and bi(X) = 0 for i ą n. (´1)nχ(X) ě 0. The characteristic varieties propagate, i.e., V1

1(X) Ď ¨ ¨ ¨ Ď Vn 1(X).

THEOREM (DENHAM–S. 2018) Let M be a quasi-projective manifold of dimension n. Suppose M has a smooth compactification M for which

1

Components of MzM form an arrangement of hypersurfaces A;

2

For each submanifold X in the intersection poset L(A), the complement of the restriction of A to X is a Stein manifold. Then M is both a duality space and an abelian duality space of dimension n.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 24 / 25

APPLICATIONS DUALITY AND PROPAGATION

LINEAR, ELLIPTIC, AND TORIC ARRANGEMENTS

THEOREM (DS18) Suppose that A is one of the following:

1

An affine-linear arrangement in Cn, or a hyperplane arrangement in CPn;

2

A non-empty elliptic arrangement in En;

3

A toric arrangement in (Cˆ)n. Then the complement M(A) is both a duality space and an abelian duality space of dimension n ´ r, n + r, and n, respectively, where r 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 25 / 25

GEOMETRY AND TOPOLOGY OF

COHOMOLOGY JUMP LOCI

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) COHOMOLOGY JUMP LOCI MIMS SUMMER 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) COHOMOLOGY JUMP LOCI MIMS SUMMER 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 characteristic 0. That is:

A “ À

iě0 Ai, where Ai are k-vector spaces.

The multiplication ¨: Ai b Aj Ñ Ai`j 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); dimk Ai is finite for i ď q.

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

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 3 / 24

RESONANCE VARIETIES OF CDGAS COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS

A cdga morphism 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) COHOMOLOGY JUMP LOCI MIMS SUMMER 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, pA‚, δaq: A0

δ0

a

A1

δ1

a

A2

δ2

a

¨ ¨ ¨ ,

with differentials δi

apuq “ a ¨ u ` d u, for all u P Ai.

The resonance varieties of A are the sets Ri

kpAq “ ta P H1pAq | dim HipA‚, δaq ě ku.

If A is q-finite, then Ri

kpAq 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 5 / 24

RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

Fix a k-basis te1, . . . , eru for H1pAq, and let tx1, . . . , xru be the dual basis for H1pAq “ pH1pAqq˚. Identify SympH1pAqq with S “ krx1, . . . , xrs, the coordinate ring of the affine space H1pAq. Define a cochain complex of free S-modules, LpAq :“ pA‚ bk S, δq, ¨ ¨ ¨

Ai b S

δi

Ai`1 b S

δi`1 Ai`2 b S

¨ ¨ ¨ ,

where δipu b fq “ řn

j“1 eju b fxj ` d u b f.

The specialization of pA bk S, δq at a P A1 coincides with pA, δaq. Hence, Ri

kpAq is the zero-set of the ideal generated by all minors

• f size bipAq ´ k ` 1 of the block-matrix δi`1 ‘ δi.

In particular, R1

kpAq “ VpIr´kpδ1qq, the zero-set of the ideal of

codimension k minors of δ1.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 6 / 24

RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

EXAMPLE (EXTERIOR ALGEBRA) Let E “ Ź V, where V “ kn, and S “ SympVq. Then LpEq is the Koszul complex on V. E.g., for n “ 3: S

δ1“ ˆ x1 x2 x3 ˙

S3

δ2“ ˜ x2 x3 ´x1 x3 ´x1 ´x2 ¸

S3 δ3“p x3 ´x2 x1 q S .

Hence, Ri

kpEq “

# t0u if k ď `n

i

˘ , H

• therwise.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 7 / 24

RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

EXAMPLE (NON-ZERO RESONANCE) Let A “ Źpe1, e2, e3q{xe1e2y, and set S “ krx1, x2, x3s. Then LpAq : S

δ1“ ˆ x1 x2 x3 ˙

S3

δ2“ ˆ x3 0 ´x1 0 x3 ´x2 ˙

S2 .

R1

kpAq “

$ & % tx3 “ 0u if k “ 1, t0u if k “ 2 or 3, H if k ą 3. EXAMPLE (NON-LINEAR RESONANCE) Let A “ Źpe1, . . . , e4q{xe1e3, e2e4, e1e2 ` e3e4y. Then LpAq : S

δ1“ ¨ ˝ x1 x2 x3 x4 ˛ ‚

S4

δ2“ ˜ x4 ´x1 x3 ´x2 ´x2 x1 x4 ´x3 ¸

S3 .

R1

1pAq “ tx1x2 ` x3x4 “ 0u

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 8 / 24

RESONANCE VARIETIES OF CDGAS RESONANCE VARIETIES

EXAMPLE (NON-HOMOGENEOUS RESONANCE) Let A “ Źpa, bq with d a “ 0, d b “ b ¨ a. H1pAq “ C, generated by a. Set S “ Crxs. Then: LpAq : S

δ1“p 0 x q

S2 δ2“p x´1 0 q S .

Hence, R1pAq “ t0, 1u, a non-homogeneous subvariety of C. Let A1 be the sub-CDGA generated by a. The inclusion map, A1 ã Ñ A, induces an isomorphism in cohomology. But R1pA1q “ t0u, and so the resonance varieties of A and A1 differ, although A and A1 are quasi-isomorphic. PROPOSITION If A »q A1, then Ri

kpAqp0q – Ri kpA1qp0q, for all i ď q and k ě 0.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 9 / 24

RESONANCE VARIETIES OF CDGAS TANGENT CONE INCLUSION

TANGENT CONE INCLUSION

THEOREM (BUDUR–RUBIO, DENHAM–S. 2018) If A is a connected k-CDGA A with locally finite cohomology, then TC0pRi

kpAqq Ď Ri kpH‚pAqq.

In general, we cannot replace TC0pRi

kpAqq by Ri kpAq.

EXAMPLE Let A “ Źpa, bq with d a “ 0 and d b “ b ¨ a. Then H‚pAq “ Źpaq, and so R1

1pAq “ t0u.

Hence R1

1pAq “ t0, 1u is not contained in R1 1pAq, though

TC0pR1pAqq “ t0u is.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 10 / 24

RESONANCE VARIETIES OF CDGAS TANGENT CONE INCLUSION

In general, the inclusion TC0pRi

kpAqq Ď Ri kpH‚pAqq is strict.

EXAMPLE Let A “ Źpa, b, cq with d a “ d b “ 0 and d c “ a ^ b. Writing S “ krx, ys, we have: LpAq : S

δ1“ ˆ x y ˙

S3

δ2“ ¨ ˝ y ´x 1 ´x ´y ˛ ‚

S3 .

Hence R1

1pAq “ t0u.

But H‚pAq “ Źpa, bq{pabq, and so R1

1pH‚pAqq “ k2.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 11 / 24

RESONANCE VARIETIES OF SPACES ALGEBRAIC MODELS FOR SPACES

ALGEBRAIC MODELS FOR SPACES

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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 12 / 24

RESONANCE VARIETIES OF SPACES GERMS OF JUMP LOCI

GERMS OF JUMP LOCI

THEOREM (DIMCA–PAPADIMA 2014) Let X be a q-finite space, and suppose X admits a q-finite, q-model A. Then the map exp: H1pX, Cq Ñ H1pX, C˚q induces a local analytic isomorphism H1pAqp0q Ñ CharpXqp1q, which identifies the germ at 0 of Ri

kpAq with the germ at 1 of Vi kpXq, for all i ď q and k ě 0.

COROLLARY If X is a q-formal space, then Vi

kpXqp1q – Ri kpXqp0q, for i ď q and k 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 13 / 24

RESONANCE VARIETIES OF SPACES TANGENT CONES AND EXPONENTIAL MAPS

TANGENT CONES AND EXPONENTIAL MAPS

The map exp: Cn Ñ pCˆqn, pz1, . . . , znq ÞÑ pez1, . . . , eznq is a homomorphism taking 0 to 1. For a Zariski-closed subset W “ VpIq inside pCˆqn, define:

The tangent cone at 1 to W as TC1pWq “ VpinpIqq. The exponential tangent cone at 1 to W as τ1pWq “ tz P Cn | exppλzq P W, @λ P Cu

These sets are homogeneous subvarieties of Cn, which depend

• nly on the analytic germ of W at 1.

Both commute with finite unions and arbitrary intersections. τ1pWq Ď TC1pWq.

“ if all irred components of W are subtori. ‰ in general.

(DPS 2009) τ1pWq is a finite union of rationally defined subspaces.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER 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 all i ď q and k ě 0, τ1pVi

kpXqq Ď TC1pVi kpXqq Ď Ri kpXq.

THEOREM (DPS-2009, DP-2014) Suppose X is a q-formal space. Then, for all i ď q and k ě 0, τ1pVi

kpXqq “ TC1pVi kpXqq “ Ri kpXq.

In particular, all irreducible components of Ri

kpXq are rationally defined

linear subspaces of H1pX, Cq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 15 / 24

RESONANCE VARIETIES OF SPACES DETECTING NON-FORMALITY

DETECTING NON-FORMALITY

EXAMPLE Let π “ xx1, x2 | rx1, rx1, x2ssy. Then V1

1pπq “ tt1 “ 1u, and so

τ1pV1

1pπqq “ TC1pV1 1pπqq “ tx1 “ 0u.

On the other hand, R1

1pπq “ C2, and so π is not 1-formal.

EXAMPLE Let π “ xx1, . . . , x4 | rx1, x2s, rx1, x4srx´2

2 , x3s, rx´1 1 , x3srx2, x4sy. Then

R1

1pπq “ tz P C4 | z2 1 ´ 2z2 2 “ 0u.

This is a quadric hypersurface which splits into two linear subspaces

• ver R, but is irreducible over Q. Thus, π is not 1-formal.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 16 / 24

RESONANCE VARIETIES OF SPACES DETECTING NON-FORMALITY

EXAMPLE Let π be a finitely presented group with πab “ Z3 and V1

1pπq “

• pt1, t2, t3q P pC˚q3 | 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, τ1pV1

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

Hence, π is not 1-formal.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 17 / 24

RESONANCE VARIETIES OF SPACES DETECTING NON-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 find that R1

1pConfnpEqq is equal to

" px, yq P Cn ˆ Cn ˇ ˇ ˇ ˇ řn

i“1 xi “ řn i“1 yi “ 0,

xiyj ´ xjyi “ 0, for 1 ď i ă j ă n * For n ě 3, this is an irreducible, non-linear variety (a rational normal scroll). Hence, ConfnpEq is not 1-formal.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 18 / 24

INFINITESIMAL FINITENESS OBSTRUCTIONS SPACES WITH FINITE MODELS

SPACES WITH FINITE MODELS

THEOREM (EXPONENTIAL AX–LINDEMANN THEOREM) Let V Ď Cn and W Ď pC˚qn be irreducible algebraic subvarieties.

1

Suppose dim V “ dim W and exppVq Ď W. Then V is a translate

• f a linear subspace, and W is a translate of an algebraic

subtorus.

2

Suppose the exponential map exp: Cn Ñ pC˚qn induces a local analytic isomorphism Vp0q Ñ Wp1q. Then Wp1q is the germ of an algebraic subtorus. THEOREM (BUDUR–WANG 2017) If X is a q-finite space which admits a q-finite q-model, then, for all i ď q and k ě 0, the irreducible components of Vi

kpXq passing through

1 are algebraic subtori of CharpXq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 19 / 24

INFINITESIMAL FINITENESS OBSTRUCTIONS SPACES WITH FINITE MODELS

EXAMPLE Let G be a f.p. group with Gab “ Zn and V1

1pGq “

• t P pCˆqn | řn

i“1 ti “ n

( . Then G admits no 1-finite 1-model. THEOREM (PAPADIMA–S. 2017) Suppose X is pq ` 1q finite, or X admits a q-finite q-model. Let MqpXq be Sullivan’s q-minimal model of X. Then bipMqpXqq ă 8, @i ď q ` 1. COROLLARY Let G be a f.g. group. Assume that either G is finitely presented, or G has a 1-finite 1-model. Then b2pM1pGqq ă 8. EXAMPLE Let G “ Fn { F2

n with n ě 2. We have V1 1pGq “ V1 1pFnq “ pCˆqn, and so

G passes the Budur–Wang test. But b2pM1pGqq “ 8, and so G admits no 1-finite 1-model (and is not finitely presented).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 20 / 24

INFINITESIMAL FINITENESS OBSTRUCTIONS ASSOCIATED GRADED LIE ALGEBRAS

ASSOCIATED GRADED LIE ALGEBRAS

The lower central series of a group G is defined inductively by γ1G “ G and γk`1G “ rγkG, Gs. This forms a filtration of G by characteristic subgroups. The LCS quotients, γkG{γk`1G, are abelian groups. The group commutator induces a graded Lie algebra structure on grpG, kq “ à

kě1pγkG{γk`1Gq bZ k.

Assume G is finitely generated. Then grpGq is also finitely generated (in degree 1) by gr1pGq “ H1pG, kq. For instance, grpFnq is the free graded Lie algebra Ln :“ Liepknq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 21 / 24

INFINITESIMAL FINITENESS OBSTRUCTIONS HOLONOMY LIE ALGEBRAS

HOLONOMY LIE ALGEBRAS

Let A be a 1-finite cdga. Set Ai “ pAiq˚ “ HomkpAi, kq. Let µ˚ : A2 Ñ A1 ^ A1 be the dual to the multiplication map µ: A1 ^ A1 Ñ A2. Let d˚ : A2 Ñ A1 be the dual of the differential d : A1 Ñ A2. 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 22 / 24

INFINITESIMAL FINITENESS OBSTRUCTIONS MALCEV LIE ALGEBRAS

MALCEV LIE ALGEBRAS

The group-algebra kG has a natural Hopf algebra 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, x kG “ lim Ð Ýk kG{Ik, is a filtered, complete Hopf algebra. An element x P x kG is called primitive if p ∆x “ x p b1 ` 1p

• bx. The set
• f all such elements, with bracket rx, ys “ xy ´ yx, and endowed

with the induced filtration, is a complete, filtered Lie algebra. We then have mpGq – Primp x kGq and grpmpGqq – grpGq. (Sullivan 1977) G is 1-formal ð ñ mpGq is quadratic, namely: mpGq “ { hpH‚pG, kq.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 23 / 24

INFINITESIMAL FINITENESS OBSTRUCTIONS FINITENESS OBSTRUCTIONS FOR GROUPS

FINITENESS OBSTRUCTIONS FOR GROUPS

THEOREM (PS 2017) A f.g. group G admits a 1-finite 1-model A if and only if mpGq is the lcs completion of a finitely presented Lie algebra, namely, mpGq – z hpAq. THEOREM (PS 2017) Let G 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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 24 / 24

GEOMETRY AND TOPOLOGY OF

COHOMOLOGY JUMP LOCI

LECTURE 3: FUNDAMENTAL GROUPS AND JUMP LOCI

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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 1 / 18

OUTLINE

1

FUNDAMENTAL GROUPS IN GEOMETRY

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

2

COMPARING CLASSES OF GROUPS

Kähler groups vs other groups Quasi-projective groups vs other groups

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 2 / 18

FUNDAMENTAL GROUPS IN GEOMETRY FUNDAMENTAL GROUPS OF MANIFOLDS

FUNDAMENTAL GROUPS OF MANIFOLDS

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-manifold group π “ π1pM3q.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 3 / 18

FUNDAMENTAL GROUPS IN GEOMETRY KÄHLER GROUPS

KÄHLER GROUPS

A Kähler manifold is a compact, connected, complex manifold, with a Hermitian metric h such that ω “ imphq is a closed 2-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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 4 / 18

FUNDAMENTAL GROUPS IN GEOMETRY KÄHLER GROUPS

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

π is finitely presented. b1pπq is even. [by Hodge theory] π is 1-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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 5 / 18

FUNDAMENTAL GROUPS IN GEOMETRY QUASI-PROJECTIVE GROUPS

QUASI-PROJECTIVE GROUPS

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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 6 / 18

FUNDAMENTAL GROUPS IN GEOMETRY QUASI-PROJECTIVE GROUPS

RESONANCE OF QUASI-KÄHLER MANIFOLDS

THEOREM (DIMCA–PAPADIMA–S. 2009) Let X be a quasi-Kähler manifold, and G “ π1pXq. Let tLαuα be the non-zero irreducible components of R1

• 1pGq. If G is 1-formal, then

Each Lα is a linear subspace of H1pG, Cq. Each Lα is p-isotropic (i.e., restriction of YG to Lα has rank p), with dim Lα ě 2p ` 2, for some p “ ppαq P t0, 1u. If α ‰ β, then Lα X Lβ “ t0u. R1

kpGq “ t0u Y Ť α:dim Lαąk`ppαq Lα.

Furthermore, If X is compact, then G is 1-formal, and each Lα is 1-isotropic. If W1pH1pX, Cqq “ 0, then G is 1-formal, and each Lα is 0-isotropic.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 7 / 18

FUNDAMENTAL GROUPS IN GEOMETRY COMPLEMENTS OF HYPERSURFACES

COMPLEMENTS OF HYPERSURFACES

A subclass of quasi-projective groups consists of fundamental groups of complements of hypersurfaces in CPn, π “ π1pCPnztf “ 0uq, f P Crz0, . . . , zns homogeneous. All such groups are 1-formal. [Kohno 1983] By the Lefschetz hyperplane sections theorem, π “ π1pCP2zCq, for some plane 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) Is π “ π1pCP2zCq residually finite, i.e., is the map to the profinite completion, π Ñ πalg :“ lim Ð ÝGŸf.i.π π{G, injective?

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 8 / 18

FUNDAMENTAL GROUPS IN GEOMETRY LINE ARRANGEMENTS

HYPERPLANE ARRANGEMENTS

Even more special are the arrangement groups, i.e., the fundamental groups of complements of complex hyperplane arrangements (or, equivalently, complex line arrangements). Let A be an arrangement of lines in CP2, defined by a polynomial f “ ś

LPA fL, with fL linear forms so that L “ PpkerpfLqq.

The combinatorics of A is encoded in the intersection poset, LpAq, with L1pAq “ tlinesu and L2pAq “ tintersection pointsu. L1 L2 L3 L4 P1 P2 P3 P4 L1 L2 L3 L4 P1 P2 P3 P4

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 9 / 18

FUNDAMENTAL GROUPS IN GEOMETRY LINE ARRANGEMENTS

Let UpAq “ CP2z Ť

LPA L. The group π “ π1pUpAqq has a finite

presentation with

Meridional generators x1, . . . , xn, where n “ |A|, and ś xi “ 1. Commutator relators xiαjpxiq´1, where α1, . . . αs P Pn Ă AutpFnq, and s “ |L2pAq|.

Let γ1pπq “ π, γ2pπq “ π1 “ rπ, πs, γkpπq “ rγk´1pπq, πs, be the lower central series of π. Then:

πab “ π{γ2 equals Zn´1. 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 Kpπ, 1q. This happens if the cone on A is a simplicial arrangement (Deligne), or supersolvable (Terao).

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 10 / 18

FUNDAMENTAL GROUPS IN GEOMETRY ARTIN GROUPS

ARTIN GROUPS

Let Γ “ pV, Eq be a finite, simple graph, and let ℓ: E Ñ Zě2 be an edge-labeling. The associated Artin group: AΓ,ℓ “ xv P V | vwv ¨ ¨ ¨ loomoon

ℓpeq

“ wvw ¨ ¨ ¨ looomooon

ℓpeq

, for e “ tv, wu P Ey. If pΓ, ℓq is Dynkin diagram of type An´1 with ℓpti, i ` 1uq “ 3 and ℓpti, juq “ 2 otherwise, then AΓ,ℓ is the braid group Bn. If ℓpeq “ 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. Γ – Γ1 ô AΓ – AΓ1 [Kim–Makar-Limanov–Neggers–Roush 80 / Droms 87]

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 11 / 18

FUNDAMENTAL GROUPS IN GEOMETRY ARTIN GROUPS

The corresponding Coxeter group, WΓ,ℓ “ AΓ,ℓ{xv2 “ 1 | v P Vy, fits into exact sequence 1

PΓ,ℓ AΓ,ℓ WΓ,ℓ 1 .

THEOREM (BRIESKORN 1971) If WΓ,ℓ is finite, then GΓ,ℓ is quasi-projective. Idea: let AΓ,ℓ “ reflection arrangement of type WΓ,ℓ (over C) XΓ,ℓ “ Cnz Ť

HPAΓ,ℓ H, where n “ |AΓ,ℓ|

PΓ,ℓ “ π1pXΓ,ℓq then: AΓ,ℓ “ π1pXΓ,ℓ{WΓ,ℓq “ π1pCnztδΓ,ℓ “ 0uq THEOREM (KAPOVICH–MILLSON 1998) There exist infinitely many pΓ, ℓq such that AΓ,ℓ is not quasi-projective.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 12 / 18

COMPARING CLASSES OF GROUPS KÄHLER GROUPS VS OTHER GROUPS

KÄHLER GROUPS VS OTHER GROUPS

QUESTION (DONALDSON–GOLDMAN 1989) Which 3-manifold groups are Kähler groups? Reznikov gave a partial solution in 2002. THEOREM (DIMCA–S. 2009) Let G be the fundamental group of a closed 3-manifold. Then G is a Kähler group ð ñ π is a finite subgroup of Op4q, acting freely on S3. Idea of our proof: compare the resonance varieties of 3-manifolds to those of Kähler manifolds. By passing to a suitable index-2 subgroup of G, we may assume that the closed 3-manifold is orientable.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 13 / 18

COMPARING CLASSES OF GROUPS KÄHLER GROUPS VS OTHER GROUPS

PROPOSITION Let M be a closed, orientable 3-manifold. Then:

1

H1pM, Cq is not 1-isotropic.

2

If b1pMq is even, then R1

1pMq “ H1pM, Cq.

On the other hand, it follows from a previous theorem that: PROPOSITION Let M be a compact Kähler manifold with b1pMq ‰ 0. If R1

1pMq “ H1pM, Cq, then H1pM, Cq is 1-isotropic.

If G is a Kähler, then b1pGq even. Thus, if G is both a 3-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) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 14 / 18

COMPARING CLASSES OF GROUPS KÄHLER GROUPS VS OTHER GROUPS

Alternative proofs have later been given by Kotschick (2012) and Biswas, Mj, and Seshadri (2012). THEOREM (FRIEDL–S. 2014) Let N be a 3-manifold with non-empty, toroidal boundary. If π1pNq is a Kähler group, then N – S1 ˆ S1 ˆ I. Subsequent generalization by Kotschick (dropping the toroidal boundary assumption): If G is both an infinite 3-manifold group and a Kähler group, then G is a surface group.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 15 / 18

COMPARING CLASSES OF GROUPS KÄHLER GROUPS VS OTHER GROUPS

THEOREM (DPS 2009) Let Γ be a finite simple graph, and le AΓ be the corresponding RAAG. The following are equivalent:

1

AΓ is a Kähler group.

2

AΓ is a free abelian group of even rank.

3

Γ is a complete graph on an even number of vertices. THEOREM (S. 2011) Let A be an arrangement of lines in CP2, 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 general position.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 16 / 18

COMPARING CLASSES OF GROUPS QUASI-PROJECTIVE GROUPS VS OTHER GROUPS

QUASI-PROJECTIVE GROUPS VS OTHER GROUPS

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 manifold X.

2

mpπq – mpπ1pNqq, where N is either S3, #nS1 ˆ S2, or S1 ˆ Σg. THEOREM (FRIEDL–S. 2014) Let N be a 3-mfd with empty or toroidal boundary. If π1pNq is a quasi- projective group, then all prime components of N are graph manifolds. In particular, the fundamental group of a hyperbolic 3-manifold with empty or toroidal boundary is never a qp-group.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 17 / 18

COMPARING CLASSES OF GROUPS QUASI-PROJECTIVE GROUPS VS OTHER GROUPS

THEOREM (DPS 2009) A right-angled Artin group AΓ is a quasi-projective group if and only if Γ is a complete multipartite graph Kn1,...,nr “ K n1 ˚ ¨ ¨ ¨ ˚ K nr , in which case AΓ “ 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 least 3; edges: tP, Qu if P, Q P L, for some L P A.

ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 18 / 18