Complex geometry and low-dimensional topology · FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY KÄHLER...

COMPLEX GEOMETRY AND LOW-DIMENSIONAL

TOPOLOGY

Alex Suciu

Northeastern University

Geometry and Topology Seminar

Max Planck Institute, Bonn

October 28, 2013

ALEX SUCIU (NORTHEASTERN) CX GEOMETRY AND LOW-DIM TOPOLOGY MPIM G&T SEMINAR 1 / 41

FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY FUNDAMENTAL GROUPS OF MANIFOLDS

FUNDAMENTAL GROUPS OF MANIFOLDS

Every finitely presented group π can be realized as π = π1(M),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 = π1(M3).


FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY KÄHLER GROUPS & 3-MANIFOLD GROUPS

KÄHLER GROUPS & 3-MANIFOLD GROUPS

A Kähler manifold is a compact, connected, complex manifold,with a Hermitian metric h such that ω = im(h) is a closed 2-form.Examples: smooth, complex projective varieties.If M is a Kähler manifold, π = π1(M) is called a Kähler group.This also puts strong restrictions on π, e.g.:

b1(π) is even (Hodge theory)

π is 1-formal: Malcev Lie algebra m(π) is quadratic (DGMS 1975)

π cannot split non-trivially as a free product (Gromov 1989)

π finite ñ π projective group (Serre 1958).

QUESTION (DONALDSON–GOLDMAN 1989)

Which 3-manifold groups are Kähler groups?

Reznikov (2002) gave a partial solution.ALEX SUCIU (NORTHEASTERN) CX GEOMETRY AND LOW-DIM TOPOLOGY MPIM G&T SEMINAR 3 / 41

FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY KÄHLER GROUPS & 3-MANIFOLD GROUPS

THEOREM (DIMCA–S. 2009)

Let π be the fundamental group of a closed 3-manifold. Then π is aKähler group ðñ π is a finite subgroup of O(4), acting freely on S3.

Alternative proofs have since been given by Kotschick (2012) and byBiswas, Mj and Seshadri (2012).

THEOREM (FRIEDL–S. 2013)

Let N be a 3-manifold with non-empty, toroidal boundary. If π1(N) is aKähler group, then N – S1 ˆS1 ˆ I.

Since then, Kotschick has generalized this result, by dropping thetoroidal boundary assumption:

THEOREM (KOTSCHICK 2013)

If π1(N) is an infinite Kähler group, then π1(N) is a surface group.


FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY QUASI-PROJECTIVE GROUPS & 3-MANIFOLD GROUPS

QUASI-PROJECTIVE GROUPS & 3-MANIFOLD GROUPS

A group π is called a quasi-projective group if π = π1(MzD),where M is a smooth, projective variety and D is a divisor withnormal crossings.

Qp groups are finitely presented. The class of qp groups is closedunder direct products and passing to finite-index subgroups.

Examples: Free groups Fn, arrangement groups, fundamentalgroups of complements of plane algebraic curves, fundamentalgroups of Brieskorn manifolds.

For qp groups, b1(π) can be arbitrary, π may be non-1-formal,and π can split as a non-trivial free product.


FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY QUASI-PROJECTIVE GROUPS & 3-MANIFOLD GROUPS

QUESTION (DIMCA–S. 2009)

Which 3-manifold groups are quasi-projective 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 m(π) – m(π1(X )), for some quasi-projective manifold X.

2 m(π) – m(π1(N)), where N is either S3, #nS1 ˆS2, or S1 ˆ Σg .

THEOREM (FRIEDL–S. 2013)

Let N be a 3-mfd with empty or toroidal boundary. If π1(N) is a quasi-projective group, then all prime components of N are graph manifolds.

In particular, the fundamental group of a hyperbolic 3-manifold withempty or toroidal boundary is never a qp-group.


FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY ALEXANDER POLYNOMIALS

ALEXANDER POLYNOMIALS

Let H be a finitely generated, free abelian group.

Let M be a finitely generated module over Λ = Z[H ]. Pick apresentation Λp α // Λs // M // 0 with p ě s.

Let Ek (M) be the ideal of minors of size s´ k of α, and set

ordk (M) := gcd(Ek (M)) P Λ

(well-defined up to units in Λ).

M = Λr ‘ Tors(M) and set

∆rM := ord0(Tors M).

Define the thickness of M as

th(M) = dim Newt(∆rM).



Let X be a finite, conn. CW-complex. Write H := H1(X ;Z)/ Tors.

Alexander invariant: AX = H1(X ;Z[H ]).Alexander polynomials: ∆k

X = ordk (AX ); usual one: ∆ = ∆0.Set th(X ) := th(AX ). Note: th(X ) = th(π1(X )).

Let pH = Hom(H,C˚) be the character torus. Define hypersurfaces

V (∆kX ) = tρ P

pH | ∆kX (ρ) = 0u.

If X = S3zK , then ∆X is the classical Alexander polynomial of K ,and V (∆k

X ) Ă C˚ is the set of roots of ∆X , of multiplicity at least k .

Also define the (degree 1) characteristic varieties of X as

Vk (X ) = tρ P pH | dim H1(X ,Cρ) ě ku,

where Cρ = C, viewed as a module over ZH, via g ¨ x = ρ(g)x .

We then have: Vk (X )zt1u = V (Ek´1(AX ))zt1u.



Let Vk (X ) be the union of all codim 1 irreducible components of Vk (X ).

LEMMA (DPS08 FOR k = 0, FS13 FOR k ą 0)

1 ∆k´1X = 0 if and only if Vk (X ) = pH, in which case Vk (X ) = H.

2 Suppose b1(X ) ě 1 and ∆k´1X ‰ 0. Then at least away from 1,

Vk (X ) = V (∆k´1X ).

THEOREM (DPS, FS)

Suppose b1(X ) ě 2. Then ∆k´1X

.= const if and only if Vk (X ) = H.

Otherwise, the following are equivalent:

1 The Newton polytope of ∆k´1X is a line segment.

2 All irreducible components of Vk (X ) are parallel, codim 1 subtoriof pH.



The next theorem is due to Arapura (1997), with improvements by DPS(2008, 2009) and Artal-Bartolo, Cogolludo, Matei (2010).

THEOREM

Let π be a quasi-projective group. Then, for each k ě 1,The irreducible components of Vk (π) are (possiblytorsion-translated) subtori of the character torus pH.Any two distinct components of Vk (π) meet in a finite set.

Using this theorem, we prove

THEOREM (DPS08 FOR k = 0, FS13 FOR k ą 0)

Let π be a quasi-projective group, and assume b1(π) ‰ 2. Then, foreach k ě 0, the polynomial ∆k

π is either zero, or the Newton polytope of∆k

π is a point or a line segment. In particular, th(π) ď 1.


3-MANIFOLD GROUPS THURSTON NORM AND ALEXANDER NORM

THURSTON NORM AND ALEXANDER NORM

Let N be a 3-manifold with either empty or toroidal boundary.

A class φ P H1(N;Z) = Hom(π1(N),Z) is fibered if there exists afibration p : N Ñ S1 such that p˚ : π1(N)Ñ Z coincides with φ.

Given a surface Σ with connected components Σ1, . . . ,Σs, putχ´(Σ) =

řsi=1 maxt´χ(Σi),0u.

Thurston norm: }φ}T = min

χ´(Σ)u, where Σ runs through all theproperly embedded surfaces dual to φ.

} ´ }T defines a (semi)norm on H1(N;Z), which can be extendedto a (semi)norm } ´ }T on H1(N;Q).

The unit norm ball, BT = tφ P H1(N;Q) | }φ}T ď 1u, is a rationalpolyhedron with finitely many sides, symmetric in the origin.



The set of fibered classes form a cone on certain open,top-dimensional faces of BT , called the fibered faces of BT .

Two faces F and G are equivalent if F = ˘G. Clearly, F is fiberedif and only if ´F is fibered.

We say φ P H1(N;Q) is quasi-fibered if it lies on the boundary of afibered face of BT . Results Stallings (1962) and Gabai (1983) imply

COROLLARY (FS13)

Let p : N 1 Ñ N be a finite cover. Then:

1 φ P H1(N;Q) quasi-fibered ñ p˚(φ) P H1(N 1;Q) quasi-fibered.

2 Pull-backs of inequivalent faces of the Thurston norm ball of N lieon inequivalent faces of the Thurston norm ball of N 1.



Let ∆N =ř

hPH ahh P Z[H ] be the Alexander polynomial of N.

Define a (semi)norm } ´ }A on H1(N;Q) by

}φ}A := max tφ(ah)´ φ(ag) | g,h P H with ag ‰ 0 and ah ‰ 0u.

THEOREM (MCMULLEN 2002)

Let N be a 3-manifold with empty or toroidal boundary and such thatb1(N) ě 2. Then }φ}A ď }φ}T , for any φ P H1(N;Q). Furthermore,equality holds for any quasi-fibered class.

COROLLARY (FS13)

Let N be a 3-manifold with empty or toroidal boundary.

If there is a fibration F Ñ N Ñ S1 with χ(F ) ă 0, then th(N) ě 1.

If N has at least two non-equivalent fibered faces, then th(N) ě 2.


3-MANIFOLD GROUPS THE RFRS PROPERTY

THE RFRS PROPERTY

DEFINITION (AGOL 2008)

A group π is called residually finite rationally solvable (RFRS) if thereis a filtration π = π0 ě π1 ě π2 ě ¨ ¨ ¨ such that

Ş

i πi = t1u, andEach group πi is a normal, finite-index subgroup of π.Each map πi Ñ πi /πi+1 factors through πi Ñ H1(πi ;Z)/ Tors.

E.g., free groups and surface groups are RFRS.

THEOREM (AGOL 2008)

Let N be an irreducible 3-manifold such that π1(N) is virtually RFRS.Let φ P H1(N;Q) be a non-fibered class. There exists then a finitecover p : N 1 Ñ N such that p˚(φ) P H1(N 1;Q) is quasi-fibered.


3-MANIFOLD GROUPS THE RFRS PROPERTY

Assume N is an irreducible 3-manifold with empty or toroidal boundary.

THEOREM (AGOL, WISE, PRZYTYCKI– WISE, . . . )

If N is not a closed graph manifold, then π1(N) is virtually RFRS.

COROLLARY

If N is not a closed graph manifold, then N is virtually fibered.

THEOREM (AGOL, WISE, . . . )

Suppose N is neither S1 ˆD2, nor T 2 ˆ I, nor finitely cover by a torusbundle. Then, @k P N, there is a finite cover N 1 Ñ N s.t. b1(N 1) ě k.

THEOREM

Suppose N is not a graph manifold. Given any k P N, there exists afinite cover N 1 Ñ N such that the Thurston norm ball of N 1 has at leastk non-equivalent fibered faces.

Next, we upgrade the statement about the Thurston unit ball to astatement about the thickness of the Alexander ball.


3-MANIFOLD GROUPS QUASI-PROJECTIVE 3-MANIFOLD GROUPS

QUASI-PROJECTIVE 3-MANIFOLD GROUPS

THEOREM (FS13)

Suppose N is not a graph manifold. There exists then a finite coverN 1 Ñ N with th(N 1) ě 2 and b1(N 1) ě 3.

PROOF.Since N is not a graph manifold, it admits finite covers witharbitrarily large first Betti numbers.We can thus assume that b1(N) ě 3.There exists a finite cover N 1 Ñ N such that the Thurston normball of N 1 has at least 2 non-equivalent fibered faces.A transfer argument shows that b1(N 1) ě b1(N) ě 3.Hence, th(N 1) ě 2.


3-MANIFOLD GROUPS QUASI-PROJECTIVE 3-MANIFOLD GROUPS

We can now prove our theorem in the case when N is irreducible.

THEOREM (FS13)

Let N be an irreducible 3-manifold with empty or toroidal boundary. IfN is not a graph manifold, then π1(N) is not a quasi-projective group.

PROOF.Suppose π1(N) is a qp group.We know there is a finite cover N 1 Ñ N with th(N 1) ě 2 andb1(N 1) ě 3.On the other hand, π1(N 1) is also a qp group.Hence, either b1(N 1) = 2, or th(N 1) ď 1.This is a contradiction.

The case when N has several prime factors is more complicated, butcan be handled with similar techniques.


HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS HYPERPLANE ARRANGEMENTS

HYPERPLANE ARRANGEMENTS

An arrangement of hyperplanes is a finite set A of codimension-1linear subspaces in Cd+1.

Intersection lattice L(A): poset of all intersections of A, orderedby reverse inclusion, and ranked by codimension.

Complement: M(A) = Cd+1zŤ

HPA H.

The Boolean arrangement BnBn: all coordinate hyperplanes zi = 0 in Cn.L(Bn): lattice of subsets of t0,1un.M(Bn): complex algebraic torus (C˚)n.

The braid arrangement An (or, reflection arr. of type An´1)An: all diagonal hyperplanes zi ´ zj = 0 in Cn.L(An): lattice of partitions of [n] = t1, . . . ,nu.M(An): configuration space of n ordered points in C (a classifyingspace for the pure braid group on n strings).



‚ ‚

‚

‚

x2 ´ x4 x1 ´ x2

x1 ´ x4

x2 ´ x3

x1 ´ x3 x3 ´ x4

FIGURE : A planar slice of the braid arrangement A4

M = M(A) has the homotopy type of a connected, finiteCW-complex of dimension d + 1.

In fact, M admits a minimal cell structure. Consequently, H˚(M,Z)is torsion-free.



The Betti numbers bq(M) := rank Hq(M,Z) are given by

d+1ÿ

q=1

bq(M)tq =ÿ

XPL(A)

µ(X )(´t)rank(X ),

where µ : L(A)Ñ Z is the Möbius function, defined recursively byµ(Cd+1) = 1 and µ(X ) = ´

ř

YĽX µ(Y ).

The Orlik–Solomon algebra A = H˚(M,Z) is the quotient of theexterior algebra on generators dual to the meridians, by an idealdetermined by the circuits in the matroid of A.

On the other hand, the fundamental group π = π1(M(A)) is notdetermined by L(A), in general (Rybnikov 1994).


HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS MULTINETS AND RESONANCE

MULTINETS AND RESONANCE

Let A be an arrangement of planes in C3. Its projectivization, A, isan arrangement of lines in CP2.L1(A)ÐÑ lines of A, L2(A)ÐÑ intersection points of A,poset structure of Lď2(A)ÐÑ incidence structure of A.A flat X P L2(A) has multiplicity q if the point X has exactly q linesfrom A passing through it.

DEFINITION (FALK AND YUZVINSKY)

A multinet on A is a partition of the set A into k ě 3 subsetsA1, . . . ,Ak , together with an assignment of multiplicities, m : AÑ N,and a subset X Ď L2(A), called the base locus, such that:

1 There is an integer d such thatř

HPAαmH = d , for all α P [k ].

2 If H and H 1 are in different classes, then H XH 1 P X .3 For each X P X , the sum nX =

ř

HPAα :HĄX mH is independent of α.4 Each set

(Ť

HPAαH)zX is connected.



A multinet as above is also called a (k ,d)-multinet, or a k -multinet.

Reduced multinet: mH = 1, for all H P A.

Net: reduced multinet with nX = 1, for all X P X . In this case,|Aα| = |A|/k = d , for all α. Moreover, X has size d2, and isencoded by a (k ´ 2)-tuple of orthogonal Latin squares.

‚ ‚

‚

‚

2

2

2

A (3,2)-net on the A3 arrangement A (3,4)-multinet on the B3 arrangementX consists of 4 triple points (nX = 1) X consists of 4 triple points (nX = 1)

and 3 triple points (nX = 2)ALEX SUCIU (NORTHEASTERN) CX GEOMETRY AND LOW-DIM TOPOLOGY MPIM G&T SEMINAR 22 / 41


A (3,3)-net on the Ceva matroid. A (4,3)-net on the Hessian matroid.

If A has no flats of multiplicity kr , for some r ą 1, then everyreduced k -multinet is a k -net.(Yuzvinsky and Pereira–Yuzvinsky): If A supports a k -multinetwith |X | ą 1, then k = 3 or 4; moreover, if the multinet is notreduced, then k = 3.Conjecture (Yuz): The only 4-multinet is the Hessian (4,3)-net.


HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS RESONANCE VARIETIES

Let X be a connected, finite CW-complex, and let A = H˚(X ,k).

If chark = 2, assume that H1(X ,Z) has no 2-torsion. Then:a P A1 ñ a2 = 0. Thus, we get a cochain complex,

(A, ¨a) : A0 a // A1 a // A2 // ¨ ¨ ¨ ,

Define the (degree 1) resonance varieties of X as

Rs(X , k) = ta P A1 | dimk H1(A, ¨a) ě su,

In particular, R1(X , k) = ta P A1 | Db P A1zk ¨ a, ab = 0u.

EXAMPLE

R1(T n,k) = t0u, for all n ą 0.R1(Cztn pointsu,k) = kn, for all n ą 1.R1(Σg ,k) = k2g , for all g ą 1.



Let A = tH1, . . . ,Hnu be an arrangement in C3. IdentifyH1(M(A), k) = kn, and fix basis dual to the meridians.

The resonance varieties Rs(A,k) := Rs(M(A),k) Ă kn lie in thehyperplane tx P kn | x1 + ¨ ¨ ¨+ xn = 0u.

R1(A,C) is a union of linear subspaces in Cn.

Each subspace has dimension at least 2, and each pair ofsubspaces meets transversely at 0.

Rs(A,C) is the union of those linear subspaces that havedimension at least s + 1.



Each flat X P L2(A) of multiplicity k ě 3 gives rise to a localcomponent of R1(A,C), of dimension k ´ 1.

More generally, every k -multinet of a sub-arrangement B Ď Agives rise to a component of dimension k ´ 1, and all componentsof R1(A,C) arise in this way.

The resonance varieties R1(A, k) can be more complicated, e.g.,they may have non-linear components.



EXAMPLE (BRAID ARRANGEMENT A4)

��

��

��

HHHH

HH

AAAAAAA

42

1 3 56

R1(A,C) Ă C6 has 4 local components (from triple points), and onenon-local component, from the (3,2)-net:

L124 = tx1 + x2 + x4 = x3 = x5 = x6 = 0u,L135 = tx1 + x3 + x5 = x2 = x4 = x6 = 0u,L236 = tx2 + x3 + x6 = x1 = x4 = x5 = 0u,L456 = tx4 + x5 + x6 = x1 = x2 = x3 = 0u,L = tx1 + x2 + x3 = x1 ´ x6 = x2 ´ x5 = x3 ´ x4 = 0u.



Let Hom(π1(M), k˚) = (k˚)n be the character torus.

The characteristic varieties Vs(A,k) := Vs(M(A),k) Ă (k˚)n lie inthe subtorus tt P (k˚)n | t1 ¨ ¨ ¨ tn = 1u.

Each variety Vs(A,C) is a finite union of torsion-translates ofalgebraic subtori of (C˚)n.

If a linear subspace L Ă Cn is a component of Rs(A,C), then thealgebraic torus T = exp(L) is a component of Vs(A,C).

All components of Vs(A,C) passing through the origin 1 P (C˚)n

arise in this way (and thus, are combinatorially determined).


HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS THE MILNOR FIBRATION OF AN ARRANGEMENT

THE MILNOR FIBRATION OF AN ARRANGEMENT

For each H P A, let fH : Cd+1 Ñ C be a linear form with kernel H

Let Q(A) =ś

HPA fH , a homogeneous polynomial of deg n = |A|.

The map Q : Cd+1 Ñ C restricts to a map Q : M(A)Ñ C˚.

This is the projection of a smooth, locally trivial bundle, known asthe Milnor fibration of the arrangement.

The typical fiber, F (A) = Q´1(1), is a Stein manifold; it has thehomotopy type of a connected, finite CW-complex of dim d .

The monodromy of the bundle is the diffeomorphismh : F Ñ F , z ÞÑ e2πi/nz.


HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS THE MILNOR FIBRATION OF AN ARRANGEMENT

EXAMPLE

Let A be a pencil of 3 lines through the origin of C2. Then F (A) is athrice-punctured torus, and h is an automorphism of order 3:

A

F (A)

h

F (A)

More generally, if A is a pencil of n lines in C2, then F (A) is aRiemann surface of genus (n´1

2 ), with n punctures.

EXAMPLE

Let Bn be the Boolean arrangement, with Q = z1 ¨ ¨ ¨ zn. ThenM(Bn) = (C˚)n and F (Bn) = ker(Q) – (C˚)n´1.


HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS THE HOMOLOGY OF THE MILNOR FIBER

THE HOMOLOGY OF THE MILNOR FIBER

Two basic questions about the Milnor fiber of an arrangement:

(Q1) Are the homology groups H˚(F (A),C) and the characteristicpolynomial of the algebraic monodromy determined by L(A)?

(Q2) Are the homology groups H˚(F (A),Z) torsion-free? If so, doesF (A) admit a minimal cell structure?

The Milnor fiber F (A) is a regular Zn-cover of the projectivizedcomplement U = M(A)/C˚.This cover classified by the homomorphism δ : π1(U) � Zn thatsends each meridian to 1.Let pδ : Hom(Zn, k˚)Ñ Hom(π1(U), k˚). If char(k) - n, then

dimk H1(F (A),k) =ÿ

sě1

ˇ

ˇ

ˇVs(U,k)X im(pδ)

ˇ

ˇ

ˇ.



Recall: the monodromy h : F (A)Ñ F (A) has order n = |A|.

Thus, the characteristic polynomial of h˚ acting on H1(F (A),C)can be written as

∆(t) := det(h˚ ´ t ¨ id) =ź

d |n

Φd (t)ed (A),

where Φ1 = t ´ 1, Φ2 = t + 1, Φ3 = t2 + t + 1, Φ4 = t2 + 1, . . .are the cyclotomic polynomials, and ed (A) P Zě0.

Easy to see: e1(A) = n´ 1. Thus, in degree 1, question (Q1) isequivalent to: are the integers ed (A) determined by Lď2(A)?

Work of Falk–Yuzvinsky and many others implies:

THEOREM

If A admits a reduced k-multinet, then ek (A) ě k ´ 2.



Let A˚ = H˚(M(A),k), where k is a field of characteristic p ą 0.

Let σ =ř

HPA eH P A1 be the “diagonal" vector.

Define the mod-p Aomoto-Betti number of A as

βp(A) = dimk H1(A, ¨σ).

βp(A) depends only on L(A) and p, and 0 ď βp(A) ď |A| ´ 2.

(Cohen–Orlik 2000, Papadima–S. 2010) eps(A) ď βp(A).

THEOREM (PAPADIMA–S. 2013)

Suppose L2(A) has no flats of multiplicity 3r , for some r ą 1. Thenβ3(A) ď 2. Moreover, e3(A) = β3(A), and thus e3(A) iscombinatorially determined.

A similar result holds for e2(A).



THEOREM (DENHAM–S. 2013)For every prime p ě 2, there is an arrangement A such thatHq(F (A),Z) has non-zero p-torsion, for some q ą 1.

Simplest example: the arrangement of 27 hyperplanes in C8 withQ(A) = xy(x2 ´ y2)(x2 ´ z2)(y2 ´ z2)w1w2w3w4w5(x2 ´ w2

1 )(x2 ´ 2w2

1 )(x2 ´ 3w2

1 )(x ´ 4w1)¨

((x ´ y)2 ´ w22 )((x + y)2 ´ w2

3 )((x ´ z)2 ´ w24 )((x ´ z)2 ´ 2w2

4 ) ¨ ((x + z)2 ´ w25 )((x + z)2 ´ 2w2

5 ).

Then H6(F (A),Z) has 2-torsion (of rank 108).ALEX SUCIU (NORTHEASTERN) CX GEOMETRY AND LOW-DIM TOPOLOGY MPIM G&T SEMINAR 34 / 41

HYPERPLANE ARRANGEMENTS AND MILNOR FIBRATIONS BOUNDARY STRUCTURES

BOUNDARY STRUCTURES

Let A be an arrangement of hyperplanes in Cd+1 (d ě 1).

Let P(A) = tP(H)uHPA, and let ν(W ) be a regular neighborhoodof the algebraic hypersurface W =

Ť

HPA P(H) inside CPd .

Let U = CPdz int(ν(W )) be the exterior of P(A).

The boundary manifold of A is BU = Bν(W ): a compact,orientable, smooth manifold of dimension 2d ´ 1.

EXAMPLE

Let A be a pencil of n hyperplanes in Cd+1, defined by Q = zn1 ´ zn

2 .If n = 1, then BU = S2d´1. If n ą 1, then BU = 7n´1S1 ˆS2(d´1).

EXAMPLE

Let A be a near-pencil of n planes in C3, defined byQ = z1(zn´1

2 ´ zn´13 ). Then BU = S1 ˆ Σn´2, where Σg = 7gS1 ˆS1.



By Lefschetz duality: Hq(BU,Z) – Hq(U,Z)‘H2d´q´1(U,Z)

Let A = H˚(U,Z); then A = HomZ(A,Z) is an A-bimodule, with(a ¨ f )(b) = f (ba) and (f ¨ a)(b) = f (ab).

THEOREM (COHEN–S. 2006)

The ring pA = H˚(BU,Z) is the “double" of A, that is: pA = A‘ A, withmultiplication given by (a, f ) ¨ (b,g) = (ab,ag + fb), and gradingpAq = Aq ‘ A2d´q´1.

Now assume d = 2. Then BU is a graph-manifold of dimension 3,modeled on a graph Γ based on the poset Lď2(A).

THEOREM (COHEN–S. 2008)

The manifold BU admits a minimal cell structure. Moreover,

V11 (BU) =

ď

vPV(Γ) : dvě3

ttv ´ 1 = 0u,

where dv denotes the degree of the vertex v, and tv =ś

iPv ti .ALEX SUCIU (NORTHEASTERN) CX GEOMETRY AND LOW-DIM TOPOLOGY MPIM G&T SEMINAR 36 / 41


Let A be an arrangement in Cd+1.

Define F (A) = F (A)XD2(d+1) to be the closed Milnor fiber of A.Clearly, F (A) deform-retracts onto F (A).

The boundary of the Milnor fiber of A is the compact, smooth,orientable, (2d ´ 1)-manifold BF (A) = F (A)XS2d+1.

The pair (F , BF ) is (d ´ 1)-connected. In particular, if d ě 2, thenBF is connected, and π1(BF )Ñ π1(F ) is surjective.

FIGURE : Closed Milnor fiber for Q(A) = xy



EXAMPLE

Let Bn be the Boolean arrangement in Cn. Recall F = (C˚)n´1.Hence, F = T n´1 ˆDn´1, and so BF = T n´1 ˆSn´2.Let A be a near-pencil of n planes in C3. Then BF = S1 ˆ Σn´2.

The Hopf fibration π : Cd+1zt0u Ñ CPd restricts to regular, cyclicn-fold covers, π : F Ñ U and π : BF Ñ BU, which fit into the ladder

Zn

��

Zn

��

Zn //

��

C˚

��

C˚

��BF

π��

// F

π��

» // F

π

��

// M //

π

��

Cd+1zt0u

π��

BU // U » // U U // CPd



Assume now that d = 2. The group π1(BU) has generatorsx1, . . . , xn´1 corresponding to the meridians around the first n´ 1 linesin P(A), and generators y1, . . . , ym corresponding to the cycles in theassociated graph Γ.

PROPOSITION (SU13)

The Zn-cover π : BF Ñ BU is classified by the homomorphismπ1(BU) � Zn given by xi ÞÑ 1 and yi ÞÑ 0.

EXAMPLE

Let A be a pencil of n + 1 planes in C3. Since BU = 7nS1 ˆS2, andBF Ñ BU is a cover with n + 1 sheets, we see that BF = 7n

2S1 ˆS2.



THEOREM (NÉMETHI–SZILARD 2012)

Let A be an arrangement of n planes in C3. The characteristicpolynomial of the algebraic monodromy acting on H1(BF ,C) is given by

∆(t) =ź

XPL2(A)

(t ´ 1)(tgcd(µ(X )+1,n) ´ 1)µ(X )´1.

This shows that b1(BF ) is a much less subtle invariant than b1(F ):it depends only on the number and type of multiple points ofP(A), but not on their relative position.

On the other hand, the torsion in H1(BF ,Z) is still not understood.

For a generic arrangement of n planes in C3, I expect thatH1(BF ,Z) = Zn(n´1)/2 ‘Z

(n´2)(n´3)/2n .

In general, it would be interesting to see whether the torsion inH1(BF (A),Z) consists only of Zn-summands, where n = |A|.



REFERENCES

Stefan Friedl and Alexander Suciu, Kähler groups, quasi-projective groups, and3-manifold groups, Journal of the London Mathematical Society 89 (2014), no. 1,151–168.

Alex Suciu, Characteristic varieties and Betti numbers of free abelian covers, Int.Math. Res. Notices 2014 (2014), no. 4, 1063–1124.

Graham Denham and Alex Suciu, Multinets, parallel connections, and Milnorfibrations of arrangements, Proc. London Math. Soc. 108 (2014), no. 6,1435–1470.

Alex Suciu, Hyperplane arrangements and Milnor fibrations, Ann. Fac. Sci.Toulouse Math. 23 (2014), no. 2, 417–481.

Stefan Papadima and Alex Suciu, The Milnor fibration of a hyperplanearrangement: from modular resonance to algebraic monodromy,arXiv:1401.0868.


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