Torsion in the homology of Milnor fibers of hyperplane arrangements
aa r X i v : . [ m a t h . AG ] O c t Torsion in the homology of Milnorfibers of hyperplane arrangements
Graham Denham and Alexander I. Suciu
Abstract
As is well-known, the homology groups of the complement of acomplex hyperplane arrangement are torsion-free. Nevertheless, as we showedin a recent paper [2], the homology groups of the Milnor fiber of such an ar-rangement can have non-trivial integer torsion. We give here a brief accountof the techniques that go into proving this result, outline some of its appli-cations, and indicate some further questions that it brings to light.
Introduction
This talk reported on the main results of [2]. Here, we give an outline of ourapproach and a summary of our conclusions. Our main result gives a con-struction of a family of projective hypersurfaces for which the Milnor fiberhas torsion in homology. The hypersurfaces we use are hyperplane arrange-ments, for which techniques are available to examine the homology of finitecyclic covers quite explicitly, by reducing to rank 1 local systems.The parameter spaces for rank 1 local systems with non-vanishing ho-mology are known as characteristic varieties. In the special case of complexhyperplane arrangement complements, the combinatorial theory of multinetslargely elucidates their structure, at least in degree 1. We make use of aniterated parallel connection construction to build arrangements with suitablecharacteristic varieties, then vary the characteristic of the field of definition
Graham DenhamDepartment of Mathematics, University of Western Ontario, London, ON N6A 5B7e-mail: [email protected]
Supported by NSERCAlexander I. SuciuDepartment of Mathematics, Northeastern University, Boston, MA 02115e-mail: [email protected]
Supported in part by NSF grant DMS–1010298 and NSA grant H98230-13-1-0225 1 Graham Denham and Alexander I. Suciu in order to construct finite cyclic covers with torsion in first homology. Thesecovers include the Milnor fiber. We now give some detail about each step.
The Milnor fibration
A classical construction due to J. Milnor associates to every homogeneouspolynomial f ∈ C [ z , . . . , z ℓ ] a fiber bundle, with base space C ∗ = C \ { } ,total space the complement in C ℓ to the hypersurface defined by f , andprojection map f : C ℓ \ f − (0) → C ∗ .The Milnor fiber F = f − (1) has the homotopy type of a finite, ( ℓ − h : F → F ,is given by h ( z ) = e πi/n z , where n is the degree of f . If f has an isolatedsingularity at the origin, then F is homotopic to a bouquet of ( ℓ − f completely factors into distinct linear forms:that is, when the hypersurface { f = 0 } is a hyperplane arrangement.Building on our previous work with D. Cohen [1], we show there existprojective hypersurfaces (indeed, hyperplane arrangements) whose comple-ments have torsion-free homology, but whose Milnor fibers have torsion inhomology. Our main result can be summarized as follows. Theorem 1.
For every prime p ≥ , there is a hyperplane arrangementwhose Milnor fiber has non-trivial p -torsion in homology. This resolves a problem posed by Randell [7, Problem 7], who conjecturedthat Milnor fibers of hyperplane arrangements have torsion-free homology.Our examples also give a refined answer to a question posed by Dimca andN´emethi [3, Question 3.10]: torsion in homology may appear even if the hyper-surface is defined by a reduced equation. We note the following consequence:
Corollary 1.
There are hyperplane arrangements whose Milnor fibers do nothave a minimal cell structure.
This stands in contrast with arrangement complements, which always ad-mit perfect Morse functions. Our method also allows us to compute the ho-momorphism induced in homology by the monodromy, with coefficients in afield of characteristic not dividing the order of the monodromy.It should be noted that our approach produces only examples of arrange-ments A for which the Milnor fiber F ( A ) has torsion in q -th homology, forsome q >
1. This leaves open the following question.
Question 1.
Is there an arrangement A such that H ( F ( A ) , Z ) has non-zerotorsion? ilnor fibrations of arrangements 3 Since our methods rely on complete reducibility, it is also natural to ask:do there exist projective hypersurfaces of degree n for which the Milnor fiberhas homology p -torsion, where p divides n ? If so, is there a hyperplane ar-rangement with this property?A much-studied question in the subject is whether the Betti numbers of theMilnor fiber of an arrangement A are determined by the intersection lattice, L ( A ). While we do not address this question directly, our result raises arelated, and arguably even more subtle problem. Question 2.
Is the torsion in the homology of the Milnor fiber of a hyperplanearrangement combinatorially determined?As a preliminary question, one may also ask: can one predict the existenceof torsion in the homology of F ( A ) simply by looking at L ( A )? As it turnsout, under fairly general assumptions, the answer is yes: if L ( A ) satisfiescertain very precise conditions, then automatically H ∗ ( F ( A ) , Z ) will havenon-zero torsion, in a combinatorially determined degree. Hyperplane arrangements
Let A be a (central) arrangement of n hyperplanes in C ℓ , defined by a poly-nomial Q ( A ) = Q H ∈A f H , where each f H is a linear form whose kernel is H . The starting point of our study is the well-known observation that theMilnor fiber of the arrangement, F ( A ), is a cyclic, n -fold regular cover of theprojectivized complement, U ( A ); this cover is defined by the homomorphism δ : π ( U ( A )) ։ Z n , taking each meridian generator x H to 1.Now, if k is an algebraically closed field whose characteristic does notdivide n , then H q ( F ( A ) , k ) decomposes as a direct sum, L ρ H q ( U ( A ) , k ρ ),where the rank 1 local systems k ρ are indexed by characters ρ : π ( U ( A )) → k ∗ that factor through δ . Thus, if there is such a character ρ for which H q ( U ( A ) , k ρ ) = 0, but there is no corresponding character in characteris-tic 0, then the group H q ( F ( A ) , Z ) will have non-trivial p -torsion.To find such characters, we first consider multi-arrangements ( A , m ),with positive integer weights m H attached to each hyperplane H ∈ A .The corresponding Milnor fiber, F ( A , m ), is defined by the homomorphism δ m : π ( U ( A )) ։ Z N , x H m H , where N denotes the sum of the weights.Fix a prime p . Starting with an arrangement A supporting a suitable multi-net, we find a deletion A ′ = A \ { H } , and a choice of multiplicities m ′ on A ′ such that H ( F ( A ′ , m ′ ) , Z ) has p -torsion. Finally, we construct a “polarized”arrangement B = A ′ k m ′ , and show that H ∗ ( F ( B ) , Z ) has p -torsion. Graham Denham and Alexander I. Suciu
Characteristic varieties
Our arguments depend on properties of the jump loci of rank 1 local systems.The characteristic varieties of a connected, finite CW-complex X are thesubvarieties V qd ( X, k ) of the character group b G = Hom( G, k ∗ ), consisting ofthose characters ρ for which H q ( X, k ρ ) had dimension at least d .Suppose X χ → X is a regular cover, defined by an epimorphism χ : G → A to a finite abelian group, and if k is an algebraically closed field of character-istic p , where p ∤ |A| , then dim H q ( X χ , k ) = P d ≥ | im( b χ k ) ∩ V qd ( X, k ) | , where b χ k : b A → b G is the induced morphism between character groups. Theorem 2.
Let X χ → X be a regular, finite cyclic cover. Suppose that im( b χ C )
6⊆ V q ( X, C ) , but im( b χ k ) ⊆ V q ( X, k ) , for some field k of characteristic p not dividing the order of the cover. Then H q ( X χ , Z ) has non-zero p -torsion. Multinets
In the case when X = M ( A ) is the complement of a hyperplane arrangement,the positive-dimensional components of the characteristic variety V ( X, C )have a combinatorial description, for which we refer in particular to work ofFalk and Yuzvinsky in [5].A multinet consists of a partition of A into at least 3 subsets A , . . . , A k ,together with an assignment of multiplicities, m : A → N , and a subset X ofthe rank 2 flats, such that any two hyperplanes from different parts intersectat a flat in X , and several technical conditions are satisfied: for instance,the sum of the multiplicities over each part A i is constant, and for each flat Z ∈ X , the sum n Z := P H ∈A i : H ⊃ Z m H is independent of i . Each multinetgives rise to an orbifold fibration X → P \ { k points } ; in turn, such a mapyields by pullback an irreducible component of V ( X, C ).We say that a multinet on A is pointed if for some hyperplane H , wehave m H > m H | n Z for each flat Z ⊂ H in X . We show that thecomplement of the deletion A ′ := A\ { H } supports an orbifold fibration withbase C ∗ and at least one multiple fiber. Consequently, for any prime p | m H ,and any sufficiently large integer r not divisible by p , there exists a regular, r -fold cyclic cover Y → U ( A ′ ) such that H ( Y, Z ) has p -torsion.Furthermore, we also show that any finite cyclic cover of an arrangementcomplement is dominated by a Milnor fiber corresponding to a suitable choiceof multiplicities. Putting things together, we obtain the following result. Theorem 3.
Suppose A admits a pointed multinet, with distinguished hyper-plane H and multiplicity vector m . Let p be a prime dividing m H . There isthen a choice of multiplicity vector m ′ on the deletion A ′ = A \ { H } suchthat H ( F ( A ′ , m ′ ) , Z ) has non-zero p -torsion. For instance, if A is the reflection arrangement of type B , defined bythe polynomial Q = xyz ( x − y )( x − z )( y − z ), then A satisfies the ilnor fibrations of arrangements 5 conditions of the theorem, for m = (2 , , , , , , , ,
1) and H = { z =0 } . Choosing then multiplicities m ′ = (2 , , , , , , ,
1) on A ′ shows that H ( F ( A ′ , m ′ ) , Z ) has non-zero 2-torsion.Similarly, for primes p >
2, we use the fact that the reflection arrangementof the full monomial complex reflection group, A ( p, , p -torsion in the first homology of the Milnor fiber of asuitable multi-arrangement on the deletion. Parallel connections and polarizations
The last step of our construction replaces multi-arrangements with simplearrangements. We add more hyperplanes and increase the rank by means ofsuitable iterated parallel connections. The complement of the parallel con-nection of two arrangements is diffeomorphic to the product of the respectivecomplements, by a result of Falk and Proudfoot [4]. Then the characteristicvarieties of the result are given by a formula due to Papadima and Suciu [6].We organize the process by noting that parallel connection of matroids hasan operad structure, and we analyze a special case which we call the polariza-tion of a multi-arrangement ( A , m ). By analogy with a construction involvingmonomial ideals, we use parallel connection to attach to each hyperplane H the central arrangement of m H lines in C , to obtain a simple arrangement wedenote by Ak m . A crucial point here is the connection between the respectiveMilnor fibers: the pullback of the cover F ( Ak m ) → U ( Ak m ) along the canon-ical inclusion U ( A ) → U ( Ak m ) is equivalent to the cover F ( A , m ) → U ( A ).Using this fact, we prove the following. Theorem 4.
Suppose A admits a pointed multinet, with distinguished hyper-plane H and multiplicity m . Let p be a prime dividing m H . There is thena choice of multiplicities m ′ on the deletion A ′ = A \ { H } such that theMilnor fiber of the polarization A ′ k m ′ has p -torsion in homology, in degree |{ K ∈ A ′ : m ′ K ≥ }| . For instance, if A ′ is the deleted B arrangement as above, then choosing m ′ = (8 , , , , , , ,
1) produces an arrangement B = A ′ k m ′ of 27 hyper-planes in C , such that H ( F ( B ) , Z ) has 2-torsion of rank 108. References
1. D.C. Cohen, G. Denham, A.I. Suciu,
Torsion in Milnor fiber homology , Alg. Geom.Topology (2003), 511–535. MR1997327 (2004d:32043)2. G. Denham, A.I. Suciu, Multinets, parallel connections, and Milnor fibrations of ar-rangements , Proc. London Math. Soc. (2014), no. 6, 1435–1470. MR3218315 Graham Denham and Alexander I. Suciu3. A. Dimca, A. N´emethi,
Hypersurface complements, Alexander modules and mon-odromy , in: Real and complex singularities, 19–43, Contemp. Math., vol. 354, Amer.Math. Soc., Providence, RI, 2004. MR2087802 (2005h:32076)4. M. Falk, N. Proudfoot,
Parallel connections and bundles of arrangements , TopologyAppl. (2002), no. 1-2, 65–83. MR1877716 (2002k:52033)5. M. Falk, S. Yuzvinsky,
Multinets, resonance varieties, and pencils of plane curves ,Compositio Math. (2007), no. 4, 1069–1088. MR2339840 (2009e:52043)6. S. Papadima, A.I. Suciu,
Bieri–Neumann–Strebel–Renz invariants and homol-ogy jumping loci , Proc. London Math. Soc. (2010), no. 3, 795–834.MR2640291 (2011i:55006)7. R. Randell,