On Betti numbers of Milnor fiber of hyperplane arrangements

On Betti numbers of Milnor fiber of hyperplane arrangements

KaiHo Tommy Wong  and  Yun Su Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 wkhtommy@gmail.com ysu@math.wisc.edu
Abstract.

We find a combinatorial upper bound for the first Betti number of the Milnor fiber for central hyperplane arrangements, which improves previous similar results in [DM] and in [CDO]. In particular, we obtain a combinatorial obstruction for trivial algebraic monodromy of the first homology of Milnor fiber. Calculations and comparisons to known examples in [CS] will be provided.

1. Introduction

Let be a central hyperplane arrangement in and be the defining equations of the hyperplanes of . Let and . There is a -locally trivial fibration

where is called the Milnor fiber [SP]. can be identified as the affine hypersurface in .

Many open questions have been raised subject to . For instance, it is known that the integral cohomology ring of is given by the Orlik-Soloman algebra, which only depends on the intersection lattice of [OS]. It has been conjectured that the integral homology, or the characteristic polynomial, hence the Betti numbers, of are also determined by .

There are active work on this conjecture in the case of with controls of the local multiplicities of the 1-flats, complex line given by intersection of the hyperplane components. Assume that the multiplicities of the 1-flats are at most 3 111projective line arrangement with only double and triple points as singularities. Libgober showed that the characteristic polynomial of can be computed explicitly using superabundance ([EM]). Moreover the characteristic polynomial is combinatorially determined by , by the work of Papadima and Suciu in [3C]. We refer to [HS] for a survey of the study of Milnor fiber of arrangements.

While the general conjecture is still open, in the case of , Massey obtained an upper bound for the Betti numbers of the Milnor fiber with the combinatoris in using vanishing cycles [DM]. A better combinatorial bound is obtained in [CDO] by Cohen, Dimca, and Orlik, using perverse sheaves theory. In this paper, we improve these previous results by method similar to those in [EM]and [TT]. Specifically, we prove the following theorem.

Theorem 1.1.

Suppose is a central hyperplane arrangement. Let be the Milnor fiber of . For each 1-flat , let denote the number of hyperplanes containing . Then

where is the greatest common factor of and . Moreover, the characteristic polynomial of the monodromy action on has the form

where divides

The structure of this paper is as follows. In section 2, we survey some basic facts about the Euler characteristic of the Milnor fiber an arrangement in and the monodromy of the first homology. The conjecture in the case of reduces to the study of non-trivial eigenspaces of the first homology of the Milnor fiber. In section 3, the notion of Alexander modules will be introduced for hyperplane arrangement, as another perspective for homology of the Milnor fiber. A divisibility theorem for Alexander modules for arrangements will be stated and the essential parts of the proof will be recalled. Then we prove theorem 1.1 in section 4. Section 5 consists of calculations and comparisons to examples in [CS]. In section 6, we introduce the notion of Milnor fiber with multiplicities, and extend our results to this generalized notion of Milnor fiber. In the last section, we compute an upper bound of the second Betti number of the Milnor fiber of central hyperplane arrangements in .

2. Basics about Milnor Fiber of Hyperplane Arrangement in

Let be a central arrangement of hyperplanes in and be the defining equations of the hyperplanes in . The Milnor fiber can be identified with the hypersurface . A 1-flat of is formed by the intersection of two or more hyperplanes in . For each 1-flat , let denote the number of hyperplanes in the arrangement which contain .

Since is homogeneous, there is a fibration, called the Milnor fibration,

The geometric monodromy of the fibration is given by coordinate-wise multiplication of , where is a primitive -root of unity. This monodromy automorphism generates a cyclic free action on . The quotient space can be identified to the projective complement , where is the image of in under the Hopf bundle [CS]. Moreover, the Hopf bundle factors through this regular -fold covering . The covering map is the composition .

Therefore, the Euler characteristic equals to and can be explicitly computed using local multiplicities of the flats of .

Proposition 2.1.

Proof.

Recall that Euler characteristic is additive in the complex algebraic setting. Let be the projective lines in and be the number of intersection points between lines on . Denote these intersection points by and their multiplicities by . The set of intersection points are in one to one correspondence with the set of 1-flat in the affine arrangement . The multiplicities ’s are in correspondence with ’s.

Proposition 2.2.

(cf [DM])

Remark 2.3.

Proposition 2.2 implies that the first and the second Betti numbers of differ by some known quantity, which only involve the degree of the arrangements and the multiplicities of the flats. It was proved by Massey in [DM] using L numbers. On the other hand, it is also a quick corollary of proposition 2.1.

The (geometric) monodromy of has finite order , so as the algebraic monodromy on the first homology

is diagonalizable and its eigenvalues are -roots of unity. We will denote the dimension of the eigenspace by . The homology Wang exact sequence

yields that . As a result, the study of reduces to the study of the eigenspaces of associated to non-trivial eigenvalues, which will be referred as non-trivial eigenspaces in this paper.

3. Divisibility result for Alexander Modules

In this section, we regard the Milnor fiber as an infinite cyclic cover of , up to homotopy. We then can identify the homology of the Milnor fiber as an Alexander module over a Laurent polynomial ring, whose order is called the Alexander polynomial.

Theorem 3.1.

([RR]) The first Alexander polynomial of is the characteristic polynomial of the monodromy .

The degree of the (first) Alexander polynomial is equal to . Moreover, the multiplicities of the roots of the Alexander polynomial, which are the eigenvalues of , indicate the dimensions of the associated eigenspaces. Here we introduce the notion of Alexander modules for a central arrangement in , while the definition applies in general to path connected finite CW-complex with free first integral homology.

3.1. Alexander module of

([MT])

Let be a central hyperplane arrangement in and . The first integral homology is isomorphic to and is generated by the homology class of the meridians at each hyperplane component in . By [HS], the Milnor fiber is homotopy equivalent to the regular infinite cyclic cover of defined by the linking number homomorphism

where sends each generator of to . defines a automorphism by multiplication with . Hence, the homology with local coefficients , called the Alexander module, has the following -modules identification ([AH])

Since is an complex -dimensional affine hypersurface, it has a -dimensional finite CW complex structure ([DB]). is a torsion -module and a finite dimensional -vector space. Since is a principal ideal domain, the Alexander module has the identification . The order of the Alexander module, called the Alexander polynomial, is defined up to units in . The Alexander polynomial is the same as the characteristic polynomial of the Milnor fiber ([RR]). We have a remark regarding Alexander modules in general.

Remark 3.2.

[DN] Let be finite CW-complexes and their first integral homology are free. Let and be the local coefficient defined by . Then is a -module homomorphism and

3.2. Stratification of and local Alexander polynomials

([EM],[TT])

In this section, we introduce a natural stratification of and define the notion of local polynomials.

Define two points in to be equivalent if the two collections of hyperplanes in containing each point coincide. The equivalence classes form stratification of and can be explicitly described: if , then the corresponding strata is

Let be a regular tubular neighborhood of in and . According to the stratification, can be decomposed into subspaces; each of which corresponds to exactly one strata in , such that it trivially fibers over the strata. Denote as the -th dimensional strata in and as the associated subspace in the tube . The fiber of the trivial fibration is the complement of in a small complex disk , transversal to . Moreover, is a central hyperplane arrangement in . In short, the space has a decomposition , where each is the product of and its corresponding fiber.

Example 3.3.

Let be given by . The only 0-strata is the origin and the complements of the origin in the -axis and -axis are the 1-stratas. Let . Then the strata in are and being the complement of the origin in the -axis. is the complement of in a small ball at the origin. is the complement of in a small tube around the -axis. fibers over with fiber .

Example 3.4.

Let be a central hyperplane arrangement in . will be the complement of 1-flats in . The ’s are the 1-flats on , minus the origin. On each , the fiber is the complement of in a small complex 2-dimensional normal disk, which is an affine central line arrangement complement.

If the intersection is non-empty and , then this intersection is a product space of a submanifold of and the same fiber from .

Definition 3.5.

For each , the local Alexander module is defined as

Note that all local meridians in are mapped to the meridian around via the induced map on fundamental group by inclusion. The induced local system is compatible to each local linking number homomorphisms. As -modules,

The local Alexander polynomials are the same as the characteristics polynomials of the local Milnor fiber for .

3.3. Divisibility theorem of Alexander Module

The following theorem is a previous work of the first author in [TT], rephrased in the context of Alexander modules of hyperplane arrangement. There are three main steps in the proof, where the first two will be recalled here. Please refer to [TT] for details.

Theorem 3.6.

([TT]) For , the -th global Alexander polynomial divides the product of the local Alexander polynomials associated to the strata in one of the hyperplanes, say , with:

Proof.

Step 1:

First prove that is a -module epimorphism.

An Lefschetz hyperplane theorem argument yields the following results, similar as in [RF]

is an isomorphism for and is an epimorphism for .It follows that for . Hence, has the homotopy type of attached with some cells of dimension and the lower skeleton of and are homotopic. Moreover, the same is true for their coverings. As a result, there are isomorphisms of -modules

for , and an injection

which prove the first step.

Step 2:

The main tool to analyze is the homology Mayer-Vietoris spectral sequence ([DB2],[EM])

with .

Recall that , where is a submanifold. The fact that is the constant sheaf and the Kunneth formula of homology yield

From above calculation, we will focus on local homology with because the local Milnor fiber has the homotopy type of a () dimensional CW complex. In addition, we also have , that is . Since is the number of subspaces of involved in the calculation, . As a result, the interesting local homology are the in range .

Step 3: The final step is another application of the Lefschetz hyperplane theorem, which leads to the desired ranges and (for details, please see [TT]).

4. On Arrangements in

In this section, we apply the divisibility theorem to a central hyperplane arrangement .

Theorem 4.1.

Suppose is a central hyperplane arrangement. Let be the Milnor fiber of . For each 1-flat , let denote the number of hyperplanes containing . Then

where is the greatest common factor of and . Moreover, the characteristic polynomial of the Milnor monodromy action on has the form

where divides

The below two corollaries of theorem 4.1 are previous known results by Massey and by Cohen, Dimca, and Orlik, aiming to get combinatorial upper bounds for .

Corollary 4.2 (Massey [Dm]).

Suppose is a central hyperplane arrangement. Let be the Milnor fiber of . For each 1-flat , let denote the number of hyperplanes containing . Then

where is the greatest common factor of and . Moreover, the characteristic polynomial of the Milnor monodromy action on has the form

where divides

Corollary 4.3 (Cohen, Dimca, Orlik [Cdo]).

For each and ,

An explicit example will be used to illustrate the proof of theorem 4.1, from which the general proof follows. First we state a well known formula which is essential to the proof.

Theorem 4.4 ([Ac]).

Let be a line arrangement with lines all passing through the origin. Then the Alexander polynomial of is

4.1. Illustrating Example

Let and . Denote . Let . We have . The Mayer-Vietoris spectral sequence applied to yields . The following picture shows all partitions in and the table shows all the -modules summands in and .

By the restriction ranges in the divisibility result with , only the summands with or with will contribute . Moreover, summands with are out of the consideration because they are isomorphic to a finite sum of , hence essentially only possibly contribute to the eigenspaces associated to the trivial eigenvalues.

Only the three terms in the form are considered in the calculation of the upper bound for non-trivial eigenspaces. Each of them corresponds to a single 1-flat in . Recall that and, in this case, . Therefore, .

Now write the first characteristic polynomial of as , where . So, divides the product of the orders of , which is, by theorem 4.4,

Since and that the characteristic polynomial of only has as -roots of unity as zeroes, divides

In this example, , so . It is clear that the values are not used in the proof. Above argument actually holds in general.

4.2. The general case

For general central arrangement in , fix . Only the terms in the form in the Mayer-Vietoris spectral sequence are considered in the calculation of the upper bound for non-trivial eigenspaces. Each . Denote the first characteristic polynomial of by , where . Then divides the order of

As a result, divides Since above is true for all hyperplane component in , divides

5. Examples

We now apply our results on specific examples. We try to detect whether the Milnor fiber monodromy is trivial. Most of the examples provided below have known Betti numbers computed by Fox calculus from the fundamental group of ([CS]).

From the upper bound theorem, to obtain non-vanishing eigenspaces associated to non trivial eigenvalues, some local multiplicities (the ’s) must not be coprime to in all components. Certainly not all hyperplane arrangement would satisfy this condition.

Example 5.1.

Let be the arrangement with defining equation

On the component , the multiplicities of the 1-flats are 4,4,2,2. Note that is coprime to the mattered multiplicities. By the upper bound theorem, . Hence, and . This result is confirmed by explicit calculations in [CS].

As suggested in [EM], non-vanishing eigenspaces with non-trivial eigenvalues only exist when the local multiplicities on different components have a nontrivial common divisor. This combinatorial obstruction can hardly met by many arrangements.

Example 5.2.

Let be the arrangement with defining equation

On the component , the multiplicities of the 1-flats are 4,2,2.
On the component , the multiplicities of the 1-flats are 3,2,2,2.

Then the upper bound theorem implies that and that has no non-trivial eigenvalues . As a comparison, Cohen, Dimca, and Orlik upper bound in [CDO] gives that , where is the primitive third root of unity.

The degree of and all component-wise local multiplicities (besides 2) are not coprime in the following three examples. In addition, the local multiplicities are at most 3, one can use superabundance ([EM]) or the combinatorial formula in [3C] to compute them.

Example 5.3.

Let be the arrangement with defining equation

and , where is the primitive third root of unity. The multiplicities of the 1-flats in all components are 3,3,2, so the global Alexander polynomial of divides

Therefore, and .

Example 5.4.

Let be the Pappus configuration arrangement with defining equation

and , where is the primitive third root of unity. The multiplicities of the 1-flats in all components is 3,3,3,2,2, so the global Alexander polynomial of divides

Hence, and .

Example 5.5.

Let be the Pappus configuration arrangement with defining equation

This is a counterexample showing that the upper bound theorem does not always detect trivial mondromy. It is known that , and hence the mondromy is trivial. However, the multiplicities of the 1-flats in all components is 3,3,3,2,2. The upper bound theorem implies that the Alexander polynomial of divides

which suggests possible room for non-trivial eigenspaces.

Note that in the last three examples, since the multiplicities of the 1-flats do not change across hyperplane components, Cohen, Dimca, and Orlik upper bound gives the same result.

6. On Milnor Fibers with Multiplicities

We can extend the upper bound formula to the case of Milnor fiber with multiplicities for central arrangements in .

Definition 6.1.

Let be a central hyperplane arrangement. Denote the defining equation of by . Let be positive integers. Then is called a multi-arrangement with defining equation The corresponding Milnor fiber is called the Milnor fiber with multiplicities

Proposition 6.2 (cf [Tt]).

Let be a multi-arrangement in . Define

sending the meridian about to . Then defines a local coefficients ; and as -modules,

Remark 6.3.

The notion of is a special case of twisted Alexander modules, which are studied intensively in [TT].

Theorem 6.4.

(theorem 2.8 in [TT]) Let be a multi-arrangement in . The first characteristic polynomial of is

Theorem 6.5.

Let be a multi-arrangement in . Let be the number of hyperplane in that contains the 1-flat . Denote the number of 1-flats in by . The first characteristic polynomial of divides

Proof.

The divisibility theorem holds for Alexander modules with -coefficients. In particular,

and the Mayer-Vietoris spectral sequence with -coefficients implies that . By the divisibility result, is bounded above by the direct sum of local modules from , with . Let us study the summands in the page of the corresponding spectral sequence:

  • : not considered because

  • : not considered because

The proof is completed by the following two computations. By theorem 6.4, the order of is

where is the 1-flat sitting in . On the other hand,

7. On Arrangements in

In this section, we demonstrate such a Mayer-Vietoris argument shown in this paper can be applied to obtain combinatorial upper bound for the second Betti numbers of the Milnor fiber of central hyperplane arrangement in . In principle, bounds for higher Betti numbers can obtained through similar way.

Proposition 7.1.

Suppose is a central hyperplane arrangement. For each flat of rank 1, denote be the number of hyperplanes which contain and be the number of 2-flats which are contained in . Let be the Milnor fiber of . Then

Theorem 7.2.

Suppose is a central hyperplane arrangement. Let be the Milnor fiber of . Then . For each 1-flat , denote be the number of hyperplanes which contain and be the number of 2-flats which are contained in .

where is the sum of upper bounds of the second Betti numbers of the Milnor fiber of local central hyperplane arrangements for the 1-flats in .

Proof.

Since , it is clear that .

By the Wang sequence:

Then

We find an upper bound for the non-trivial eigenspaces of , using a Mayer-Vietoris argument. We only consider the summands in the Mayer-Vietoris spectral sequence which possibly contribute the non-trivial eigenspaces. By the divisibility result of Alexander modules, is bounded above by the direct sum of local Alexander modules from only one component, say , of , with . Now analyze the summands in the page of the corresponding spectral sequence:

  • :

  • :

    1. :

    2. : will only consider the summand with

    3. : , only consider summand with

    4. : will only consider the summand with

    5. : , only consider summand with

    6. : , only consider summand with

  • :

    1. :

    2. : will only consider the summand with

    3. : will only consider the summand with

    4. : , only consider summand with

Only the four terms in red are involved in the upper bound of non-trivial eigenspaces.

  • The (2,1) part of is isomorphic to

  • The (2,1) part of is isomorphic to