Computing the Monodromy and Pole Order Filtration On

Computing the monodromy and pole order ﬁltration on Milnor ﬁber cohomology of plane curves

Alexandru Dimca 1 Gabriel Sticlaru 2

1Université Côte d’Azur, LJAD

2Ovidius University

MEGA 2017 NICE

Acknowledgement: This work has been supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 1 / 24 References

A. Dimca, M. Saito, Koszul complexes and spectra of projective hypersurfaces with isolated singularities, arXiv:1212.1081 A. Dimca, G. Sticlaru, A computational approach to Milnor fiber cohomology, Forum Mathematicum, DOI: 10.1515/forum-2016-0044. A. Dimca, G. Sticlaru, Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves, arXiv: 1609.06818. A. Dimca, G. Sticlaru, Computing Milnor fiber monodromy for projective hypersurfaces, arXiv:1703.07146. SINGULAR codes available at http://math1.unice.fr/ dimca/singular.html

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 2 / 24 Outline

1 Main characters and main question Milnor ﬁbers and monodromy Basic facts: Alexander polynomial and Hodge Theory Main question

2 Our computational approach A general spectral sequence Back to Alexander polynomials The algorithm

3 Example of results via the algorithm Zariski’s sextic with 6 cusps on a conic A free curve with non-w. h. singularities

4 Two conjectures and a higher dimensional example

Dimca, Sticlaru (UCA, Ovidius) Milnor fiber cohomology for plane curves 12 June 2017 3 / 24 Main characters and main question Milnor fibers and monodromy Basic definitions

Let C : f (x, y, z) = 0 be a reduced plane curve in the complex projective plane P2, defined by a degree d homogeneous polynomial f in the graded polynomial ring S = C[x, y, z]. The smooth affine surface F : f (x, y, z) = 1 in C3 is called the Milnor fiber of f . The mapping h : F → F given by (x, y, z) 7→ (θx, θy, θz) for θ = exp(2πi/d) is called the monodromy of f . There are induced monodromy operators hj : Hj (F, Q) → Hj (F, Q), hj (ω) = (h−1)∗(ω), for j = 0, 1, 2, and we can look at the corresponding characteristic polynomials

j j j ∆C(t) = det(t · Id − h |H (F, Q)).

Dimca, Sticlaru (UCA, Ovidius) Milnor fiber cohomology for plane curves 12 June 2017 4 / 24 Main characters and main question Milnor fibers and monodromy Basic definitions

j j j ∆C(t) = det(t · Id − h |H (F, Q)).

Dimca, Sticlaru (UCA, Ovidius) Milnor fiber cohomology for plane curves 12 June 2017 4 / 24 Main characters and main question Milnor fibers and monodromy Basic definitions

j j j ∆C(t) = det(t · Id − h |H (F, Q)).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 4 / 24 Main characters and main question Basic facts: Alexander polynomial and Hodge Theory Basic facts: Alexander polynomial

Proposition

1 One has hd = Id, and hence the monodromy operators hj are semisimple.

2 j The roots of the characteristic polynomials ∆C(t) are d-th roots of unity. 3 H0(F, Q) = Q and h0 = Id. 4 0 1 −1 2 d χ(U) ∆C(t)∆C(t) ∆C(t) = (t − 1) , where χ(U) denotes the topological Euler characteristic of the complement U = P2 \ C.

1 Hence the polynomial ∆C(t) = ∆C(t), also called the Alexander 2 polynomial of C, determines the remaining polynomial ∆C(t).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 5 / 24 Main characters and main question Basic facts: Alexander polynomial and Hodge Theory Basic facts: Alexander polynomial

Proposition

1 One has hd = Id, and hence the monodromy operators hj are semisimple.

1 Hence the polynomial ∆C(t) = ∆C(t), also called the Alexander 2 polynomial of C, determines the remaining polynomial ∆C(t).

Theorem ( A.D., G. Lehrer, S. Papadima, M. Saito) There is a direct sum decomposition

1 1 1 H (F, Q) = H (F, Q)1 ⊕ H (F, Q)6=1

according to eigenspaces of the monodromy operator h1 such that 1 1 1 H (F, Q)1 = H (U, Q) is a pure Hodge structure of type (1, 1), of dimension r − 1, where r is the number of irreducible components of the curve C; 1 2 H (F, Q)6=1 is a pure Hodge structure of weight 1. In particular, for λ 6= 1 an eigenvalue of h1, we have 1 1,0 0,1 H (F, C)λ = H (F, C)λ ⊕ H (F, C)λ 1,0 0,1 and dim H (F, C)λ = dim H (F, C)λ. Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 6 / 24 Main characters and main question Basic facts: Alexander polynomial and Hodge Theory Basic facts:Hodge theory

Theorem ( A.D., G. Lehrer, S. Papadima, M. Saito) There is a direct sum decomposition

1 1 1 H (F, Q) = H (F, Q)1 ⊕ H (F, Q)6=1

Question Given the degree d reduced plane curve C : f = 0 and a d-th root of unity λ 6= 1, determine the multiplicity m(λ) of λ as a root of the Alexander polynomial ∆C(t).

A more precise question: ﬁnd a basis (e.g. in terms of rational 1 1,0 differential forms) for the eigenspace H (F, C)λ or for H (F, C)λ. The complete answer is known in the case C smooth and for 2 H (F, C)λ (in fact for any smooth projective hypersurface) by the work of Ph. Grifﬁths (1969) and J. Steenbrink (1977). For singular curves, several answers have been given by H. Esnault (1982), A. Libgober (1982), E. Artal-Bartolo (1990) and many others.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 7 / 24 Main characters and main question Main question Main question

Question Given the degree d reduced plane curve C : f = 0 and a d-th root of unity λ 6= 1, determine the multiplicity m(λ) of λ as a root of the Alexander polynomial ∆C(t).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 7 / 24 Main characters and main question Main question Main question

Question Given the degree d reduced plane curve C : f = 0 and a d-th root of unity λ 6= 1, determine the multiplicity m(λ) of λ as a root of the Alexander polynomial ∆C(t).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 7 / 24 Main characters and main question Main question Main question

Question Given the degree d reduced plane curve C : f = 0 and a d-th root of unity λ 6= 1, determine the multiplicity m(λ) of λ as a root of the Alexander polynomial ∆C(t).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 7 / 24 Our computational approach A general spectral sequence A general spectral sequence

Let Ωj denote the graded S-module of (polynomial) differential j-forms on C3, for 0 ≤ j ≤ 3. For instance f ∈ Ω0 = S and

1 df = fx dx + fy dy + fz dz ∈ Ω .

∗ ∗ The complex Kf = (Ω , df ∧) is nothing else but the Koszul complex in S of the partial derivatives fx , fy and fz of the polynomial f . Fix an integer k such that 1 ≤ k ≤ d and set from now on

λ = exp(−2πik/d).

Then one has the following results.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 8 / 24 Our computational approach A general spectral sequence

Theorem (A.D. (1990), A.D. and M. Saito (2012), M. Saito (2016))

With the above notation, for any integer k with 1 ≤ k ≤ d, there is an E1- spectral sequence E∗(f )k such that

s,t s+t+1 ∗ E1 (f )k = H (Kf )td+k and converging to

s,t s s+t E∞ (f )k = GrP H (F, C)λ where P∗ is a decreasing filtration on the Milnor fiber cohomology, called the pole order filtration. Moreover • E1(f )k = E∞(f )k for all k’s if and only if C is smooth. • E2(f )k = E∞(f )k for all k’s if and only if C has only weighted homogeneous singularities.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 9 / 24 Our computational approach A general spectral sequence

−1,2 0,2 d1 E1 (f )k E1 (f )k

0,1 d1 1,1 E1 (f )k E1 (f )k

1,0 2,0 E1 (f )k d1 E1 (f )k

Figure: The E1-term of the spectral sequence E∗(f )k , d1([ω]) = [d(ω)].

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 10 / 24 Our computational approach Back to Alexander polynomials Back to Alexander polynomials

What about the curves with non weighted homogeneous singularities? Then it is known that E2(f )k 6= E∞(f )k for any 1 ≤ k ≤ d. However we have the following. Theorem Let C : f = 0 be a reduced degree d curve , and let λ = exp(−2πik/d), with k ∈ (0, d) an integer. Then λ is a root of the Alexander polynomial ∆C(t) of multiplicity m(λ) given by

1,0 1,0 m(λ) = dim E2 (f )k + dim E2 (f )k 0 .

Main idea: show that the pole order ﬁltration on H1(F, C) coincides with the Hodge ﬁltration.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 11 / 24 Our computational approach Back to Alexander polynomials Back to Alexander polynomials

1,0 1,0 m(λ) = dim E2 (f )k + dim E2 (f )k 0 .

Main idea: show that the pole order ﬁltration on H1(F, C) coincides with the Hodge ﬁltration.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 11 / 24 Our computational approach The algorithm The algorithm The following steps can be performed using the software SINGULAR for instance. 1 Deﬁne Syz(f ) := ker{df ∧ :Ω2 → Ω3}, and determine a C-vector basis of the q-th homogeneous component Syz(f )q, for 2 ≤ q ≤ 2d. This is particularly easy when C is a free curve.

2 Determine the dimension κq of the kernel of the morphism δq : Syz(f )q → M(f )q−3, where M(f ) = S/(fx , fy , fz ) is the Milnor algebra of f and

δq(ady ∧ dz − bdx ∧ dz + cdx ∧ dy) = [ax + by + cz ] ∈ M(f )q−3.

δq is the differential d1 in the spectral sequence. 1 3 Compute q = κq − dim(df ∧ Ω )q, where the last dimension depends only on the degree d, being exactly 3 dim Sq−d−1.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 12 / 24 Our computational approach The algorithm

Finally note that 1−t,t q = dim E2 (f )k where k ∈ [1, d], q − k is divisible by d and t := (q − k)/d. This gives us all the E2-terms of the spectral sequence that occur in our main results above. Moreover, let ι : F → C3 denote the inclusion of the Milnor fiber F into − C3 and ∆ : Ωj → Ωj 1 denote the contraction with the Euler vector field E = x∂x + y∂y + z∂z . If a 2-form η satisfies both

df ∧ η = 0 and dη = 0,

∗ then it gives rise to an element α = ι (∆(η)) in H1(F, C). In this way, the above algorithm can give a basis for the cohomology group H1(F, C).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 13 / 24 Our computational approach The algorithm

df ∧ η = 0 and dη = 0,

∗ then it gives rise to an element α = ι (∆(η)) in H1(F, C). In this way, the above algorithm can give a basis for the cohomology group H1(F, C).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 13 / 24 Example of results via the algorithm Zariski’s sextic with 6 cusps on a conic Zariski’s sextic with 6 cusps on a conic

Consider the sextic curve

C : f = (x2 + y 2)3 + (y 3 + z3)2 = 0,

having 6 cusps on a conic and Alexander polynomial 2 ∆C(t) = t − t + 1. This curve is not free, the module Syz(f ) has a minimal set of generators consisting of one syzygy of degree 5, namely

2 2 2 ω1 = yz dy ∧ dz + xz dx ∧ dz + xy dx ∧ dy,

and 3 other syzygies of degree 7, among which

3 2 5 5 3 2 4 ω2 = (y z + z )dy ∧ dz − (x + 2x y + xy )dx ∧ dy.

It is clear that d(ω1) = d(ω2) = 0.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 14 / 24 Example of results via the algorithm Zariski’s sextic with 6 cusps on a conic

Theorem Let C : f = (x2 + y 2)3 + (y 3 + z3)2 = 0 be the above sextic curve and set λ = exp(πi/3). Then the following hold. ∗ 1 α = ι (∆(ω1)) spans the 1-dimensional vector space 1,0 1 H (F, C)λ = H (F, C)λ. ∗ 2 β = ι (∆(ω2)) spans the 1-dimensional vector space 0,1 1 H (F, C)λ = H (F, C)λ. Hence the pair α, β gives a basis for the cohomology group

1 1 1 H (F, C) = H (F, C)λ ⊕ H (F, C)λ.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 15 / 24 Example of results via the algorithm Zariski’s sextic with 6 cusps on a conic A SINGULAR output for Zariski sextic

2 1−t,t For the he term E2 = E3 = E∞, we set q = dim E2 (f )k and 2 2−t,t µq = dim E2 (f )k , where q = td + k. q : 3 4 56 7 8 9 10 11 12

2 q : 0 0 10100000 2 µq : 1 3 67 10 9 8 6 4 1

2 ∞ The sequence q = q is symmetric with respect to the center q = d, 2 ∞ while the sequence µq = µq does not have this property.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 16 / 24 Example of results via the algorithm A free curve with non-w. h. singularities A free curve with non-w. h. singularities Consider the following conic pencil with one point base locus:

tx2 + s(xz + y 2) = 0

Using this pencil we construct the following curves:

2m 2 m 2m 2 m C2m : f = x +(xz+y ) = 0 and C2m+1 : f = x(x +(xz+y ) ) = 0.

These curves have been essentially introduced by C.T.C. Wall and independently by Arkadiusz Płoski.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 17 / 24 Example of results via the algorithm A free curve with non-w. h. singularities

Theorem (A.D. 2017)

Consider the curves Cd deﬁned above, for d ≥ 3. Then the following holds.

1 The curves Cd are free with exponents d1 = 1 and d2 = d − 2.

2 The complement U satisﬁes b2(U) = 0 and the Euler characteristic χ(U) is given by

d χ(U) = 2 − d + b c. 2

3 When d is odd, then the Milnor ﬁber F is homotopy equivalent to a bouquet of circles ∨S1, and hence the corresponding Alexander polynomial ∆(t) of Cd is given by

∆(t) = (t − 1)(td − 1)−χ(U).

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 18 / 24 Example of results via the algorithm A free curve with non-w. h. singularities A SINGULAR output for d = 9

The term E2

q : 3 4 5 6 7 89 10 11 12 13 14 15

2 q : 1 1 2 2 3 34333333 2 µq : 1 1 2 2 3 33333333

The term E3 = E∞

q : 3 4 5 6 7 89 10 11 12 13 14 15

∞ q : 1 1 2 2 3 34332211 ∞ µq : 0 0 0 0 0 00000000

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 19 / 24 Two conjectures and a higher dimensional example

Conjecture (A.D. and G.S. 2016)

With the above notation, for any reduced degree d plane curve C : f = 0 and any integer k with 1 ≤ k ≤ d, the E1- spectral sequence E∗(f )k degenerates at the third page, i.e.

E3(f )k = E∞(f )k .

Note that there are similar spectral sequences for a reduced degree d hypersurface V : f = 0 in Pn, for n > 2, whose complexity increases according to the dimension of the singular set of V .

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 20 / 24 Two conjectures and a higher dimensional example

Conjecture (A.D. and G.S. 2016)

With the above notation, for any reduced degree d plane curve C : f = 0 and any integer k with 1 ≤ k ≤ d, the E1- spectral sequence E∗(f )k degenerates at the third page, i.e.

E3(f )k = E∞(f )k .

Note that there are similar spectral sequences for a reduced degree d hypersurface V : f = 0 in Pn, for n > 2, whose complexity increases according to the dimension of the singular set of V .

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 20 / 24 Two conjectures and a higher dimensional example

d1 d1

Figure: The spectral sequence E∗(f )k for a surface with 1-dim singular set.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 21 / 24 Two conjectures and a higher dimensional example

Conjecture (A.D. and G.S. 2017)

For any hyperplane arrangement V : f = 0 in Pn and any integer k with 1 ≤ k ≤ d, d being the number of hyperplanes in V , the E1- spectral sequence E∗(f )k degenerates at the second page, i.e.

E2(f )k = E∞(f )k .

s,t Moreover, one has E2 (f )k = 0 for t > 1. s,t One knows that E∞ (f )k = 0 for t > 1 using results by M. Saito on the Bernstein-Sato polynomials of hyperplane arrangements. A similar result holds for free locally quasi-homogeneous hypersurfaces V : f = 0 using results by L. Narvéz Macarro.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 22 / 24 Two conjectures and a higher dimensional example

Deﬁnition

Let V : f = 0 be a hypersurface of degree d in Pn and let k be a positive integer satisfying 1 ≤ k ≤ d. We say that the hypersurface V , or the deﬁning polynomial f , is k-top-computable if

n,0 n−1,1 n dim E2 (f )k + dim E2 (f )k = dim H (F, C)λ, where λ = exp(−2πik/d) 6= 1, and respectively n,0 n dim E2 (f )d = dim H (F, C)1 if k = d.

Conjecture

For any arrangement A : f = 0 of d hyperplanes in Pn, and for any free locally quasi-homogeneous divisor V : f = 0 of degree d in Pn, the deﬁning polynomial f is k-top-computable for any positive integer k satisfying 1 ≤ k ≤ d.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 23 / 24 Two conjectures and a higher dimensional example An Example

Example (The Coxeter arrangement D4)

4 The arrangement D4 is deﬁned in C by the equation of degree 12

A : f = (x2 − y 2)(x2 − z2)(x2 − w 2)(y 2 − z2)(y 2 − w 2)(z2 − w 2) = 0.

We have the following formulas for the Alexander polynomials:

1 11 ∆ (A) = Φ1 Φ3,

2 39 4 9 4 ∆ (A) = Φ1 · Φ2 · Φ3 · Φ6, and 3 45 20 24 16 20 16 ∆ (A) = Φ1 · Φ2 · Φ3 · Φ4 · Φ6 · Φ12, where Φj is the j-th cyclotomic polynomial. Any value k 6= 2, 4, 6, 12 is non-resonant with respect to the arrangement D4.

Dimca, Sticlaru (UCA, Ovidius) Milnor ﬁber cohomology for plane curves 12 June 2017 24 / 24