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 (!) = (h−1)∗(!), for j = 0; 1; 2, and we can look at the corresponding characteristic polynomials j j j ∆C(t) = det(t · Id − h jH (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 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 (!) = (h−1)∗(!), for j = 0; 1; 2, and we can look at the corresponding characteristic polynomials j j j ∆C(t) = det(t · Id − h jH (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 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 (!) = (h−1)∗(!), for j = 0; 1; 2, and we can look at the corresponding characteristic polynomials j j j ∆C(t) = det(t · Id − h jH (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 n 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. 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 n 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: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 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 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 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). 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). 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).