Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements

Transcription

1 Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements Pauline Bailet Hokkaido University Computational Geometric Topology in Arrangement Theory July 6-10, / 15

2 Reference Joint work with M. Yoshinaga Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements, Geometriae Dedicata, 2014, /s / 15

3 1 2 Orlik-Solomon algebra 2 key results 3 Total degeneration Directional degeneration 3 / 15

4 Let Ā = { L 0, L 1,..., L n } P 2 C be a projective line arrangement. Let α j be the linear form dening L j, i.e. L j := {(x : y : z) P 2 C α j(x, y, z) = 0}, and Q(x, y, z) = n j=0 α j(x, y, z) be the dening polynomial of Ā. 4 / 15

5 The intersection lattice L(Ā) = { j I L j } codies the combinatorics. 5 / 15

6 The intersection lattice L(Ā) = { j I L j } codies the combinatorics. The complement M(Ā) = P 2 C \ n j=0 L j. 5 / 15

7 The intersection lattice L(Ā) = { j I L j } codies the combinatorics. The complement M(Ā) = P 2 C \ n j=0 L j. The Milnor ber F = {(x, y, z) C 3 Q(x, y, z) = 1} C 3. 5 / 15

8 Let h be the monodromy of the Milnor bration where λ = exp(2 1π/n + 1), h : F F (x, y, z) λ (x, y, z), 6 / 15

9 Let h be the monodromy of the Milnor bration where λ = exp(2 1π/n + 1), and h be the monodromy operator h : F F (x, y, z) λ (x, y, z), h : H (F, C) H (F, C). 6 / 15

10 Let h be the monodromy of the Milnor bration where λ = exp(2 1π/n + 1), and h be the monodromy operator h : F F (x, y, z) λ (x, y, z), h : H (F, C) H (F, C). We have the following decomposition H (F, C) = H (F, C) β, β n+1 =1 where H (F, C) β = ker{h βid}. 6 / 15

11 Open questions Are the H (F, C) determined by the arrangement's combinatorics? Are the h determined by the arrangement's combinatorics? 7 / 15

12 Notation: Let k > 1 be an integer. We denote by µ( L j, k) the number of intersection points on L j with multiplicities divisible by k. 8 / 15

13 Motivation Theorem (Libgober, 2002) Let k > 1 and β 1 be a non trivial eigenvalue of order k. If µ( L j, k) = 0 for some L j Ā, then H 1 (F, C) β = 0. 9 / 15

14 Motivation Theorem (Libgober, 2002) Let k > 1 and β 1 be a non trivial eigenvalue of order k. If µ( L j, k) = 0 for some L j Ā, then H 1 (F, C) β = 0. Theorem (Yoshinaga, 2013) Assume that Ā is dened over R. Let k > 1 and β 1 be a non trivial eigenvalue of order k. If µ( L j, k) 1 for some L j Ā, then H 1 (F, C) β = 0. 9 / 15

15 Main vanishing result Theorem (Yoshinaga, B.) Let β 1 be a non trivial eigenvalue of order p s, p prime, s 1. Assume that Ā is essential. If µ( L j, p) 1 for some L j Ā, then H 1 (F, C) β = / 15

16 Orlik Solomon Algebra Orlik-Solomon Algebra 2 key results We consider A = {L 1,..., L n } C 2 the deconing of Ā. Let R be a commutative ring. Let A R (A) H (M(A), R) be the Orlik-Solomon algebra of A. Let ω 1 = e 1 + e e n A 1 R (A), where e j = 1 dα j 2 1π α j. We consider the Aomoto complex: (A R (A), ω 1 ) = { A R (A) ω 1 A +1 R (A) } / 15

17 Key results Orlik-Solomon Algebra 2 key results Theorem (Papadima, Suciu, 2010) Let p Z be a prime, and β 1 be an eigenvalue of order p s, s 1. Then dim H 1 (F, C) β dim H 1 (A F p (A), ω 1 ). 12 / 15

18 Key results Orlik-Solomon Algebra 2 key results Theorem (Papadima, Suciu, 2010) Let p Z be a prime, and β 1 be an eigenvalue of order p s, s 1. Then dim H 1 (F, C) β dim H 1 (A F p (A), ω 1 ). Theorem (Yoshinaga, B.) Let p Z be a prime. Assume that Ā is essential. If µ( L j, p) 1 for some L j Ā, then H 1 (A F p (A), ω 1 ) = / 15

19 Total degeneration Total degeneration Directional degeneration Let A = {L 1,..., L n } C 2 be the deconing of A and its partition in t classes of parallel lines: A = A 1 A 2 A t. Theorem (Yoshinaga, B.) There exists a surjective homomorphism, called total degeneration: tot : A R (A) A R (C t), where C t is a central arrangement of t lines in C / 15

20 Directional degeneration Total degeneration Directional degeneration Let A = {L 1,..., L n } C 2 be the deconing of A and its partition in t classes of parallel lines: A = A 1 A 2 A t. Let us x a class A α. Assume A α = r. Theorem (Yoshinaga, B.) There exists surjective homomorphism, called directional degeneration with respect to A α dir : A R (A) A R (P r ), where P r is composed of r parallel lines and an other line transversal to them. 14 / 15

21 Total degeneration Directional degeneration Thank you for your attention! 15 / 15