T OPOLOGY AND COMBINATORICS OF M ILNOR FIBRATIONS OF HYPERPLANE ARRANGEMENTS Alex Suciu Northeastern University Conference on Hyperplane Arrangements and Characteristic Classes Research Institute for Mathematical Sciences, Kyoto November 13, 2013 A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 1 / 30
R EFERENCES G. Denham, A. Suciu, Multinets, parallel connections, and Milnor fibrations of arrangements , arxiv:1209.3414, to appear in Proc. London Math. Soc. A. Suciu, Hyperplane arrangements and Milnor fibrations , arxiv:1301.4851, to appear in Ann. Fac. Sci. Toulouse Math. S. Papadima, A. Suciu, The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy , arxiv:1401.0868. A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 2 / 30
H YPERPLANE ARRANGEMENTS H YPERPLANE ARRANGEMENTS A (central) arrangement of hyperplanes in C ℓ . A : L ( A ) . Intersection lattice: M ( A ) = C ℓ z Ť Complement: H P A H . The Boolean arrangement B n B n : all coordinate hyperplanes z i = 0 in C n . L ( B n ) : lattice of subsets of t 0 , 1 u n . M ( B n ) : complex algebraic torus ( C ˚ ) n . The braid arrangement A n (or, reflection arr. of type A n ´ 1 ) A n : all diagonal hyperplanes z i ´ z j = 0 in C n . L ( A n ) : lattice of partitions of [ n ] = t 1 , . . . , n u . M ( A n ) : configuration space of n ordered points in C (a classifying space for the pure braid group on n strings). A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 3 / 30
H YPERPLANE ARRANGEMENTS ‚ x 2 ´ x 4 x 1 ´ x 2 ‚ x 2 ´ x 3 ‚ ‚ x 1 ´ x 3 x 3 ´ x 4 x 1 ´ x 4 F IGURE : A planar slice of the braid arrangement A 4 Let A be an arrangement of planes in C 3 . Its projectivization, ¯ A , is an arrangement of lines in CP 2 . Ñ lines of ¯ Ñ intersection points of ¯ L 1 ( A ) Ð A , L 2 ( A ) Ð A . Ñ incidence structure of ¯ Poset structure of L ď 2 ( A ) Ð A . A flat X P L 2 ( A ) has multiplicity q if A X = t H P A | X Ą H u has size q , i.e., there are exactly q lines from ¯ A passing through ¯ X . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 4 / 30
H YPERPLANE ARRANGEMENTS If A is essential, then M = M ( A ) is a (very affine) subvariety of ( C ˚ ) n , where n = | A | . M has the homotopy type of a connected, finite CW-complex of dimension ℓ . In fact, M admits a minimal cell structure. In particular, H ˚ ( M , Z ) is torsion-free. The Betti numbers b q ( M ) : = rank H q ( M , Z ) are given by ÿ ℓ ÿ b q ( M ) t q = µ ( X )( ´ t ) rank ( X ) . q = 0 X P L ( A ) The Orlik–Solomon algebra A = H ˚ ( M , Z ) is the quotient of the exterior algebra on generators dual to the meridians, by an ideal determined by the circuits in the matroid of A . On the other hand, the group π 1 ( M ) is not determined by L ( A ) . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 5 / 30
T HE M ILNOR FIBRATION OF AN ARRANGEMENT T HE M ILNOR FIBRATION OF AN ARRANGEMENT For each H P A , let f H : C ℓ Ñ C be a linear form with kernel H Let Q ( A ) = ś H P A f H , a homogeneous polynomial of degree n . The map Q : C ℓ Ñ C restricts to a map Q : M ( A ) Ñ C ˚ . This is the projection of a smooth, locally trivial bundle, known as the Milnor fibration of the arrangement. The typical fiber, F ( A ) = Q ´ 1 ( 1 ) , is a very affine variety, with the homotopy type of a connected, finite CW-complex of dim ℓ ´ 1. The monodromy of the bundle is the diffeomorphism z ÞÑ e 2 π i/ n z . h : F Ñ F , A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 6 / 30
T HE M ILNOR FIBRATION OF AN ARRANGEMENT E XAMPLE Let A be a pencil of 3 lines through the origin of C 2 . Then F ( A ) is a thrice-punctured torus, and h is an automorphism of order 3: h A F ( A ) F ( A ) More generally, if A is a pencil of n lines in C 2 , then F ( A ) is a Riemann surface of genus ( n ´ 1 2 ) , with n punctures. E XAMPLE Let B n be the Boolean arrangement, with Q = z 1 ¨ ¨ ¨ z n . Then M ( B n ) = ( C ˚ ) n and F ( B n ) = ker ( Q ) – ( C ˚ ) n ´ 1 . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 7 / 30
T HE M ILNOR FIBRATION OF AN ARRANGEMENT Two basic questions about the Milnor fibration of an arrangement: (Q1) Are the Betti numbers b q ( F ( A )) and the characteristic polynomial of the algebraic monodromy, h q : H q ( F ( A ) , C ) Ñ H q ( F ( A ) , C ) , determined by L ( A ) ? (Q2) Are the homology groups H ˚ ( F ( A ) , Z ) torsion-free? If so, does F ( A ) admit a minimal cell structure? Recent progress on both questions: A partial, positive answer to (Q1). A negative answer to (Q2). A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 8 / 30
T HE M ILNOR FIBRATION OF AN ARRANGEMENT Let ∆ A ( t ) : = det ( h 1 ´ t ¨ id ) . Then b 1 ( F ( A )) = deg ∆ A . T HEOREM (P APADIMA –S. 2013) Suppose all flats X P L 2 ( A ) have multiplicity 2 or 3 . Then ∆ A ( t ) , and thus b 1 ( F ( A )) , are combinatorially determined. T HEOREM (D ENHAM –S. 2013) For every prime p ě 2 , there is an arrangement A such that H q ( F ( A ) , Z ) has non-zero p-torsion, for some q ą 1 . In both results, we relate the cohomology jump loci of M ( A ) in characteristic p with those in characteristic 0. In the first result, the bridge between the two goes through the representation variety Hom Lie ( h ( A ) , sl 2 ) . A key combinatorial ingredient in both proofs is the notion of multinet. A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 9 / 30
R ESONANCE VARIETIES R ESONANCE VARIETIES AND THE β p - INVARIANTS Let A = H ˚ ( M ( A ) , k ) — an algebra that depends only on L ( A ) (and the field k ). For each a P A 1 , we have a 2 = 0. Thus, we get a cochain a a � A 1 � A 2 � ¨ ¨ ¨ complex, ( A , ¨ a ) : A 0 The (degree 1) resonance varieties of A are the cohomology jump loci of this “Aomoto complex": R s ( A , k ) = t a P A 1 | dim k H 1 ( A , ¨ a ) ě s u , In particular, a P A 1 belongs to R 1 ( A , k ) iff there is b P A 1 not proportional to a , such that a Y b = 0 in A 2 . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 10 / 30
R ESONANCE VARIETIES Now assume k has characteristic p ą 0. Let σ = ř H P A e H P A 1 be the “diagonal" vector, and define β p ( A ) = dim k H 1 ( A , ¨ σ ) . That is, β p ( A ) = max t s | σ P R 1 s ( A , k ) u . Clearly, β p ( A ) depends only on L ( A ) and p . Moreover, 0 ď β p ( A ) ď | A | ´ 2. T HEOREM (PS) If L 2 ( A ) has no flats of multiplicity 3 r with r ą 1 , then β 3 ( A ) ď 2 . For each m ě 1, there is a matroid M m with all rank 2 flats of multiplicity 3, and such that β 3 ( M m ) = m . M 1 : pencil of 3 lines. M 2 : Ceva arrangement. M m with m ą 2: not realizable over C . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 11 / 30
T HE HOMOLOGY OF THE M ILNOR FIBER T HE HOMOLOGY OF THE M ILNOR FIBER The monodromy h : F ( A ) Ñ F ( A ) has order n = | A | . Thus, ź Φ d ( t ) e d ( A ) , ∆ A ( t ) = d | n where Φ 1 = t ´ 1, Φ 2 = t + 1, Φ 3 = t 2 + t + 1, Φ 4 = t 2 + 1, . . . are the cyclotomic polynomials, and e d ( A ) P Z ě 0 . Easy to see: e 1 ( A ) = n ´ 1. Hence, H 1 ( F ( A ) , C ) , when viewed as a module over C [ Z n ] , decomposes as à ( C [ t ] / ( t ´ 1 )) n ´ 1 ‘ ( C [ t ] / Φ d ( t )) e d ( A ) . 1 ă d | n In particular, b 1 ( F ( A )) = n ´ 1 + ř 1 ă d | n ϕ ( d ) e d ( A ) . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 12 / 30
T HE HOMOLOGY OF THE M ILNOR FIBER Thus, in degree 1, question (Q1) is equivalent to: are the integers e d ( A ) determined by L ď 2 ( A ) ? Not all divisors of n appear in the above formulas: If d does not divide | A X | , for some X P L 2 ( A ) , then e d ( A ) = 0 (Libgober). In particular, if L 2 ( A ) has only flats of multiplicity 2 and 3, then ∆ A ( t ) = ( t ´ 1 ) n ´ 1 ( t 2 + t + 1 ) e 3 . If multiplicity 4 appears, then also get factor of ( t + 1 ) e 2 ¨ ( t 2 + 1 ) e 4 . T HEOREM (C OHEN –O RLIK 2000, P APADIMA –S. 2010) e p s ( A ) ď β p ( A ) , for all s ě 1 . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 13 / 30
T HE HOMOLOGY OF THE M ILNOR FIBER T HEOREM (PS13) Suppose L 2 ( A ) has no flats of multiplicity 3 r, with r ą 1 . Then e 3 ( A ) = β 3 ( A ) , and thus e 3 ( A ) is combinatorially determined. A similar result holds for e 2 ( A ) and e 4 ( A ) , under some additional hypothesis. C OROLLARY A is an arrangement of n lines in P 2 with only double and triple If ¯ points, then ∆ A ( t ) = ( t ´ 1 ) n ´ 1 ( t 2 + t + 1 ) β 3 ( A ) is combinatorially determined. C OROLLARY (L IBGOBER 2012) A is an arrangement of n lines in P 2 with only double and triple If ¯ points, then the question whether ∆ A ( t ) = ( t ´ 1 ) n ´ 1 or not is combinatorially determined. A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 14 / 30
T HE HOMOLOGY OF THE M ILNOR FIBER C ONJECTURE Let A be an essential arrangement in C 3 . Then ∆ A ( t ) = ( t ´ 1 ) | A |´ 1 ( t 2 + t + 1 ) β 3 ( A ) [( t + 1 )( t 2 + 1 )] β 2 ( A ) . A LEX S UCIU (N ORTHEASTERN ) M ILNOR FIBRATIONS OF ARRANGEMENTS RIMS C ONFERENCE 2013 15 / 30
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