H YPERPLANE ARRANGEMENTS AT THE CROSSROADS OF TOPOLOGY AND COMBINATORICS Alex Suciu Northeastern University PIMS Distinguished Lecture University of Regina August 14, 2015 A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 1 / 32
O UTLINE H YPERPLANE ARRANGEMENTS 1 Complement and intersection lattice Cohomology ring Fundamental group C OHOMOLOGY JUMP LOCI 2 Characteristic varieties Resonance varieties The Tangent Cone theorem J UMP LOCI OF ARRANGEMENTS 3 Resonance varieties Multinets Characteristic varieties T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT 4 The Milnor fibrations of an arrangement The homology of the Milnor fiber Modular inequalities Torsion in homology A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 2 / 32
H YPERPLANE ARRANGEMENTS C OMPLEMENT AND INTERSECTION LATTICE H YPERPLANE ARRANGEMENTS An arrangement of hyperplanes is a finite collection A of codimension 1 linear subspaces in C ℓ . Intersection lattice L p A q : poset of all intersections of A , ordered by reverse inclusion, and ranked by codimension. Complement : M p A q “ C ℓ z Ť H P A H . L 4 P 1 P 2 P 3 P 4 L 3 P 4 L 2 P 3 L 1 P 1 P 2 L 1 L 2 L 3 L 4 A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 3 / 32
H YPERPLANE ARRANGEMENTS C OMPLEMENT AND INTERSECTION LATTICE E XAMPLE (T HE B OOLEAN ARRANGEMENT ) B n : all coordinate hyperplanes z i “ 0 in C n . L p B n q : Boolean lattice of subsets of t 0 , 1 u n . M p B n q : complex algebraic torus p C ˚ q n . E XAMPLE (T HE BRAID ARRANGEMENT ) A n : all diagonal hyperplanes z i ´ z j “ 0 in C n . L p A n q : lattice of partitions of r n s : “ t 1 , . . . , n u , ordered by refinement. M p A n q : configuration space of n ordered points in C (a classifying space for P n , the pure braid group on n strings). A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 4 / 32
H YPERPLANE ARRANGEMENTS C OMPLEMENT AND INTERSECTION LATTICE ‚ z 2 ´ z 4 z 1 ´ z 2 ‚ z 2 ´ z 3 ‚ ‚ z 1 ´ z 3 z 3 ´ z 4 z 1 ´ z 4 F IGURE : A planar slice of the braid arrangement A 4 A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 5 / 32
H YPERPLANE ARRANGEMENTS C OMPLEMENT AND INTERSECTION LATTICE We may assume that A is essential, i.e., Ş H P A H “ t 0 u . Fix an ordering A “ t H 1 , . . . , H n u , and choose linear forms f i : C ℓ Ñ C with ker p f i q “ H i . Define an injective linear map ι : C ℓ Ñ C n , z ÞÑ p f 1 p z q , . . . , f n p z qq . This map restricts to an inclusion ι : M p A q ã Ñ M p B n q . Hence, M p A q “ ι p C ℓ q X p C ˚ q n is a Stein manifold. Therefore, M “ M p A q has the homotopy type of a connected, finite cell complex of dimension ℓ . In fact, M has a minimal cell structure (Dimca–Papadima, Randell, Salvetti, Adiprasito,. . . ). Consequently, H ˚ p M , Z q is torsion-free. A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 6 / 32
H YPERPLANE ARRANGEMENTS C OHOMOLOGY RING C OHOMOLOGY RING The Betti numbers b q p M q : “ rank H q p M , Z q are given by ÿ ℓ ÿ q “ 0 b q p M q t q “ X P L p A q µ p X qp´ t q rank p X q , where µ : L p A q Ñ Z is the Möbius function, defined recursively by µ p C ℓ q “ 1 and µ p X q “ ´ ř Y Ľ X µ p Y q . Let E “ Ź p A q be the Z -exterior algebra on degree 1 classes e H dual to the meridians around the hyperplanes H P A . Let B : E ‚ Ñ E ‚´ 1 be the differential given by Bp e H q “ 1, and set e B “ ś H P B e H for each B Ă A . The cohomology ring H ˚ p M p A q , Z q is isomorphic to the Orlik–Solomon algebra A p A q “ E { I , where ˇ A E č ˇ I “ ideal B e B ˇ codim H ă | B | . H P B A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 7 / 32
H YPERPLANE ARRANGEMENTS F UNDAMENTAL GROUP F UNDAMENTAL GROUP Given a generic projection of a generic slice of A in C 2 , the fundamental group π “ π 1 p M p A qq can be computed from the resulting braid monodromy α “ p α 1 , . . . , α s q , where α r P P n . π has a (minimal) finite presentation with Meridional generators x 1 , . . . , x n , where n “ | A | . Commutator relators x i α j p x i q ´ 1 , where each α j acts on F n via the Artin representation. Let π { γ k p π q be the p k ´ 1 q th nilpotent quotient of π . Then: π ab “ π { γ 2 equals Z n . π { γ 3 is determined by A ď 2 p A q , and thus by L ď 2 p A q . π { γ 4 (and thus, π ) is not determined by L p A q . (Rybnikov). A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 8 / 32
C OHOMOLOGY JUMP LOCI C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a connected, finite cell complex, and let π “ π 1 p X , x 0 q . Let k be an algebraically closed field, and let Hom p π, k ˚ q be the affine algebraic group of k -valued, multiplicative characters on π . The characteristic varieties of X are the jump loci for homology with coefficients in rank-1 local systems on X : V q s p X , k q “ t ρ P Hom p π, k ˚ q | dim k H q p X , k ρ q ě s u . Here, k ρ is the local system defined by ρ , i.e, k viewed as a k π -module, via g ¨ x “ ρ p g q x , and H i p X , k ρ q “ H i p C ˚ p r X , k q b k π k ρ q . These loci are Zariski closed subsets of the character group. The sets V 1 s p X , k q depend only on π { π 2 . A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 9 / 32
C OHOMOLOGY JUMP LOCI C HARACTERISTIC VARIETIES E XAMPLE (C IRCLE ) We have Ă S 1 “ R . Identify π 1 p S 1 , ˚q “ Z “ x t y and kZ “ k r t ˘ 1 s . Then: C ˚ p Ă t ´ 1 � k r t ˘ 1 s � k r t ˘ 1 s � 0 . S 1 , k q : 0 For ρ P Hom p Z , k ˚ q “ k ˚ , we get ρ ´ 1 � k C ˚ p Ă � k � 0 , S 1 , k q b kZ k ρ : 0 which is exact, except for ρ “ 1, when H 0 p S 1 , k q “ H 1 p S 1 , k q “ k . Hence: V 0 1 p S 1 , k q “ V 1 1 p S 1 , k q “ t 1 u and V i s p S 1 , k q “ H , otherwise. E XAMPLE (P UNCTURED COMPLEX LINE ) Identify π 1 p C zt n points uq “ F n , and x F n “ p k ˚ q n . Then: $ p k ˚ q n & if s ă n , V 1 s p C zt n points u , k q “ t 1 u if s “ n , % H if s ą n . A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 10 / 32
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES R ESONANCE VARIETIES Let A “ H ˚ p X , k q , where char k ‰ 2. Then: a P A 1 ñ a 2 “ 0. We thus get a cochain complex a a � A 1 � A 2 � ¨ ¨ ¨ . p A , ¨ a q : A 0 The resonance varieties of X are the jump loci for the cohomology of this complex s p X , k q “ t a P A 1 | dim k H q p A , ¨ a q ě s u R q 1 p X , k q “ t a P A 1 | D b P A 1 , b ‰ λ a , ab “ 0 u . E.g., R 1 These loci are homogeneous subvarieties of A 1 “ H 1 p X , k q . E XAMPLE R 1 1 p T n , k q “ t 0 u , for all n ą 0. R 1 1 p C zt n points u , k q “ k n , for all n ą 1. A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 11 / 32
C OHOMOLOGY JUMP LOCI T HE T ANGENT C ONE THEOREM T HE T ANGENT C ONE THEOREM Given a subvariety W Ă p C ˚ q n q , let τ 1 p W q “ t z P C n | exp p λ z q P W , @ λ P C u . (Dimca–Papadima–S. 2009) τ 1 p W q is a finite union of rationally defined linear subspaces, and τ 1 p W q Ď TC 1 p W q . (Libgober 2002/DPS 2009) τ 1 p V i s p X qq Ď TC 1 p V i s p X qq Ď R i s p X q . (DPS 2009/DP 2014): Suppose X is a k -formal space. Then, for each i ď k and s ą 0, τ 1 p V i s p X qq “ TC 1 p V i s p X qq “ R i s p X q . Consequently, R i s p X , C q is a union of rationally defined linear subspaces in H 1 p X , C q . A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 12 / 32
J UMP LOCI OF ARRANGEMENTS R ESONANCE VARIETIES J UMP LOCI OF ARRANGEMENTS Work of Arapura, Falk, D.Cohen–A.S., Libgober, and Yuzvinsky, completely describes the varieties R s p A q : “ R 1 s p M p A q , C q : R 1 p A q is a union of linear subspaces in H 1 p M p A q , C q – C | A | . Each subspace has dimension at least 2, and each pair of subspaces meets transversely at 0. R s p A q is the union of those linear subspaces that have dimension at least s ` 1. (Falk–Yuzvinsky 2007) Each k-multinet on a sub-arrangement B Ď A gives rise to a component of R 1 p A q of dimension k ´ 1. Moreover, all components of R 1 p A q arise in this way. A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 13 / 32
J UMP LOCI OF ARRANGEMENTS M ULTINETS M ULTINETS To compute R 1 p A q , we may assume A is an arrangement in C 3 . Its projectivization, ¯ A , is an arrangement of lines in CP 2 . Ñ lines of ¯ Ñ intersection points of ¯ L 1 p A q Ð A , L 2 p A q Ð A . A flat X P L 2 p A q has multiplicity q if the point ¯ X has exactly q lines from ¯ A passing through it. A p k , d q -multinet on A is a partition into k ě 3 subsets, A 1 , . . . , A k , together with an assignment of multiplicities, m : A Ñ N , and a subset X Ď L 2 p A q , such that (basically): D d P N such that ř H P A α m H “ d , for all α P r k s . 1 If H and H 1 are in different classes, then H X H 1 P X . 2 @ X P X , the sum n X “ ř H P A α : H Ą X m H is independent of α . 3 The multinet is reduced if m H “ 1, for all H P A . A net is a reduced multinet with n X “ 1, for all X P X . A LEX S UCIU H YPERPLANE ARRANGEMENTS U NIVERSITY OF R EGINA , 2015 14 / 32
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