A RRANGEMENT COMPLEMENTS AND M ILNOR FIBRATIONS Alex Suciu Northeastern University Special Session Advances in Arrangement Theory Mathematical Congress of the Americas Montréal, Canada July 25, 2017 A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 1 / 18
C OMPLEMENTS OF HYPERPLANE ARRANGEMENTS I NTERSECTION LATTICE AND COMPLEMENT I NTERSECTION LATTICE AND COMPLEMENT An arrangement of hyperplanes is a finite set A of codimension 1 linear subspaces in a finite-dimensional C -vector space V . The intersection lattice , L p A q , is the poset of all intersections of A , ordered by reverse inclusion, and ranked by codimension. The complement , M p A q “ V z Ť H P A H , is a connected, smooth quasi-projective variety, and also a Stein manifold. It has the homotopy type of a minimal CW-complex of dimension p M p A q , Z q is torsion-free. equal to dim V . In particular, H . The fundamental group π “ π 1 p M p A qq admits a finite presentation, with generators x H for each H P A . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 2 / 18
C OMPLEMENTS OF HYPERPLANE ARRANGEMENTS C OHOMOLOGY RING C OHOMOLOGY RING For each H P A , let α H be a linear form s.t. H “ ker p α H q . The 1 logarithmic 1-form ω H “ 2 π i d log α H P Ω dR p M q is a closed form, representing a class e H P H 1 p M , Z q . Let E be the Z -exterior algebra on t e H | H P A u , and let B : E ‚ Ñ E ‚´ 1 be the differential given by Bp e H q “ 1. (Orlik–Solomon 1980). The cohomology ring A “ H . p M p A q , Z q is determined by the intersection lattice: A “ E { I , where ¯ ˇ ! ´ ź ) č I “ ideal B e H ˇ B Ď A and codim H ă | B | . ˇ H P B H P B The map e H ÞÑ ω H extends to a cdga quasi-isomorphism, » p H . p M , R q , d “ 0 q Ý Ñ Ω . dR p M q . Therefore, M p A q is formal. Also, M p A q is minimally pure (i.e., H k p M p A q , Q q is pure of weight 2 k , for all k ), which again implies formality (Dupont 2016). A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 3 / 18
C OMPLEMENTS OF HYPERPLANE ARRANGEMENTS R ESONANCE VARIETIES R ESONANCE VARIETIES For a connected, finite CW-complex X , set A “ H . p X , C q . For each a P A 1 , we have a cochain complex ¨ a ¨ a � A 1 � A 2 � ¨ ¨ ¨ . p A , ¨ a q : A 0 The resonance varieties of X are defined as s p X q “ t a P A 1 | dim H q p A , ¨ a q ě s u . R q They are Zariski closed, homogeneous subsets of affine space A 1 . Now let M “ M p A q . Since M is formal, its resonance varieties are unions of linear subspaces of H 1 p M , C q – C | A | . (Falk–Yuzvinsky 2007) The irreducible components of R 1 1 p M q arise from multinets on sub-arrangements of A : each such k -multinet yields a (linear) component of dimension k ´ 1 ě 2. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 4 / 18
C OMPLEMENTS OF HYPERPLANE ARRANGEMENTS C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a connected, finite cell complex, and set π “ π 1 p X , x 0 q . The characteristic varieties of X are the jump loci for homology with coefficients in rank-1 local systems, V q s p X q “ t ρ P Hom p π, C ˚ q | dim H q p X , C ρ q ě s u . These loci are Zariski closed subsets of the character group. For q “ 1, they depend only on π { π 2 . They determine the characteristic polynomial of the algebraic monodromy of every regular Z n -cover Y Ñ X . Now let M “ M p A q be an arrangement complement. Since M is a smooth, quasi-projective variety, the characteristic varieties of M are unions of torsion-translated algebraic subtori of the character torus, Hom p π, C ˚ q – p C ˚ q | A | . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 5 / 18
T HE M ILNOR FIBRATION M ILNOR FIBER AND MONODROMY M ILNOR FIBER AND MONODROMY h A F F Let A be an arrangement of n hyperplanes in C d ` 1 , d ě 1. H P A α H : C d ` 1 Ñ C (Milnor 1968). The polynomial map f : “ ś restricts to a smooth fibration, f : M p A q Ñ C ˚ . Define the Milnor fiber of A as F p A q : “ f ´ 1 p 1 q . The monodromy diffeo, h : F Ñ F , is given by h p z q “ e 2 π i { n z . F is a Stein manifold. It has the homotopy type of a connected, finite cell complex of dimension d . In general, F is neither formal, nor minimal. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 6 / 18
� � �� � � T HE M ILNOR FIBRATION A REGULAR Z n - COVER A REGULAR Z n - COVER Ñ CP d restricts to a trivial The Hopf fibration C ˚ Ñ C d ` 1 zt 0 u π Ý fibration C ˚ Ñ M p A q π Ý Ñ U p A q : “ P p M p A qq . In turn, this fibration restricts to a regular Z n -cover π : F Ñ U , classified by the homomorphism ϕ : π 1 p U q ։ Z n taking each meridional loop x H to 1. Z � � � � ˆ n i 7 f 7 π 1 p F q � � π 1 p M q � � Z � � π 7 π 7 � � π 1 p U q Z n ϕ � � Z n A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 7 / 18
T HE M ILNOR FIBRATION T HE ALGEBRAIC MONODROMY T HE ALGEBRAIC MONODROMY Let ∆ A p t q be the characteristic polynomial of h ˚ : H 1 p F , C q � . WLOG, we may assume ¯ A “ P p A q is a line arrangement in CP 2 . Let β p p A q “ dim F p H 1 p H . p M p A q , F p q , ¨ σ q , where σ “ ř H P A e H . (An integer depending only on L p A q and on the prime p .) T HEOREM ( P APADIMA –S. 2017) If ¯ A has only points of multiplicity 2 and 3 , then ∆ A p t q “ p t ´ 1 q n ´ 1 p t 2 ` t ` 1 q β 3 p A q . C ONJECTURE If rank p A q ě 3, then ∆ A p t q “ p t ´ 1 q | A |´ 1 pp t ` 1 qp t 2 ` 1 qq β 2 p A q p t 2 ` t ` 1 q β 3 p A q . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 8 / 18
B OUNDARY STRUCTURES T HE BOUNDARY MANIFOLD T HE BOUNDARY MANIFOLD As before, let A be a central arrangement of hyperplanes in V “ C d ` 1 ( d ě 1). Let U p A q “ CP d z int p N q , where N is a (closed) regular H P A P p H q Ă CP d . neighborhood of the hypersurface Ť The boundary manifold of the arrangement, B U “ B N , is a compact, orientable, smooth manifold of dimension 2 d ´ 1. E XAMPLE Let A be a pencil of n hyperplanes in C d ` 1 . If n “ 1, then B U “ S 2 d ´ 1 . If n ą 1, then B U “ 7 n ´ 1 S 1 ˆ S 2 p d ´ 1 q . Let A be a near-pencil of n planes in C 3 . Then B U “ S 1 ˆ Σ n ´ 2 , where Σ g “ 7 g S 1 ˆ S 1 . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 9 / 18
B OUNDARY STRUCTURES T HE BOUNDARY MANIFOLD When d “ 2, the boundary manifold B U is a 3-dimensional graph-manifold M Γ , where Γ is the incidence graph of A , with V p Γ q “ L 1 p A q Y L 2 p A q and E p Γ q “ tp L , P q | P P L u . Vertex manifolds M v “ S 1 ˆ ` S 2 z Ť t v , w uP E p Γ q D 2 ˘ are glued along v , w edge manifolds M e “ S 1 ˆ S 1 via flip maps. T HEOREM ( J IANG –Y AU 1993) U p A q – U p A 1 q ñ M Γ – M Γ 1 ñ Γ – Γ 1 ñ L p A q – L p A 1 q . T HEOREM ( C OHEN –S. 2008) V 1 1 p M Γ q “ Ť v P V p Γ q : deg p v qě 3 t ś i P v t i “ 1 u . Moreover, TFAE: M Γ is formal. TC 1 p V 1 1 p M Γ qq “ R 1 1 p M Γ q . A is a pencil or a near-pencil. A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 10 / 18
B OUNDARY STRUCTURES T HE BOUNDARY OF THE M ILNOR FIBER T HE BOUNDARY OF THE M ILNOR FIBER Let F p A q “ F p A q X D 2 p d ` 1 q be the closed Milnor fiber of A . The boundary of the Milnor fiber of A is the compact, smooth, orientable, p 2 d ´ 1 q -manifold B F “ F X S 2 d ` 1 . The pair p F , B F q is p d ´ 1 q -connected. In particular, if d ě 2, then B F is connected, and π 1 pB F q Ñ π 1 p F q is surjective. If A is the Boolean arrangement in C n , then F “ p C ˚ q n ´ 1 . Hence, F “ T n ´ 1 ˆ D n ´ 1 , and so B F “ T n ´ 1 ˆ S n ´ 2 . If A is a near-pencil of n planes in C 3 , then B F “ S 1 ˆ Σ n ´ 2 . A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 11 / 18
B OUNDARY STRUCTURES T HE BOUNDARY OF THE M ILNOR FIBER The Hopf fibration π : C d ` 1 zt 0 u Ñ CP d restricts to regular, cyclic n -fold covers, π : F Ñ U and π : B F Ñ B U . Assume now that d “ 2. The fundamental group of B U “ M Γ has generators x H for H P A and generators y c for the cycles of Γ . P ROPOSITION ( S. 2014) The Z n -cover π : B F Ñ B U is classified by the homomorphism π 1 pB U q ։ Z n given by x H ÞÑ 1 and y c ÞÑ 0 . T HEOREM ( N ÉMETHI –S ZILARD 2012) The characteristic polynomial of h ˚ : H 1 pB F , C q � is given by p t ´ 1 qp t gcd p m v , n q ´ 1 q m v ´ 2 . ź δ A p t q “ v P L 2 p A q Note: H 1 pB F , Z q may have torsion. E.g., if A is generic, then H 1 pB F , Z q “ Z n p n ´ 1 q{ 2 ‘ Z p n ´ 2 qp n ´ 3 q{ 2 . n A LEX S UCIU (N ORTHEASTERN ) A RRANGEMENT AND M ILNOR FIBRATIONS M ONTRÉAL , J ULY 24, 2017 12 / 18
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