O N THE TOPOLOGY OF LINE ARRANGEMENTS Alex Suciu Northeastern University Summer Conference on Hyperplane Arrangements Hokkaido University, Sapporo, Japan August 9, 2016
C OMPLEMENTS OF HYPERPLANE ARRANGMENTS § 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 dim V . In particular, H . p M p A q , Z q is torsion-free. § The fundamental group π “ π 1 p M p A qq admits a finite presentation, with generators x H for each H P A . § Set U p A q “ P p M p A qq . Then M p A q – U p A q ˆ C ˚ .
T HE ABELIANIZATION MAP § We may assume that A is essential, i.e., Ş H P A H “ t 0 u . § For each H P A , let α H be a linear form s.t. H “ ker p α H q . § Fix an ordering A “ t H 1 , . . . , H n u . Since A is essential, the linear map α : V Ñ C n , z ÞÑ p α 1 p z q , . . . , α n p z qq is injective. § Let B n be the ‘Boolean arrangement’ of coordinate hyperplanes in C n , with M p B n q “ p C ˚ q n . § The map α restricts to an inclusion α : M p A q ã Ñ M p B n q . Thus, M p A q “ α p V q X p C ˚ q n . § The induced homomorphism, α 7 : π 1 p M p A qq Ñ π 1 p M p B n qq , coincides with the abelianization map, ab : π ։ π ab “ Z n .
C OHOMOLOGY RING 1 § The 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. § The ring H . p M p A q , Z q is isomorphic to the OS-algebra E { I , where ! ´ ź ¯ ˇ ) č I “ ideal B e H ˇ B Ď A and codim H ă | B | . ˇ H P B H P B § Hence, the map e H ÞÑ ω H extends to a cdga quasi-isomor- � Ω . phism, ω : p H . p M , R q , d “ 0 q dR p M q . § Therefore, M p A q is formal. § 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 STRATIFICATION OF THE REPRESENTATION VARIETY § Let X be a connected, finite-type CW-complex, π “ π 1 p X q . § Let G be a complex, linear algebraic group. § The representation variety Hom p π, G q is an affine variety. § Given a representation τ : π Ñ GL p V q , let V τ be the left C r π s -module V defined by g ¨ v “ τ p g q v . § The characteristic varieties of X with respect to a rational representation ι : G Ñ GL p V q are the algebraic subsets V i s p X , ι q “ t ρ P Hom p π, G q | dim H i p X , V ι ˝ ρ q ě s u . § When G “ C ˚ and ι : C ˚ » Ý Ñ GL 1 p C q , we get the rank 1 characteristic varieties, V i s p X q , sitting inside the character group Char p X q : “ Hom p π, C ˚ q .
J UMP LOCI OF SMOOTH , QUASI - PROJECTIVE VARIETIES T HEOREM ( . . . , A RAPURA , . . . , B UDUR –W ANG ) If M is a quasi-projective manifold, the varieties V i s p M q are finite unions of torsion-translates of subtori of Char p M q . § A holomorphic map f : M Ñ Σ is admissible if it surjective, its fibers are connected, and Σ is a smooth complex curve. § The map f 7 : π 1 p M q Ñ π 1 p Σ q is also surjective. Thus, the morphism f ! : “ f ˚ 7 : Char p Σ q Ñ Char p M q is injective. § Up to reparametrization at the target, there is a finite set E p M q of admissible maps with the property that χ p Σ q ă 0. T HEOREM ( A RAPURA 1997) The correspondence f � f ! Char p Σ q defines a bijection between E p M q and the set of positive-dimensional, irreducible components of V 1 1 p M q passing through 1 .
T HEOREM ( K APOVICH –M ILLSON UNIVERSALITY ) PSL 2 -representation varieties of Artin groups may have arbitrarily bad singularities away from the origin. T HEOREM ( K APOVICH –M ILLSON 1998) Let M be a quasi-projective manifold, and G be a reductive algebraic group. If ρ : π 1 p M q Ñ G is a representation with finite image, then the germ Hom p π 1 p M q , G q p ρ q is analytically isomorphic to a quasi-homogeneous cone with generators of weight 1 and 2 and relations of weight 2 , 3 , and 4 . T HEOREM ( C ORLETTE -S IMPSON 08, L ORAY -P EREIRA -T OUZET 16) If ρ : π 1 p M q Ñ SL 2 p C q is not virtually abelian, then there is an orbifold morphism f : M Ñ N such that ˜ ρ : π 1 p M q Ñ PSL 2 p C q belongs to f ! Hom p π 1 p N q , PSL 2 p C qq , where N is either a 1 -dim complex orbifold, or a polydisk Shimura modular orbifold.
SL 2 - REPRESENTATION VARIETIES OF ARRANGEMENTS § For an arrangement A , all base curves Σ have genus 0, by purity of the MHS on H . p M p A q , Q q . § Set E p A q “ E p M p A qq Y t α u . Note that all maps f P E p A q are of the form f : M p A q Ñ M p A f q , for some arrangement A f . § Write π “ π 1 p M p A qq and π f “ π 1 p M p A f qq T HEOREM ( P APADIMA –S. 2016) Let G “ SL 2 p C q and let ι : G Ñ GL p V q be a rational representation. Then, f ! Hom p π f , G q p 1 q ď Hom p π, G q p 1 q “ f P E p A q V 1 ď f ! V 1 1 p π, ι q p 1 q “ 1 p π f , ι q p 1 q f P E p A q
T HE T ANGENT C ONE THEOREM § Let X be a connected, finite-type CW-complex, let k be a field (char p k q ‰ 2), and set A “ H . p X , k q . § For each a P A 1 , we get a cochain complex a a � A 1 � A 2 � ¨ ¨ ¨ p A , ¨ a q : A 0 § The resonance varieties of X are the homogeneous algebraic sets R i s p X , k q “ t a P H 1 p X , k q | dim k H i p A , a q ě s u . T HEOREM ( D IMCA –P APADIMA –S. 2010, D IMCA –P APADIMA 2014) Let X be a formal space. Then: § The homomorphism exp : H 1 p X , C q Ñ H 1 p X , C ˚ q induces » isos of analytic germs, R i Ñ V i s p X , C q p 0 q Ý s p X q p 1 q . § All irreducible components of R i s p X , C q are rationally defined linear subspaces.
A BELIAN DUALITY AND PROPAGATION OF JUMP LOCI § X is an abelian duality space of dim n if H i p X , Z π ab q “ 0 for i ‰ n and B : “ H n p X , Z π ab q is non-zero and torsion-free. § H i p X , A q – H n ´ i p X , B b A q , for any Z π ab -module A . T HEOREM ( D ENHAM –S.–Y UZVINSKY 2015/16) Let X be an abelian duality space of dimension n. Then: § V 1 1 p X q Ď ¨ ¨ ¨ Ď V n 1 p X q . § b 1 p X q ě n ´ 1 . § If n ě 2 , then b i p X q ‰ 0 , for all 0 ď i ď n. § A cyclic, graded E -module A “ E { I has the EPY property if A ˚ p n q is a Koszul module for some integer n . § If A “ H . p X , k q has this property, we say that X has the EPY property over k .
P ROPAGATION OF RESONANCE T HEOREM ( DSY) Suppose X is a finite, connected CW-complex of dimension n with the EPY property over a field k . Then the resonance varieties of X propagate: R 1 p X , k q Ď ¨ ¨ ¨ Ď R n p X , k q . T HEOREM ( DSY) Let A be an essential arrangement in C n . Then M p A q is an abelian duality space of dimension n (and also is formal and has the EPY property). Consequently, the characteristic and resonance varieties of M p A q propagate. § All irreducible components of R i s p M p A q , C q are linear. § In general, R 1 1 p M p A q , k q may have non-linear components.
M ULTINETS AND DEGREE 1 RESONANCE 2 2 2 F IGURE : p 3 , 2 q -net; p 3 , 4 q -multinet; non-3-net, reduced p 3 , 4 q -multinet T HEOREM ( F ALK , C OHEN –S., L IBGOBER –Y UZVINSKY , Falk–Yuz ) R 1 ď ď s p M p A q , C q “ P N . B Ď A N a k-multinet on B with at least s ` 2 parts where P N is the p k ´ 1 q -dimensional linear subspace spanned by the vectors u 2 ´ u 1 , . . . , u k ´ u 1 , where u α “ ř H P B α m H e H .
M ILNOR FIBRATION h A F F § Let A be an arrangement of n hyperplanes in C d ` 1 . For each H P A let α H be a linear form with ker p α H q “ H , and let Q “ ś H P A α H . § Q : C d ` 1 Ñ C restricts to a smooth fibration, Q : M p A q Ñ C ˚ . The Milnor fiber of the arrangement is F p A q : “ Q ´ 1 p 1 q . § F is a Stein manifold. It has the homotopy type of a finite cell complex of dim d . In general, F is neither formal, nor minimal. § F “ F p A q is the regular, Z n -cover of U “ U p A q , classified by the morphism π 1 p U q ։ Z n taking each loop x H to 1.
M ODULAR INEQUALITIES § The monodromy diffeo, h : F Ñ F , is given by h p z q “ e 2 π i { n z . § Let ∆ p t q be the characteristic polynomial of h ˚ : H 1 p F , C q � . Since h n “ id, we have ź Φ r p t q e r p A q , ∆ p t q “ r | n where Φ r p t q is the r -th cyclotomic polynomial, and e r p A q P Z ě 0 . § WLOG, we may assume d “ 2, so that ¯ A “ P p A q is an arrangement of lines in CP 2 . § If there is no point of ¯ A of multiplicity q ě 3 such that r | q , then e r p A q “ 0 (Libgober 2002). § In particular, if ¯ A has only points of multiplicity 2 and 3, then ∆ p t q “ p t ´ 1 q n ´ 1 p t 2 ` t ` 1 q e 3 . If multiplicity 4 appears, then we also get factor of p t ` 1 q e 2 ¨ p t 2 ` 1 q e 4 .
§ Let A “ H . p M p A q , k q , and let σ “ ř H P A e H P A 1 . § Assume k has characteristic p ą 0, and define β p p A q “ dim k H 1 p A , ¨ σ q . That is, β p p A q “ max t s | σ P R 1 s p A , k qu . T HEOREM ( C OHEN –O RLIK 2000, P APADIMA –S. 2010) e p m p A q ď β p p A q , for all m ě 1 . T HEOREM ( P APADIMA –S. 2014) § Suppose A admits a k-net. Then β p p A q “ 0 if p ∤ k and β p p A q ě k ´ 2 , otherwise. § If A admits a reduced k-multinet, then e k p A q ě k ´ 2 .
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