Alexandru Suciu Northeastern University Session on the Geometry and - PowerPoint PPT Presentation

T OPOLOGY OF COMPLEX ARRANGEMENTS Alexandru Suciu Northeastern University Session on the Geometry and Topology of Differentiable Manifolds and Algebraic Varieties The Eighth Congress of Romanian Mathematicians Ia¸ si, Romania, June 30, 2015 A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 1 / 23

C OMBINATORIAL COVERS C OMBINATORIAL COVERS C OMBINATORIAL COVERS A combinatorial cover for a space X is a triple p C , φ, ρ q , where C is a countable cover which is either open, or closed and locally 1 finite. φ : N p C q Ñ P is an order-preserving, surjective map from the 2 nerve of the cover to a finite poset P , such that, if S ď T and φ p S q “ φ p T q , then X T ã Ñ X S admits a homotopy inverse. If S ď T and Ş S “ Ş T , then φ p S q “ φ p T q . 3 ρ : P Ñ Z is an order-preserving map whose fibers are antichains. 4 φ induces a homotopy equivalence, φ : | N p C q| Ñ | P | . 5 A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 2 / 23

C OMBINATORIAL COVERS C OMBINATORIAL COVERS Example: X “ D 2 zt 4 points u . U 1 C : U 3 U 2 t U 1 , U 2 , U 3 u ˚ N p C q : P : t U 1 , U 2 u t U 1 , U 3 u t U 2 , U 3 u 1 2 3 t U 1 u t U 2 u t U 3 u φ : N p C q Ñ P : φ pt U i uq “ i and φ p S q “ ˚ if | S | ‰ 1. ρ : P Ñ Z : ρ p˚q “ 1 and ρ p i q “ 0. X S “ X T for any S , T P φ ´ 1 p˚q . Both | N p C q| and | P | are contractible. Thus, C is a combinatorial cover. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 3 / 23

C OMBINATORIAL COVERS A RRANGEMENTS OF SUBMANIFOLDS A RRANGEMENTS OF SUBMANIFOLDS Let A be an arrangement of submanifolds in a smooth, connected manifold. Assume each submanifold is either compact or open. Let L p A q be the (ranked) intersection poset of A . Assume that every element of L p A q is smooth and contractible. T HEOREM (D ENHAM –S.–Y UZVINSKY 2014) The complement M p A q has a combinatorial cover p C , φ, ρ q over L p A q . A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 4 / 23

C OMBINATORIAL COVERS A SPECTRAL SEQUENCE A SPECTRAL SEQUENCE T HEOREM (DSY) Suppose X has a combinatorial cover p C , φ, ρ q over a poset P. For every locally constant sheaf F on X, there is a spectral sequence with ź E pq H p ´ ρ p x q´ 1 p lk | P | p x q ; H q ` ρ p x q p X , F | U x qq r “ 2 x P P converging to H p ` q p X , F q . Here, U x “ X S, where S P N p C q with φ p S q “ x. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 5 / 23

P ROPAGATION OF COHOMOLOGY JUMP LOCI D UALITY SPACES D UALITY SPACES Let X be a path-connected space, having the homotopy type of a finite-type CW-complex. Set π “ π 1 p X q . Recall a notion due to Bieri and Eckmann (1978). X is a duality space of dimension n if H i p X , Z π q “ 0 for i ‰ n and H n p X , Z π q ‰ 0 and torsion-free. Let D “ H n p X , Z π q be the dualizing Z π -module. Given any Z π -module A , we have H i p X , A q – H n ´ i p X , D b A q . If D “ Z , with trivial Z π -action, then X is a Poincaré duality space. If X “ K p π, 1 q is a duality space, then π is a duality group . A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 6 / 23

P ROPAGATION OF COHOMOLOGY JUMP LOCI A BELIAN DUALITY SPACES A BELIAN DUALITY SPACES We introduce an analogous notion, by replacing π � π ab . X is an abelian duality space of dimension n if H i p X , Z π ab q “ 0 for i ‰ n and H n p X , Z π ab q ‰ 0 and torsion-free. Let B “ H n p X , Z π ab q be the dualizing Z π ab -module. Given any Z π ab -module A , we have H i p X , A q – H n ´ i p X , B b A q . The two notions of duality are independent. Fix a field k . T HEOREM (D ENHAM –S.–Y UZVINSKY 2015) Let X be an abelian duality space of dimension n. If ρ : π 1 p X q Ñ k ˚ satisfies H i p X , k ρ q ‰ 0 , then H j p X , k ρ q ‰ 0 , for all i ď j ď n. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 7 / 23

P ROPAGATION OF COHOMOLOGY JUMP LOCI C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Consider the jump loci for cohomology with coefficients in rank-1 local systems on X , V i s p X , k q “ t ρ P Hom p π 1 p X q , k ˚ q | dim k H i p X , k ρ q ě s u , and set V i p X , k q “ V i 1 p X , k q . C OROLLARY (DSY) Let X be an abelian duality space of dimension n. Then: The characteristic varieties propagate: V 1 p X , k q Ď ¨ ¨ ¨ Ď V n p X , k q . dim k H 1 p X , k q ě n ´ 1 . If n ě 2 , then H i p X , k q ‰ 0 , for all 0 ď i ď n. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 8 / 23

P ROPAGATION OF COHOMOLOGY JUMP LOCI R ESONANCE VARIETIES R ESONANCE VARIETIES Assume char p k q ‰ 2, and set A “ H ˚ p X , k 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 the jump loci for the cohomology of these cochain complexes, 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 (P APADIMA –S. 2010) Let X be a minimal CW-complex. Then the linearization of the cellular cochain complex C ˚ p X ab , k q , evaluated at a P A 1 coincides with the cochain complex p A , a q . A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 9 / 23

P ROPAGATION OF COHOMOLOGY JUMP LOCI R ESONANCE VARIETIES T HEOREM (DSY) Let X be an abelian duality space of dimension n which admits a minimal cell structure. Then the resonance varieties of X propagate: R 1 p X , k q Ď ¨ ¨ ¨ Ď R n p X , k q . C OROLLARY (DSY) Let M be a compact, connected, orientable smooth manifold of dimension n. Suppose M admits a perfect Morse function, and R 1 p M , k q ‰ 0 . Then M is not an abelian duality space. E XAMPLE Let M be the 3-dimensional Heisenberg nilmanifold. M admits a perfect Morse function. Characteristic varieties propagate: V i p M , k q “ t 1 u for i ď 3. Resonance does not propagate: R 1 p M , k q “ k 2 but R 3 p M , k q “ 0. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 10 / 23

C OMPLEX ARRANGEMENTS H YPERPLANE ARRANGEMENTS H YPERPLANE ARRANGEMENTS Let A be a central, essential hyperplane arrangement in C n . Its complement, M p A q , is a Stein manifold. It has the homotopy type of a minimal CW-complex of dimension n . M p A q is a formal space. M p A q admits a combinatorial cover. T HEOREM (D AVIS –J ANUSZKIEWICZ –O KUN ) M p A q is a duality space of dimension n. Using the above spectral sequence, we prove: T HEOREM (D ENHAM -S.-Y UZVINSKY 2015) M p A q is an abelian duality space of dimension n. Furthermore, both the characteristic and resonance varieties of M p A q propagate. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 11 / 23

C OMPLEX ARRANGEMENTS E LLIPTIC ARRANGEMENTS E LLIPTIC ARRANGEMENTS An elliptic arrangement is a finite collection, A , of subvarieties in a product of elliptic curves E n , each subvariety being a fiber of a group homomorphism E n Ñ E . If A is essential, the complement M p A q is a Stein manifold. M p A q is minimal. M p A q may be non-formal (examples by Bezrukavnikov and Berceanu–M˘ acinic–Papadima–Popescu). T HEOREM (DSY) The complement of an essential, unimodular elliptic arrangement in E n is both a duality space and an abelian duality space of dimension n. In particular, the pure braid group of n strings on an elliptic curve is both a duality group and an abelian duality group. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 12 / 23

R ESONANCE AND MULTINETS R ESONANCE VARIETIES AND MULTINETS R ESONANCE VARIETIES AND MULTINETS Let R s p A , k q “ R 1 s p M p A q , k q . Work of Arapura, Falk, D.Cohen–A.S., Libgober–Yuzvinsky, and Falk–Yuzvinsky completely describes the varieties R s p A , C q : R 1 p A , C 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 , C q is the union of those linear subspaces that have dimension at least s ` 1. Each k-multinet on a sub-arrangement B Ď A gives rise to a component of R 1 p A , C q of dimension k ´ 1. Moreover, all components of R 1 p A , C q arise in this way. A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 13 / 23

R ESONANCE AND MULTINETS R ESONANCE VARIETIES AND MULTINETS D EFINITION (F ALK AND Y UZVINSKY ) A multinet on A is a partition of the set A 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 , called the base locus, such that: There is an integer d 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 For each X P X , the sum n X “ ř H P A α : H Ą X m H is independent of 3 α . ` Ť ˘ z X is connected. Each set H P A α H 4 A multinet as above is also called a p k , d q -multinet, or a k -multinet. 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 T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 14 / 23

R ESONANCE AND MULTINETS R ESONANCE VARIETIES AND MULTINETS ‚ ‚ ‚ ‚ F IGURE : A p 3 , 2 q -net on the A 3 arrangement: X consists of 4 triple points ( n X “ 1) 2 2 2 F IGURE : A p 3 , 4 q -multinet on the B 3 arrangement: X consists of 4 triple points ( n X “ 1) and 3 triple points ( n X “ 2) A LEX S UCIU T OPOLOGY OF COMPLEX ARRANGEMENTS I A ¸ SI , J UNE 30, 2015 15 / 23