Free abelian covers and arrangements of Schubert varieties Alex Suciu Northeastern University Centro Ennio De Giorgi Pisa, Italy May 25, 2010 Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 1 / 41
Outline Characteristic varieties and Dwyer–Fried invariants 1 Free abelian covers The Dwyer–Fried sets Characteristic varieties Computing the Ω -invariants Characteristic arrangements and Schubert varieties 2 Tangent cones Characteristic subspace arrangements Special Schubert varieties 3 Resonance varieties and straight spaces The Aomoto complex Resonance varieties Straight spaces Ω -invariants of straight spaces Examples 4 Toric complexes Hyperplane arrangements Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 2 / 41
� � � Characteristic varieties and Dwyer–Fried invariants Free abelian covers Free abelian covers Let X be a connected CW-complex, with finite k -skeleton, for some k ≥ 1. We may assume X has a single 0-cell, call it x 0 . Let G = π 1 ( X , x 0 ) . Consider the connected, regular covering spaces of X , with group of deck transformations a free abelian group of fixed rank r . Model situation: the r -dimensional torus T r and its universal cover, Z r → R r → T r . Any epimorphism ν : G ։ Z r gives rise to a Z r -cover, by pull back: X ν R r f � T r , X where f ♯ : π 1 ( X , x 0 ) → π 1 ( T r ) realizes ν . (Note: X ν is the homotopy fiber of f ). All connected, regular Z r -covers of X arise in this manner. Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 3 / 41
Characteristic varieties and Dwyer–Fried invariants Free abelian covers The map ν factors as ab ν ∗ → Z r , G − − → G ab − − where ν ∗ may be identified with the induced homomorphism f ∗ : H 1 ( X , Z ) → H 1 ( T r , Z ) . Passing to the homomorphism in Q -homology, we see that the cover X ν → X is determined by the kernel of ν ∗ : H 1 ( X , Q ) → Q r . Conversely, every codimension- r linear subspace of H 1 ( X , Q ) can be realized as ker ( ν ∗ : H 1 ( X , Q ) → Q r ) . for some ν : G ։ Z r , and thus gives rise to a cover X ν → X . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 4 / 41
Characteristic varieties and Dwyer–Fried invariants Free abelian covers Let Gr r ( H 1 ( X , Q )) be the Grassmanian of r -planes in the finite-dimensional, rational vector space H 1 ( X , Q ) . Using the dual map ν ∗ : Q r → H 1 ( X , Q ) instead, we obtain: Proposition (Dwyer–Fried 1987) The connected, regular covers of X whose group of deck transformations is free abelian of rank r are parametrized by the rational Grassmannian Gr r ( H 1 ( X , Q )) , via the correspondence � � � � Z r -covers X ν → X r-planes P ν := im ( ν ∗ ) in H 1 ( X , Q ) ← → . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 5 / 41
Characteristic varieties and Dwyer–Fried invariants The Dwyer–Fried sets The Dwyer–Fried sets Moving about the rational Grassmannian, and recording how the Betti numbers of the corresponding covers vary leads to: Definition The Dwyer–Fried invariants of X are the subsets � � � � b j ( X ν ) < ∞ for j ≤ i Ω i P ν ∈ Gr r ( H 1 ( X , Q )) r ( X ) = , defined for all i ≥ 0 and all r > 0, with the convention that Ω i r ( X ) = ∅ if r > b 1 ( X ) . For a fixed r > 0, get a descending filtration of the Grassmanian of r -planes in Q n , where n = b 1 ( X ) : Gr r ( Q n ) = Ω 0 r ( X ) ⊇ Ω 1 r ( X ) ⊇ Ω 2 r ( X ) ⊇ · · · . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 6 / 41
Characteristic varieties and Dwyer–Fried invariants The Dwyer–Fried sets The Ω -sets are homotopy-type invariants of X : Lemma Suppose X ≃ Y. For each r > 0 , there is an isomorphism Gr r ( H 1 ( Y , Q )) ∼ = Gr r ( H 1 ( X , Q )) sending each subset Ω i r ( Y ) bijectively onto Ω i r ( X ) . In view of this lemma, we may extend the definition of the Ω -sets from spaces to groups. Let G be a finitely-generated group. Pick a classifying space K ( G , 1 ) with finite k -skeleton, for some k ≥ 1. Definition The Dwyer–Fried invariants of G are the subsets Ω i r ( G ) = Ω i r ( K ( G , 1 )) of Gr r ( H 1 ( G , Q )) , defined for all i ≥ 0 and r ≥ 1. Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 7 / 41
Characteristic varieties and Dwyer–Fried invariants The Dwyer–Fried sets Especially manageable situation: r = n , where n = b 1 ( X ) > 0. In this case, Gr n ( H 1 ( X , Q )) = { pt } . This single point corresponds to the maximal free abelian cover, X α → X , where α : G ։ G ab / Tors ( G ab ) = Z n . The sets Ω i n ( X ) are then given by � if b j ( X α ) < ∞ for j ≤ i , { pt } Ω i n ( X ) = ∅ otherwise . Example Let X = S 1 ∨ S k , for some k > 1. Then X α ≃ � j ∈ Z S k j . Thus, � { pt } for i < k , Ω i n ( X ) = ∅ for i ≥ k . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 8 / 41
Characteristic varieties and Dwyer–Fried invariants The Dwyer–Fried sets Remark Finiteness of the Betti numbers of a free abelian cover X ν does not imply finite-generation of the integral homology groups of X ν . E.g., let K be a knot in S 3 , with complement X = S 3 \ K , infinite cyclic cover X ab , and Alexander polynomial ∆ K ∈ Z [ t ± 1 ] . Then H 1 ( X ab , Z ) = Z [ t ± 1 ] / (∆ K ) . Hence, H 1 ( X ab , Q ) = Q d , where d = deg ∆ K . Thus, Ω 1 1 ( X ) = { pt } . But, if ∆ K is not monic, H 1 ( X ab , Z ) need not be finitely generated. Example (Milnor 1968) Let K be the 5 2 knot, with Alex polynomial ∆ K = 2 t 2 − 3 t + 2. Then H 1 ( X ab , Z ) = Z [ 1 / 2 ] ⊕ Z [ 1 / 2 ] is not f.g., though H 1 ( X ab , Q ) = Q ⊕ Q . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 9 / 41
Characteristic varieties and Dwyer–Fried invariants Characteristic varieties Characteristic varieties Consider the group of complex-valued characters of G , � G = Hom ( G , C × ) = H 1 ( X , C × ) Let G ab = G / G ′ ∼ = H 1 ( X , Z ) be the abelianization of G . The projection ab : G → G ab induces an isomorphism � ≃ → � − G ab G . The identity component, � G 0 , is isomorphic to a complex algebraic torus of dimension n = rank G ab . The other connected components are all isomorphic to G 0 = ( C × ) n , and are indexed by the finite abelian group Tors ( G ab ) . � � G parametrizes rank 1 local systems on X : ρ : G → C × L ρ � the complex vector space C , viewed as a right module over the group ring Z G via a · g = ρ ( g ) a , for g ∈ G and a ∈ C . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 10 / 41
Characteristic varieties and Dwyer–Fried invariants Characteristic varieties The homology groups of X with coefficients in L ρ are defined as H ∗ ( X , L ρ ) = H ∗ ( L ρ ⊗ Z G C • ( � X , Z )) , where C • ( � X , Z ) is the equivariant chain complex of the universal cover of X . Definition The characteristic varieties of X are the sets V i ( X ) = { ρ ∈ � G | H j ( X , L ρ ) � = 0 , for some j ≤ i } , defined for all degrees 0 ≤ i ≤ k . Get filtration { 1 } = V 0 ( X ) ⊆ V 1 ( X ) ⊆ · · · ⊆ V k ( X ) ⊆ � G . Each V i ( X ) is a Zariski closed subset of the algebraic group � G . The characteristic varieties are homotopy-type invariants: Suppose X ≃ X ′ . There is then an isomorphism � G ′ ∼ = � G , which restricts to isomorphisms V i ( X ′ ) ∼ = V i ( X ) , for all i ≤ k . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 11 / 41
Characteristic varieties and Dwyer–Fried invariants Characteristic varieties The characteristic varieties may be reinterpreted as the support varieties for the Alexander invariants of X . Let X ab → X be the maximal abelian cover. View H ∗ ( X ab , C ) as a module over C [ G ab ] . Then (Papadima–S. 2010), � � � �� � � V i ( X ) = V X ab , C . ann H j j ≤ i Set W i ( X ) = V i ( X ) ∩ � G 0 . View H ∗ ( X α , C ) as a module over C [ G α ] ∼ = Z [ t ± 1 1 , . . . , t ± 1 n ] , where n = b 1 ( G ) . Then � � � �� � � W i ( X ) = V X α , C ann H j . j ≤ i Example Let L = ( L 1 , . . . , L n ) be a link in S 3 , with complement X = S 3 \ � n i = 1 L i and Alexander polynomial ∆ L = ∆ L ( t 1 , . . . , t n ) . Then V 1 ( X ) = { z ∈ ( C × ) n | ∆ L ( z ) = 0 } ∪ { 1 } . Alex Suciu (Northeastern University) Free abelian covers and Schubert varieties Pisa, May 2010 12 / 41
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