T OPOLOGY AND GEOMETRY OF COHOMOLOGY JUMP LOCI Alex Suciu Northeastern University Conference on Experimental and Theoretical Methods in Algebra, Geometry and Topology Eforie Nord, Romania June 24, 2013 A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 1 / 24
J UMP LOCI S UPPORT LOCI S UPPORT LOCI Let k be an algebraically closed field. Let S be a commutative, finitely generated k -algebra. Let Spec ( S ) = Hom k -alg ( S , k ) be the maximal spectrum of S . d i � E i ´ 1 � E i � ¨ ¨ ¨ � E 0 � 0 be an S -chain complex. Let E : ¨ ¨ ¨ The support varieties of E are the subsets of Spec ( S ) given by ľ d � � W i d ( E ) = supp H i ( E ) . They depend only on the chain-homotopy equivalence class of E . For each i ě 0, Spec ( S ) = W i 0 ( E ) Ě W i 1 ( E ) Ě W i 2 ( E ) Ě ¨ ¨ ¨ . If all E i are finitely generated S -modules, then the sets W i d ( E ) are Zariski closed subsets of Spec ( S ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 2 / 24
J UMP LOCI H OMOLOGY JUMP LOCI H OMOLOGY JUMP LOCI The homology jump loci of the S -chain complex E are defined as V i d ( E ) = t m P Spec ( S ) | dim k H i ( E b S S / m ) ě d u . They depend only on the chain-homotopy equivalence class of E . For each i ě 0, Spec ( S ) = V i 0 ( E ) Ě V i 1 ( E ) Ě V i 2 ( E ) Ě ¨ ¨ ¨ . (Papadima–S. 2013) Suppose E is a chain complex of free , finitely generated S -modules. Then, Each V i d ( E ) is a Zariski closed subset of Spec ( S ) . For each q , ď ď V i W i 1 ( E ) = 1 ( E ) . i ď q i ď q A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 3 / 24
J UMP LOCI R ESONANCE VARIETIES R ESONANCE VARIETIES Let A be a commutative graded k -algebra, with A 0 = k . Let a P A 1 , and assume a 2 = 0 (this condition is redundant if char ( k ) ‰ 2, by graded-commutativity of the multiplication in A ). The Aomoto complex of A (with respect to a P A 1 ) is the cochain complex of k -vector spaces, a a a � ¨ ¨ ¨ , � A 1 � A 2 ( A , a ) : A 0 with differentials given by b ÞÑ a ¨ b , for b P A i . The resonance varieties of A are the sets d ( A ) = t a P A 1 | a 2 = 0 and dim k H i ( A , a ) ě d u . R i If A is locally finite (i.e., dim k A i ă 8 , for all i ě 1), then the sets R i d ( A ) are Zariski closed cones inside the affine space A 1 . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 4 / 24
J UMP LOCI OF A SPACE C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a connected, finite-type CW-complex. Fundamental group π = π 1 ( X , x 0 ) : a finitely generated, discrete group, with π ab – H 1 ( X , Z ) . Fix a field k with k = k , and let S = k [ π ab ] . Identify Spec ( S ) with the character group y π ab = p π = Hom ( π , k ˚ ) . The characteristic varieties of X are the homology jump loci of free S -chain complex E = C ˚ ( X ab , k ) : V i d ( X , k ) = t ρ P p π | dim C H i ( X , k ρ ) ě d u . d ( X , k ) is a subvariety of p Each set V i k . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 5 / 24
J UMP LOCI OF A SPACE C HARACTERISTIC VARIETIES Homotopy invariance: If X » Y , then V i d ( Y , k ) – V i d ( X , k ) . Product formula: 1 ( X 1 ˆ X 2 , k ) = Ť p + q = i V p 1 ( X 1 , k ) ˆ V q V i 1 ( X 2 , k ) . Degree 1 interpretation: The sets V 1 d ( X , k ) depend only on π = π 1 ( X ) —in fact, only on π / π 2 . Write them as V 1 d ( π , k ) . ϕ : p Functoriality: If ϕ : π ։ G is an epimorphism, then ˆ Ñ p G ã π restricts to an embedding V 1 Ñ V 1 d ( G , k ) ã d ( π , k ) , for each d . Universality: Given any subvariety W Ă ( k ˚ ) n , there is a finitely presented group π such that π ab = Z n and V 1 1 ( π , k ) = W . Alexander invariant interpretation: Let X ab Ñ X be the maximal abelian cover. View H ˚ ( X ab , k ) as a module over S = k [ π ab ] . Then: � à ď �� V j X ab , k � 1 ( X ) = supp H j . j ď i j ď i A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 6 / 24
J UMP LOCI OF A SPACE T HE TANGENT CONE THEOREM T HE TANGENT CONE THEOREM The resonance varieties of X (with coefficients in k ) are the loci R i d ( X , k ) associated to the cohomology algebra A = H ˚ ( X , k ) . Each set R i d ( X ) : = R i d ( X , C ) is a homogeneous subvariety of H 1 ( X , C ) – C n , where n = b 1 ( X ) . Recall that V i d ( X ) : = V i d ( X , C ) is a subvariety of H 1 ( X , C ˚ ) – ( C ˚ ) n ˆ Tors ( H 1 ( X , Z )) . (Libgober 2002) TC 1 ( V i d ( X )) Ď R i d ( X ) . Given a subvariety W Ă H 1 ( X , C ˚ ) , let τ 1 ( W ) = t z P H 1 ( X , C ) | exp ( λ z ) P W , @ λ P C u . (Dimca–Papadima–S. 2009) τ 1 ( W ) is a finite union of rationally defined linear subspaces, and τ 1 ( W ) Ď TC 1 ( W ) . Thus, τ 1 ( V i d ( X )) Ď TC 1 ( V i d ( X )) Ď R i d ( X ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 7 / 24
J UMP LOCI OF A SPACE T HE TANGENT CONE THEOREM X is formal if there is a zig-zag of cdga quasi-isomorphisms from ( A PL ( X , Q ) , d ) to ( H ˚ ( X , Q ) , 0 ) . X is k-formal (for some k ě 1) if each of these morphisms induces an iso in degrees up to k , and a monomorphism in degree k + 1. X is 1-formal if and only if π = π 1 ( X ) is 1-formal, i.e., its Malcev Lie algebra, m ( π ) = Prim ( y Q π ) , is quadratic. For instance, compact Kähler manifolds and complements of hyperplane arrangements are formal. (Dimca–Papadima–S. 2009) Let X be a 1-formal space. Then, for each d ą 0, τ 1 ( V 1 d ( X )) = TC 1 ( V 1 d ( X )) = R 1 d ( X ) . Consequently, R 1 d ( X ) is a finite union of rationally defined linear subspaces in H 1 ( X , C ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 8 / 24
J UMP LOCI OF A SPACE T HE TANGENT CONE THEOREM This theorem yields a very efficient formality test. E XAMPLE Let π = x x 1 , x 2 , x 3 , x 4 | [ x 1 , x 2 ] , [ x 1 , x 4 ][ x ´ 2 2 , x 3 ] , [ x ´ 1 1 , x 3 ][ x 2 , x 4 ] y . Then 1 ( π ) = t x P C 4 | x 2 R 1 1 ´ 2 x 2 2 = 0 u splits into linear subspaces over R but not over Q . Thus, π is not 1-formal. E XAMPLE Let F ( Σ g , n ) be the configuration space of n labeled points of a Riemann surface of genus g (a smooth, quasi-projective variety). Then π 1 ( F ( Σ g , n )) = P g , n , the pure braid group on n strings on Σ g . Compute: " ˇ * ř n i = 1 x i = ř n ˇ i = 1 y i = 0 , ( x , y ) P C n ˆ C n ˇ R 1 1 ( P 1 , n ) = ˇ x i y j ´ x j y i = 0 , for 1 ď i ă j ă n For n ě 3, this is an irreducible, non-linear variety (a rational normal scroll). Hence, P 1 , n is not 1-formal. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 9 / 24
J UMP LOCI OF A SPACE T HE TANGENT CONE THEOREM P ROPAGATION OF COHOMOLOGY JUMP LOCI (Denham–S.–Yuzvinsky 2013) Assume X is an abelian duality space of dimension n , i.e., H p ( X , Z π ab ) = 0 for p ‰ n and H n ( X , Z π ab ) ‰ 0 and torsion-free. Given a character character ρ : π Ñ C ˚ , if H p ( X , C ρ ) ‰ 0, then H q ( X , C ρ ) ‰ 0 for all p ď q ď n . Thus, the characteristic varieties of X “propagate": V 1 1 ( X ) Ď V 2 1 ( X ) Ď ¨ ¨ ¨ Ď V n 1 ( X ) . Moreover, if X admits a minimal cell structure, then R 1 1 ( X ) Ď R 2 1 ( X ) Ď ¨ ¨ ¨ Ď R n 1 ( X ) . If A is an arrangement of rank d , then its complement, M ( A ) , is an abelian duality space of dim d . Thus, both the characteristic and the resonance varieties of M ( A ) propagate. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 10 / 24
J UMP LOCI OF A SPACE A PPLICATIONS A PPLICATIONS OF COHOMOLOGY JUMP LOCI Homological and geometric finiteness of regular abelian covers Bieri–Neumann–Strebel–Renz invariants Dwyer–Fried invariants Obstructions to (quasi-) projectivity Right-angled Artin groups and Bestvina–Brady groups 3-manifold groups, Kähler groups, and quasi-projective groups Resonance varieties and representations of Lie algebras Homological finiteness in the Johnson filtration of automorphism groups Homology of finite, regular abelian covers Homology of the Milnor fiber of an arrangement Rational homology of smooth, real toric varieties Lower central series and Chen Lie algebras The Chen ranks conjecture for arrangements A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 11 / 24
� � � � � F INITENESS PROPERTIES IN ABELIAN COVERS F INITENESS PROPERTIES IN ABELIAN COVERS Recall X is a connected, finite-type CW-complex, π = π 1 ( X ) . Let A be an abelian group (quotient of π ab ). Equivalence classes of Galois A -covers of X can be identified with Epi ( π , A ) / Aut ( A ) – Epi ( π ab , A ) / Aut ( A ) . p π ab � π ab X ab X ν π Ð Ñ π p ν ν p ab A X In particular, Galois Z r -covers are parametrized by the Grassmannian Gr r ( H 1 ( X , Q )) , via the correspondence Ñ P ν : = im ( ν ˚ : Q r Ñ H 1 ( X , Q )) X ν Ñ X Ð Goal: Use the cohomology jump loci of X to analyze the geometric and homological finiteness properties of regular A -covers of X . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI ETMAGT, E FORIE N ORD 12 / 24
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