C OHOMOLOGY JUMP LOCI AND DUALITY PROPERTIES Alex Suciu Northeastern University Topology Seminar Institute of Mathematics of the Romanian Academy June 1, 2018 A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 1 / 30
O UTLINE J UMP LOCI 1 Support loci Homology jump loci Resonance varieties of a cdgm P OINCARÉ DUALITY 2 Poincaré duality algebras 3-dimensional Poincaré duality algebras C HARACTERISTIC VARIETIES 3 Characteristic varieties The Tangent Cone theorem C HARACTERISTIC VARIETIES OF 3 - MANIFOLDS 4 Alexander polynomials A Tangent Cone theorem for 3-manifolds A BELIAN DUALITY 5 Duality spaces Abelian duality spaces Arrangements of smooth hypersurfaces A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 2 / 30
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 ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 3 / 30
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 . Get stratifications Spec ( S ) = V i 0 ( E ) Ě V i 1 ( E ) Ě V i 2 ( E ) Ě ¨ ¨ ¨ . T HEOREM (P APADIMA –S. 2014) 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 ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 4 / 30
J UMP LOCI R ESONANCE VARIETIES OF A CDGM R ESONANCE VARIETIES OF A CDGM Let A = ( A ‚ , d A ) be a connected, finite-type k - CDGA ( char k ‰ 2). Let M = ( M ‚ , d M ) be an A - CDGM . For each a P Z 1 ( A ) – H 1 ( A ) , we have a cochain complex, δ 0 δ 1 δ 2 � M 1 � M 2 � ¨ ¨ ¨ , ( M ‚ , δ a ) : M 0 a a a with differentials δ i a ( m ) = a ¨ m + d ( m ) , for all m P M i . The resonance varieties of A are the affine varieties R i s ( M ) = t a P H 1 ( A ) | dim k H i ( M ‚ , δ a ) ě s u . If A is a CGA (that is, d A = 0), the resonance varieties R i s ( A ) are homogeneous subvarieties of A 1 . A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 5 / 30
J UMP LOCI R ESONANCE VARIETIES OF A CDGM Fix a k -basis t e 1 , . . . , e r u for A 1 , and let t x 1 , . . . , x r u be the dual basis for A 1 = ( A 1 ) ˚ . Identify Sym ( A 1 ) with S = k [ x 1 , . . . , x r ] , the coordinate ring of the affine space A 1 . Cochain complex of free S -modules, L ( M ) : = ( M ‚ b S , δ ) : δ i δ i + 1 � M i + 2 b S � M i b S � M i + 1 b S � ¨ ¨ ¨ , ¨ ¨ ¨ where δ i ( m b f ) = ř n j = 1 e j m b fx j + d ( m ) b f . The specialization of ( M b S , δ ) at a P Z 1 ( A ) is ( M , δ a ) . Hence, R i s ( M ) is the zero-set of the ideal generated by all minors of size b i ( M ) ´ s + 1 of the block-matrix δ i + 1 ‘ δ i . In particular, R 1 s ( M ) = V ( I r ´ s ( δ 1 )) , the zero-set of the ideal of codimension s minors of δ 1 . A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 6 / 30
J UMP LOCI R ESONANCE VARIETIES OF A CDGM E XAMPLE (E XTERIOR ALGEBRA ) Let E = Ź V , where V = k n , and S = Sym ( V ) . Then L ( E ) is the Koszul complex on V . E.g., for n = 3: � x 2 ´ x 1 0 � x 1 � � x 3 0 ´ x 1 x 2 ( x 3 ´ x 2 x 1 ) 0 x 3 ´ x 2 x 3 � S 3 � S 3 � S . S This chain complex provides a free resolution ε : L ( E ) Ñ k of the trivial S -module k . Hence, # if s ď ( n t 0 u i ) , R i s ( E ) = H otherwise . A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 7 / 30
J UMP LOCI R ESONANCE VARIETIES OF A CDGM E XAMPLE (N ON - ZERO RESONANCE ) Let A = Ź ( e 1 , e 2 , e 3 ) / x e 1 e 2 y , and set S = k [ x 1 , x 2 , x 3 ] . Then � x 1 � � � x 3 0 ´ x 1 x 2 0 x 3 ´ x 2 x 3 � S 3 � S . L ( A ) : S 2 $ & t x 3 = 0 u if s = 1 , R 1 s ( A ) = t 0 u if s = 2 or 3 , % H if s ą 3 . E XAMPLE (N ON - LINEAR RESONANCE ) Let A = Ź ( e 1 , . . . , e 4 ) / x e 1 e 3 , e 2 e 4 , e 1 e 2 + e 3 e 4 y . Then x 1 � x 4 0 0 ´ x 1 � x 2 0 x 3 ´ x 2 0 x 3 ´ x 2 x 1 x 4 ´ x 3 x 4 � S 4 � S . L ( A ) : S 3 R 1 1 ( A ) = t x 1 x 2 + x 3 x 4 = 0 u A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 8 / 30
J UMP LOCI R ESONANCE VARIETIES OF A CDGM T HEOREM (D ENHAM –S. 2018) Let A be a connected k - CDGA with locally finite cohomology. For every A- CDGM M and for every i , s ě 0 s ( H . ( M )) . TC 0 ( R i s ( M )) Ď R i In general, we cannot replace TC 0 ( R i ( M )) by R i ( M ) . E XAMPLE Let M = A = Ź ( a , b ) with d a = 0, d b = b ¨ a . Then R 1 ( A ) = t 0 , 1 u is not contained in R 1 ( H . ( A )) = t 0 u , though TC 0 ( R 1 ( A )) = t 0 u is. A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 9 / 30
P OINCARÉ DUALITY P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY ALGEBRAS Let A be a graded, graded-commutative algebra over a field k . A = À i ě 0 A i , where A i are k -vector spaces. ¨ : A i b A j Ñ A i + j . ab = ( ´ 1 ) ij ba for all a P A i , b P A j . We will assume that A is connected ( A 0 = k ¨ 1), and locally finite (all the Betti numbers b i ( A ) : = dim k A i are finite). A is a Poincaré duality k -algebra of dimension n if there is a k -linear map ε : A n Ñ k (called an orientation ) such that all the bilinear forms A i b k A n ´ i Ñ k , a b b ÞÑ ε ( ab ) are non-singular. Consequently, b i ( A ) = b n ´ i ( A ) , and A i = 0 for i ą n . ε is an isomorphism. The maps PD : A i Ñ ( A n ´ i ) ˚ , PD ( a )( b ) = ε ( ab ) are isomorphisms. Each a P A i has a Poincaré dual , a _ P A n ´ i , such that ε ( aa _ ) = 1. The orientation class is defined as ω A = 1 _ , so that ε ( ω A ) = 1. A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 10 / 30
P OINCARÉ DUALITY P OINCARÉ DUALITY ALGEBRAS T HE ASSOCIATED ALTERNATING FORM Associated to a k - PD n algebra there is an alternating n -form, µ A : Ź n A 1 Ñ k , µ A ( a 1 ^ ¨ ¨ ¨ ^ a n ) = ε ( a 1 ¨ ¨ ¨ a n ) . Assume now that n = 3, and set r = b 1 ( A ) . Fix a basis t e 1 , . . . , e r u for A 1 , and let t e _ 1 , . . . , e _ r u be the dual basis for A 2 . The multiplication in A , then, is given on basis elements by ÿ r µ ijk e _ e i e _ e i e j = k , j = δ ij ω , k = 1 where µ ijk = µ ( e i ^ e j ^ e k ) . Alternatively, let A i = ( A i ) ˚ , and let e i P A 1 be the (Kronecker) dual of e i . We may then view µ dually as a trivector, ÿ µ ijk e i ^ e j ^ e k P Ź 3 A 1 , µ = which encodes the algebra structure of A . A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 11 / 30
P OINCARÉ DUALITY P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY IN ORIENTABLE MANIFOLDS If M is a compact, connected, orientable, n -dimensional manifold, then the cohomology ring A = H . ( M , k ) is a PD n algebra over k . Sullivan (1975): for every finite-dimensional Q -vector space V and every alternating 3-form µ P Ź 3 V ˚ , there is a closed 3-manifold M with H 1 ( M , Q ) = V and cup-product form µ M = µ . Such a 3-manifold can be constructed via “Borromean surgery." If M bounds an oriented 4-manifold W such that the cup-product pairing on H 2 ( W , M ) is non-degenerate (e.g., if M is the link of an isolated surface singularity), then µ M = 0. A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 12 / 30
� � � P OINCARÉ DUALITY P OINCARÉ DUALITY ALGEBRAS R ESONANCE VARIETIES OF PD- ALGEBRAS Let A be a PD n algebra. For all 0 ď i ď n and all a P A 1 , the square ( δ n ´ i ´ 1 ) ˚ � ( A n ´ i ´ 1 ) ˚ ( A n ´ i ) ˚ a PD – PD – δ i A i a A i + 1 commutes up to a sign of ( ´ 1 ) i . Consequently, � ˚ � H i ( A , δ a ) – H n ´ i ( A , δ ´ a ) . Hence, for all i and s , R i s ( A ) = R n ´ i ( A ) . s In particular, R n 1 ( A ) = t 0 u . A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 13 / 30
P OINCARÉ DUALITY 3 - DIMENSIONAL P OINCARÉ DUALITY ALGEBRAS 3 - DIMENSIONAL P OINCARÉ DUALITY ALGEBRAS Let A be a PD 3 -algebra with b 1 ( A ) = r ą 0. Then R 3 1 ( A ) = R 0 1 ( A ) = t 0 u . R 2 s ( A ) = R 1 s ( A ) for 1 ď s ď r . R i s ( A ) = H , otherwise. Write R s ( A ) = R 1 s ( A ) . Then R 2 k ( A ) = R 2 k + 1 ( A ) if r is even. R 2 k ´ 1 ( A ) = R 2 k ( A ) if r is odd. If µ A has rank r ě 3, then R r ´ 2 ( A ) = R r ´ 1 ( A ) = R r ( A ) = t 0 u . If r ě 4, and k = ¯ k , then dim R 1 ( A ) ě null ( µ A ) ě 2. Here, the rank of a form µ : Ź 3 V Ñ k is the minimum dimension of a linear subspace W Ă V such that µ factors through Ź 3 W . The nullity of µ is the maximum dimension of a subspace U Ă V such that µ ( a ^ b ^ c ) = 0 for all a , b P U and c P V . A LEX S UCIU (N ORTHEASTERN ) J UMP LOCI AND DUALITY IMAR T OPOLOGY S EMINAR 14 / 30
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