P OINCARÉ DUALITY AND COHOMOLOGY JUMP LOCI Alex Suciu Northeastern University Online Algebraic Geometry Seminar Humboldt University Berlin July 15, 2020 A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 1 / 33
R ESONANCE VARIETIES R ESONANCE VARIETIES R ESONANCE VARIETIES Let A = ( A • , d A ) be a connected, locally finite, graded-commutative graded differential graded algebra (cdga) over a field k , and let M = ( M • , d M ) be an A -cdgm. Since A 0 = k , we have Z 1 ( A ) ∼ = H 1 ( A ) . Set Q ( A ) = { a ∈ Z 1 ( A ) | a 2 = 0 ∈ A 2 } . For each a ∈ Q ( A ) , we then 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 ( m ) , for all m ∈ M i . The resonance varieties of M (in degree i ≥ 0 and depth k ≥ 0): R i k ( M ) = { a ∈ Q ( A ) | dim k H i ( M • , δ a ) ≥ k } . k ( H • ( M )) , but not = in general. TC 0 ( R i k ( M )) ⊆ R i A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 2 / 33
R ESONANCE VARIETIES R ESONANCE VARIETIES OF GRADED ALGEBRAS R ESONANCE VARIETIES OF GRADED ALGEBRAS Now let A be a graded, graded-commutative, connected, locally finite k -algebra ( char k � = 2). For each a ∈ A 1 we have a 2 = − a 2 , and so a 2 = 0. We then obtain a cochain complex of k -vector spaces, δ 0 δ 1 δ 2 � · · · , � A 1 � A 2 ( A , δ a ) : A 0 a a a with differentials δ i a ( u ) = a · u , for all u ∈ A i . The resonance varieties of A are the affine varieties k ( A ) = { a ∈ A 1 | dim k H i ( A , δ a ) ≥ k } . R i An element a ∈ A 1 belongs to R i k ( A ) if and only if there exist u 1 , . . . , u k ∈ A i such that au 1 = · · · = au k = 0 in A i + 1 , and the set { au , u 1 , . . . , u k } is linearly independent in A i , for all u ∈ A i − 1 . A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 3 / 33
R ESONANCE VARIETIES R ESONANCE VARIETIES OF GRADED ALGEBRAS Set b j = b j ( A ) . For each i ≥ 0, we have a descending filtration, A 1 = R i 0 ( A ) ⊇ R i 1 ( A ) ⊇ · · · ⊇ R i b i ( A ) = { 0 } ⊃ R i b i + 1 ( A ) = ∅ . A linear subspace U ⊂ A 1 is isotropic if the restriction of A 1 ∧ A 1 · → A 2 to U ∧ U is the zero map (i.e., ab = 0, ∀ a , b ∈ U ). − If U ⊆ A 1 is an isotropic subspace of dimension k , then U ⊆ R 1 k − 1 ( A ) . R 1 1 ( A ) is the union of all isotropic planes in A 1 . If k ⊂ K is a field extension, then the k -points on R i k ( A ⊗ k K ) coincide with R i k ( A ) . Let ϕ : A → B be a morphism of graded, connected algebras. If the map ϕ 1 : A 1 → B 1 is injective, then ϕ 1 ( R 1 k ( A )) ⊆ R 1 k ( B ) , ∀ k . A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 4 / 33
R ESONANCE VARIETIES T HE BGG CORRESPONDENCE T HE BGG CORRESPONDENCE Fix a k -basis { e 1 , . . . , e n } for A 1 , and let { x 1 , . . . , x n } be the dual basis for A 1 = ( A 1 ) ∗ . Identify Sym ( A 1 ) with S = k [ x 1 , . . . , x n ] , the coordinate ring of the affine space A 1 . The BGG correspondence yields a cochain complex of finitely generated, free S -modules, L ( A ) : = ( A • ⊗ S , δ ) , δ i δ i + 1 � A i ⊗ S A � A i + 1 ⊗ S � A i + 2 ⊗ S � · · · , A · · · A ( u ⊗ s ) = ∑ n where δ i j = 1 e j u ⊗ sx j . The specialization of ( A ⊗ S , δ ) at a ∈ A 1 coincides with ( A , δ a ) , that is, δ i � x j = a j = δ i a . � A A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 5 / 33
R ESONANCE VARIETIES T HE BGG CORRESPONDENCE By definition, an element a ∈ A 1 belongs to R i k ( A ) if and only if rank δ i − 1 + rank δ i a ≤ b i ( A ) − k . a Let I r ( ψ ) denote the ideal of r × r minors of a p × q matrix ψ with entries in S , where I 0 ( ψ ) = S and I r ( ψ ) = 0 if r > min ( p , q ) . Then: � �� δ i − 1 R i ⊕ δ i � k ( A ) = V I b i ( A ) − k + 1 A A � ∪ V � �� I s ( δ i − 1 I t ( δ i � � � = ) A ) V . A s + t = b i ( A ) − k + 1 In particular, R 1 k ( A ) = V ( I n − k ( δ 1 A )) (0 ≤ k < n ) and R 1 n ( A ) = { 0 } . The (degree i , depth k ) resonance scheme R i k ( A ) is defined by δ i − 1 ⊕ δ i ; its underlying set is R i � � the ideal I b i ( A ) − k + 1 k ( A ) . A A A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 6 / 33
P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY ALGEBRAS Let A be a connected, locally finite k -cga. A is a Poincaré duality k -algebra of dimension m if there is a k -linear map ε : A m → k (called an orientation ) such that all the bilinear forms A i ⊗ k A m − i → k , a ⊗ b �→ ε ( ab ) are non-singular. That is, A is a graded, graded-commutative Gorenstein Artin algebra of socle degree m . If A is a PD m algebra, then: b i ( A ) = b m − i ( A ) , and A i = 0 for i > m . ε is an isomorphism. The maps PD : A i → ( A m − i ) ∗ , PD ( a )( b ) = ε ( ab ) are isomorphisms. A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 7 / 33
� P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY ALGEBRAS Each a ∈ A i has a Poincaré dual , a ∨ ∈ A m − i , such that ε ( aa ∨ ) = 1. The orientation class is ω A : = 1 ∨ . We have ε ( ω A ) = 1, and thus aa ∨ = ω A . The class of k -PD algebras is closed under taking tensor products and connected sums: If A is PD m and B is PD n , then A ⊗ k B is PD m + n . If A and B are PD m , then A # B is PD m , where ω �→ ω A � � ( ω ) A ω �→ ω B � � A # B B A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 8 / 33
P OINCARÉ DUALITY ALGEBRAS T HE ASSOCIATED ALTERNATING FORM T HE ASSOCIATED ALTERNATING FORM Associated to a k - PD m algebra there is an alternating m -form, µ A : � m A 1 → k , µ A ( a 1 ∧ · · · ∧ a m ) = ε ( a 1 · · · a m ) . Assume now that m = 3, and set n = b 1 ( A ) . Fix a basis { e 1 , . . . , e n } for A 1 , and let { e ∨ 1 , . . . , e ∨ n } be the dual basis for A 2 . The multiplication in A , then, is given on basis elements by 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 ) . Let A i = ( A i ) ∗ . We may view µ dually as a trivector, µ = ∑ µ ijk e i ∧ e j ∧ e k ∈ � 3 A 1 , which encodes the algebra structure of A . For instance, µ A # B = µ A + µ B . A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 9 / 33
P OINCARÉ DUALITY ALGEBRAS C LASSIFICATION OF ALTERNATING FORMS C LASSIFICATION OF ALTERNATING FORMS (Following J. Schouten, G. Gurevich, D. Djokovi´ c, A. Cohen–A. Helminck, . . . ) Let V be a k -vector space of dimension n . The group GL ( V ) acts on � m ( V ∗ ) by ( g · µ )( a 1 ∧ · · · ∧ a m ) = µ g − 1 a 1 ∧ · · · ∧ g − 1 a m � � . The orbits of this action are the equivalence classes of alternating m -forms on V . (We write µ ∼ µ ′ if µ ′ = g · µ .) Over k , the closures of these orbits are affine algebraic varieties. There are finitely many orbits over k only if n 2 ≥ ( n m ) , that is, m ≤ 2 or m = 3 and n ≤ 8. For k = C , each complex orbit has only finitely many real forms. When m = 3, and n = 8, there are 23 complex orbits, which split into either 1, 2, or 3 real orbits, for a total of 35 real orbits. A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 10 / 33
P OINCARÉ DUALITY ALGEBRAS C LASSIFICATION OF ALTERNATING FORMS Let A and B be two PD m algebras. We say that a morphism of graded algebras ϕ : A → B has non-zero degree if the linear map ϕ m : A m → B m is non-zero. (Equivalently, ϕ is injective.) A and B are isomorphic as PD m algebras if and only if they are isomorphic as graded algebras, in which case µ A ∼ µ B . P ROPOSITION For two PD 3 algebras A and B, the following are equivalent. A ∼ = B, as PD 3 algebras. 1 A ∼ = B, as graded algebras. 2 µ A ∼ µ B . 3 We thus have a bijection between isomorphism classes of 3-dimensional Poincaré duality algebras and equivalence classes of alternating 3-forms, given by A � µ A . A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 11 / 33
P OINCARÉ DUALITY ALGEBRAS C LASSIFICATION OF ALTERNATING FORMS P OINCARÉ DUALITY IN ORIENTABLE MANIFOLDS If M is a compact, connected, orientable, m -dimensional manifold, then the cohomology ring A = H . ( M , k ) is a PD m algebra over k . Sullivan (1975): for every finite-dimensional Q -vector space V and every alternating 3-form µ ∈ � 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." E.g., 0-surgery on the Borromean rings in S 3 yields M = T 3 , with µ M = e 1 e 2 e 3 . If M is the link of an isolated surface singularity (e.g., if M = Σ ( p , q , r ) is a Brieskorn manifold), then µ M = 0. A LEX S UCIU (N ORTHEASTERN ) P OINCARÉ DUALITY AND JUMP LOCI HU B ERLIN , J ULY 15, 2020 12 / 33
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