Intermediate Jacobians and Abel-Jacobi Maps Patrick Walls April 28, 2012 Intermediate Jacobians are complex tori defined in terms of the Hodge structure of X . Abel-Jacobi Maps are maps from the groups of cycles of X to its Intermediate Jacobians. The result is that questions about cycles can be translated into questions about complex tori . Introduction Let X be a smooth projective complex variety. The result is that questions about cycles can be translated into questions about complex tori . Introduction Let X be a smooth projective complex variety. Intermediate Jacobians are complex tori defined in terms of the Hodge structure of X . Abel-Jacobi Maps are maps from the groups of cycles of X to its Intermediate Jacobians. Introduction Let X be a smooth projective complex variety. Intermediate Jacobians are complex tori defined in terms of the Hodge structure of X . Abel-Jacobi Maps are maps from the groups of cycles of X to its Intermediate Jacobians. The result is that questions about cycles can be translated into questions about complex tori . The subspaces Hp;q(X ) consist of classes [α] of differential forms that are representable by a closed form α of type (p; q) meaning that locally X α = fI ;J dzI ^ dzJ I ; J ⊆ f1;:::; ng jI j = p ; jJj = q for some choice of local holomorphic coordinates z1;:::; zn. Hodge Decomposition Let X be a smooth projective complex variety of dimension n. The Hodge decomposition is a direct sum decomposition of the complex cohomology groups k M p;q H (X ; C) = H (X ) ; 0 ≤ k ≤ 2n : p+q=k Hodge Decomposition Let X be a smooth projective complex variety of dimension n. The Hodge decomposition is a direct sum decomposition of the complex cohomology groups k M p;q H (X ; C) = H (X ) ; 0 ≤ k ≤ 2n : p+q=k The subspaces Hp;q(X ) consist of classes [α] of differential forms that are representable by a closed form α of type (p; q) meaning that locally X α = fI ;J dzI ^ dzJ I ; J ⊆ f1;:::; ng jI j = p ; jJj = q for some choice of local holomorphic coordinates z1;:::; zn. Since real forms are conjugate invariant, we may write [α] 2 H2k(X ; R) in terms of the Hodge decomposition α = α2k;0 + ··· + αk+1;k−1 + αk;k + αk+1;k−1 + ··· + α2k;0 and for [β] 2 H2k−1(X ; R) in odd degree β = β2k+1;0 + β2k;1 + ··· + βk+1;k + βk+1;k + ··· + β2k;1 + β2k+1;0 : Hodge Decomposition (Continued) The subspaces Hp;q(X ) satisfy the Hodge symmetry Hp;q(X ) = Hq;p(X ) where α 7! α is the natural action of complex conjugation on k k H (X ; C) = H (X ; R) ⊗R C : Hodge Decomposition (Continued) The subspaces Hp;q(X ) satisfy the Hodge symmetry Hp;q(X ) = Hq;p(X ) where α 7! α is the natural action of complex conjugation on k k H (X ; C) = H (X ; R) ⊗R C : Since real forms are conjugate invariant, we may write [α] 2 H2k(X ; R) in terms of the Hodge decomposition α = α2k;0 + ··· + αk+1;k−1 + αk;k + αk+1;k−1 + ··· + α2k;0 and for [β] 2 H2k−1(X ; R) in odd degree β = β2k+1;0 + β2k;1 + ··· + βk+1;k + βk+1;k + ··· + β2k;1 + β2k+1;0 : The cohomology groups of odd degree satisfy 2k−1 k 2k−1 H (X ; C) = F H (X ) ⊕ F kH2k−1(X ) and the natural map 2k−1 2k−1 k 2k−1 H (X ; R) −! H (X ; C)=F H (X ) is an isomorphism of real vector spaces. Hodge Decomposition (Continued) The Hodge decomposition defines a filtration k k k−1 k 0 k k 0 ⊆ F H (X ) ⊆ F H (X ) ⊆ · · · ⊆ F H (X ) = H (X ; C) where M F r Hk(X ) = Hp;k−p(X ) ; 0 ≤ r ≤ k : p≥r Hodge Decomposition (Continued) The Hodge decomposition defines a filtration k k k−1 k 0 k k 0 ⊆ F H (X ) ⊆ F H (X ) ⊆ · · · ⊆ F H (X ) = H (X ; C) where M F r Hk(X ) = Hp;k−p(X ) ; 0 ≤ r ≤ k : p≥r The cohomology groups of odd degree satisfy 2k−1 k 2k−1 H (X ; C) = F H (X ) ⊕ F kH2k−1(X ) and the natural map 2k−1 2k−1 k 2k−1 H (X ; R) −! H (X ; C)=F H (X ) is an isomorphism of real vector spaces. Definition The kth intermediate Jacobian of X is the complex torus k 2k−1 k 2k−1 2k−1 J (X ) = Vk=Lk = H (X ; C)=(F H (X ) ⊕ H (X ; Z)) : Intermediate Jacobians The rank of H2k−1(X ; Z) is the dimension of H2k−1(X ; R) therefore H2k−1(X ; Z) defines a full lattice 2k−1 2k−1 k 2k−1 Lk = im H (X ; Z) ! H (X ; C)=F H (X ) in the complex vector space 2k−1 k 2k−1 Vk = H (X ; C)=F H (X ) : Intermediate Jacobians The rank of H2k−1(X ; Z) is the dimension of H2k−1(X ; R) therefore H2k−1(X ; Z) defines a full lattice 2k−1 2k−1 k 2k−1 Lk = im H (X ; Z) ! H (X ; C)=F H (X ) in the complex vector space 2k−1 k 2k−1 Vk = H (X ; C)=F H (X ) : Definition The kth intermediate Jacobian of X is the complex torus k 2k−1 k 2k−1 2k−1 J (X ) = Vk=Lk = H (X ; C)=(F H (X ) ⊕ H (X ; Z)) : We see that F n−k+1H2n−2k+1(X )? = F kH2k−1(X ) since a form of type (p; 2n − 2k + 1 − p) with p ≥ n − k + 1 wedged with a form of type (p0; 2k − 1 − p0) with p0 ≥ k is a form of type (p00; 2n − p00) with p00 ≥ n + 1 which must be zero since X has complex dimension n. Intermediate Jacobians (Continued) Poincar´eduality asserts 2n−2k+1 _ 2k−1 H (X ; C) = H (X ; C) and so n−k+1 2n−2k+1 _ 2k−1 n−k+1 2n−2k+1 ? F H (X ) = H (X ; C)=F H (X ) : Intermediate Jacobians (Continued) Poincar´eduality asserts 2n−2k+1 _ 2k−1 H (X ; C) = H (X ; C) and so n−k+1 2n−2k+1 _ 2k−1 n−k+1 2n−2k+1 ? F H (X ) = H (X ; C)=F H (X ) : We see that F n−k+1H2n−2k+1(X )? = F kH2k−1(X ) since a form of type (p; 2n − 2k + 1 − p) with p ≥ n − k + 1 wedged with a form of type (p0; 2k − 1 − p0) with p0 ≥ k is a form of type (p00; 2n − p00) with p00 ≥ n + 1 which must be zero since X has complex dimension n. Definition The kth intermediate Jacobian of X is the complex torus k n−k+1 2n−2k+1 _ J (X ) = F H (X ) =H2n−2k+1(X ; Z) : We abuse notation here as Voisin does when we do not distinguish between H2k−1(X ; Z) and its image in H2k−1(X ; C) and similarly for 2n−2k−1 _ H2n−2k−1(X ; Z) and its image in H (X ; C) . Intermediate Jacobians (Continued) Poincar´eduality further states that 2k−1 H (X ; Z) = H2n−2k+1(X ; Z) therefore we have an equivalent form of Jk(X ). We abuse notation here as Voisin does when we do not distinguish between H2k−1(X ; Z) and its image in H2k−1(X ; C) and similarly for 2n−2k−1 _ H2n−2k−1(X ; Z) and its image in H (X ; C) . Intermediate Jacobians (Continued) Poincar´eduality further states that 2k−1 H (X ; Z) = H2n−2k+1(X ; Z) therefore we have an equivalent form of Jk(X ). Definition The kth intermediate Jacobian of X is the complex torus k n−k+1 2n−2k+1 _ J (X ) = F H (X ) =H2n−2k+1(X ; Z) : Intermediate Jacobians (Continued) Poincar´eduality further states that 2k−1 H (X ; Z) = H2n−2k+1(X ; Z) therefore we have an equivalent form of Jk(X ). Definition The kth intermediate Jacobian of X is the complex torus k n−k+1 2n−2k+1 _ J (X ) = F H (X ) =H2n−2k+1(X ; Z) : We abuse notation here as Voisin does when we do not distinguish between H2k−1(X ; Z) and its image in H2k−1(X ; C) and similarly for 2n−2k−1 _ H2n−2k−1(X ; Z) and its image in H (X ; C) . The image of [Z] in H2k(X ; R) is described by Poincar´eduality 2k ∼ 2n−2k _ H (X ; R) = H (X ; R) : The class [Z] is the form of degree 2k such that Z 2n−2k h[Z]; αiX = α ; for all α 2 H (X ; R) ; Zsm where Zsm is the smooth locus of Z. Cycle Classes We can attach an integral cohomology class [Z] 2 H2k(X ; Z) to a cycle Z of codimension k in X and extending by Z-linearity defines a homomorphism k 2k Z (X ) −! H (X ; Z): Z 7! [Z] called the cycle class map. Cycle Classes We can attach an integral cohomology class [Z] 2 H2k(X ; Z) to a cycle Z of codimension k in X and extending by Z-linearity defines a homomorphism k 2k Z (X ) −! H (X ; Z): Z 7! [Z] called the cycle class map.