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 (cid:88) α = f dz ∧ dz I,J I J I,J ⊆ {1,...,n} |I| = p, |J| = q for some choice of local holomorphic coordinates z , . . . , z . 1 n 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 (cid:77) k C p,q H (X, ) = 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 (cid:77) k C p,q H (X, ) = 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 (cid:88) α = f dz ∧ dz I,J I J I,J ⊆ {1,...,n} |I| = p, |J| = q for some choice of local holomorphic coordinates z , . . . , z . 1 n Since real forms are conjugate invariant, we may write [α] ∈ 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] ∈ 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 (cid:55)→ α is the natural action of complex conjugation on k C k R C H (X, ) = H (X, ) ⊗ . R Hodge Decomposition (Continued) The subspaces Hp,q(X) satisfy the Hodge symmetry Hp,q(X) = Hq,p(X) where α (cid:55)→ α is the natural action of complex conjugation on k C k R C H (X, ) = H (X, ) ⊗ . R Since real forms are conjugate invariant, we may write [α] ∈ 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] ∈ 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 H2k−1(X, C) = FkH2k−1(X) ⊕ FkH2k−1(X) and the natural map 2k−1 R 2k−1 C k 2k−1 H (X, ) −→ H (X, )/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 C 0 ⊆ F H (X) ⊆ F H (X) ⊆ · · · ⊆ F H (X) = H (X, ) where (cid:77) r k p,k−p F H (X) = H (X) , 0 ≤ r ≤ k . p≥r Hodge Decomposition (Continued) The Hodge decomposition defines a filtration k k k−1 k 0 k k C 0 ⊆ F H (X) ⊆ F H (X) ⊆ · · · ⊆ F H (X) = H (X, ) where (cid:77) r k p,k−p F H (X) = H (X) , 0 ≤ r ≤ k . p≥r The cohomology groups of odd degree satisfy H2k−1(X, C) = FkH2k−1(X) ⊕ FkH2k−1(X) and the natural map 2k−1 R 2k−1 C k 2k−1 H (X, ) −→ H (X, )/F H (X) is an isomorphism of real vector spaces.