Geometric Hodge filtered complex cobordism
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We define geometric Hodge filtered complex cobordism groups MUn(p)(X) for complex manifolds X. Refining the Pontryagin–Thom construction, we give a natural isomorphism MUn(p)(X) ƒ MUnD (p)(X), where MUnD(p)(X) are the Hodge filtered complex cobordism groups defined in . We establish a pushforward map gú : MUn(p)(X) æ MUn+2d(p + d)(Y ) for each proper holomorphic map g : X æ Y . Using gú, we get for algebraic manifolds X a map !n alg(X) æ MU2n(n)(X), where !n alg(X) denotes the algebraic cobordism group of X. This induces an Abel–Jacobi map. Using a cycle model for Deligne-cohomology similar to that of , we describe the Thom-morphism MUn(p)(X) æ HnD (X; Z(p)), and verify that our Abel–Jacobi map refines that of Gri!ths.