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We construct a geometric cycle model for a Hodge filtered extension ofcomplex cobordism for every smooth manifold with a filtration on its de Rhamcomplex with complex coefficients. Using a refinement of the Pontryagin-Thomconstruction, we construct an explicit isomorphism between our geometric modeland the abstract model of Hodge filtered complex cobordism of Hopkins-Quick forevery complex manifold with the Hodge filtration.

Geometric Hodge filtered complex cobordism