  We construct Hodge filtered cohomology groups for complex manifolds that
combine the topological information of generalized cohomology theories with
geometric data of Hodge filtered holomorphic forms. This theory provides a
natural generalization of Deligne cohomology. For smooth complex algebraic
varieties, we show that the theory satisfies a projective bundle formula and
$\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective
morphisms.
