Hodge Style Chern Classes for Vector Bundles on Schemes

In this note, we develop the formalism of Hodge style chern classes of vector bundles over arbitrary quasi-projective schemes defined over K, a field of characteristic zero. The theory of Chern classes is well known by now and without any restriction on the characteristic, can be defined in many theories with rational coefficients, like for example the Chow ring. Atiyah [1] developed the theory of Chern classes of vector bundles with values in the Hodge ring ⊕ H p (X, Ω p X) for smooth complex varieties. Grothendieck [2] remarked that Atiyah’s constructions could be transposed (“sans difficult´e”) to the case of any S-scheme X and referred to a future paper where it would appear. To the best of our knowledge, this has not occured.