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The fact that Chern classes are Hodge classes is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is the fundamental class of every algebraic variety a rational combination of them?

The fact that Chern classes are Hodge classes (and are rational combinations of algebraic cycles) is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is the fundamental class of every algebraic variety a rational combination of them?

Why is it that the Chern classes of vector bundles generate the rational cohomology of nonsingular algebraic varieties? The fact that Chern classes are Hodge classes is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is the fundamental class of every algebraic variety a rational combination of them?

The fact that Chern classes are Hodge classes is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is the fundamental class of every algebraic variety a rational combination of them?

Why is it that the Chern classes of vector bundles generate the rational cohomology of nonsingular algebraic varieties? The fact that Chern classes are Hodge classes is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is every Hodge class a rational combination of them?

Why is it that the Chern classes of vector bundles generate the rational cohomology of nonsingular algebraic varieties? The fact that Chern classes are Hodge classes is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is the fundamental class of every algebraic variety a rational combination of them?