An integration formula of Chern forms on quasi-projective varieties

In this paper, I generalize the formula that the integration of Chern forms of hermitian line bundles equals the algebraic intersection number of the underlying line bundles. Based on the theory of Yuan and Zhang, I generalize it to a formula on a quasi-projective variety over a complete valuation field which might be archimedean or non-archimedean. Our result has a close relation with the integration of Betti forms and the notion of non-degeneracy of a closed subvariety.