An integration formula of Chern forms on quasi-projective varieties

TL;DR

This paper extends the classical formula relating Chern form integration to algebraic intersection numbers to quasi-projective varieties over complete valuation fields, bridging complex and non-archimedean geometry.

Contribution

It generalizes the integration formula of Chern forms to quasi-projective varieties over various valuation fields, connecting it with Betti form integration and subvariety non-degeneracy.

Findings

Unified formula for Chern form integration on quasi-projective varieties

Relation established between Betti form integration and algebraic intersection

Applicable to both archimedean and non-archimedean fields

Abstract

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. 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.