Characteristic classes for algebraic vector bundles with Hermitian metric. II. (English) Zbl 0715.14006

Gillet, Henri; Soulé, Christophe

Ann. Math. (2) 131, No. 2, 205-238 (1990).

[For part I of this paper see ibid., No.1, 163-203 (1990; Zbl 0715.14018).]
This part II has three sections. In section 5, the case \(X={\mathbb{P}}^ n\) (the projective space) is considered; one computes the arithmetic Chern classes of the canonical rank \(n\) vector bundle on X, which are given by the \(L^ 1\quad forms\) introduced by Levine in his paper on Nevanlinna theory for maps into \({\mathbb{P}}^ n({\mathbb{C}})\) [H. I. Levine, Ann. Math., II. Ser. 71, 529-535 (1960; Zbl 0142.048)]. In section 6, one introduces \(\hat K_ 0(X)\) and one describes it by some exact sequences. In the last section, one gives a new description of the Beilinson regulator on \(K_ 1\)(X) by means of Bott-Chern forms, and one shows that ch is an isomorphism of \(\lambda\)-rings.

Reviewer: Vasile Brînzănescu

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
32H30	Value distribution theory in higher dimensions
57R20	Characteristic classes and numbers in differential topology
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry

Keywords:

Nevanlinna theory for maps into complex projective n-space; arithmetic Chern classes; Beilinson regulator on \(K_ 1\)

Citations:

Zbl 0715.14018; Zbl 0142.048