The Kähler identity for bigraded Hodge-Lefschetz modules and its application in non-archimedean Arakelov geometry. (English) Zbl 0954.14017

Künnemann, Klaus

J. Algebr. Geom. 7, No. 4, 651-672 (1998).

Summary: We prove a Kähler identity for bigraded Hodge-Lefschetz modules and discuss its applications in non-archimedean Arakelov geometry. We consider, in particular, the bigraded Hodge-Lefschetz modules associated by Bloch, Gillet, and Soulé with a principal divisor with strict normal crossings on a scheme. The Kähler identity is used to prove a refined analogue of the \(dd^c\)-lemma for differential forms for these bigraded Hodge-Lefschetz modules. We apply this refined non-archimedean \(dd^c\)-lemma to investigate the arithmetic intersection pairing between cycles supported on the special fibers of semi-stable models of curves and abelian varieties.

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14G20	Local ground fields in algebraic geometry
32K15	Differentiable functions on analytic spaces, differentiable spaces
14C15	(Equivariant) Chow groups and rings; motives
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)

Keywords:

compact Kähler manifold; mixed Hodge structures; Chow cohomology; divisor with normal crossings; non-archimedean Arakelov geometry; bigraded Hodge-Lefschetz modules; arithmetic intersection pairing