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

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)