ANACOGA: Analyse complexe et géometrie analytique

• Research unit: EU RFP
• Project number: 00.0052

Key words (English)

Arithmetic intersection theory

Alternative project number (English)

EU project number: ERBFMRXCT980163

Research programs (English)

EU-programme: 4. Frame Research Programme - 10.1 Stimulation of training and mobility

Short description (English)

See abstract

Partners and International Organizations (English)

Coordinator: Université Pierre et Marie Curie, Paris (F)

Abstract (English)

In arithmetic intersection theory, it is basic to prove compatibility of the product and linear equivalence. In the original proof of Gillet-Soulé, I found a gap in their arguments at the infinite places. The first part of my stay at HU in Berlin was dedicated to fill up this gap. The idea was to prove a generalization of the K1-moving lemma based on a new definition of the support of a K1-chain. The new arguments were completely analytical.
It was possible to present my results at the mid-term meeting of the european project complex analysis and analytic geometry in Cortona (10.10-13.10.00). The results were written down in a paper, which was submitted and accepted in Crelle journal.
In the second part, I pushed my long term research project further. The goal is to develop a refined aritmetic intersection theory. For the classical intersection product, this is well known and one needs an analogy for the product of Green currents. I prepare now a paper handling the case of metrized Cartier divisors. This case is important for the applications in diophantine geometry.
First, I generalize the Burgos approach of the starproduct to arbitrary complex spaces. No smoothness or algebraicity assumption is needed. This leads to local heights of compact algebraic spaces assuming only that the intersection of supports of the involved Cartier-divisors is empty.
Then I refined star-product of metrized Cartier divisors on compete algebraic varieties (or more generally Moishezon spaces) is presented. It is new that only the intersection of the supports of the divisors has to be empty but the intersection hasn't to be proper. This leads to a theory of local heights of subvarieties.
A similar analysis at non-archimedean places will be done using rigid and formal geometry. Here, the interesting point is to work not only over discrete valuation rings but in complete generality.
Then global heights of subvarieties are studied by integrating the local heights over all places. In this way, an uniform presentation of the cases of number fields, function fields and Nevanlinna theory is achieved. In this frame work, I give an application to canonical heights of subvarieties of abelian varieties.

References in databases (English)

Swiss Database: Euro-DB of the
State Secretariat for Education and Research
Hallwylstrasse 4
CH-3003 Berne, Switzerland
Tel. [phone]
Swiss Project-Number: 00.0052