[Paper Review] Intersection pairing for arithmetic cycles with degenerate Green currents

Atsushi Moriwaki|ArXiv.org|Mar 13, 1998

Algebraic Geometry and Number Theory11 references5 citations

TL;DR

This paper introduces an extended arithmetic Chow group for codimension one cycles on regular projective arithmetic varieties, incorporating degenerate Green currents. It establishes the Hodge index theorem in this setting and proves an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles, providing a foundational framework for arithmetic intersection theory with singular Green currents.

ABSTRACT

In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties.

Motivation & Objective

• To extend the arithmetic Chow group of codimension one cycles to include degenerate Green currents.
• To ensure the Hodge index theorem holds in the extended framework.
• To establish an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on regular projective arithmetic varieties.
• To provide a consistent intersection pairing in the presence of degenerate Green currents.

Proposed method

• Extends the arithmetic Chow group by allowing degenerate Green currents, which are singular in the sense of not being smooth or strictly positive.
• Uses the theory of Green currents and arithmetic intersection theory on arithmetic surfaces and higher-dimensional arithmetic varieties.
• Applies the theory of arithmetic intersection numbers via the use of Green currents, even when they are degenerate.
• Employs the arithmetic Riemann-Roch theorem and properties of first Chern classes in the arithmetic setting.
• Introduces a refined intersection pairing that accounts for degeneracy in Green currents.
• Relies on the structure of regular projective arithmetic schemes and the behavior of line bundles with singular metrics.

Experimental results

Research questions

• RQ1 Can the arithmetic Chow group be extended to include cycles with degenerate Green currents while preserving key theorems like the Hodge index theorem?
• RQ2 Does an arithmetic analogue of Bogomolov's instability theorem hold for rank 2 vector bundles on regular projective arithmetic varieties?
• RQ3 How can intersection pairings be consistently defined when Green currents are degenerate or singular?
• RQ4 What is the role of degenerate Green currents in the arithmetic Riemann-Roch formula and arithmetic intersection theory?
• RQ5 Can the Hodge index theorem be generalized to arithmetic cycles with non-smooth or degenerate Green currents?

Key findings

• The extended arithmetic Chow group for codimension one cycles supports a well-defined intersection pairing even with degenerate Green currents.
• The Hodge index theorem holds in the extended arithmetic Chow group, ensuring positivity and signature constraints on arithmetic intersection numbers.
• An arithmetic analogue of Bogomolov's instability theorem is proven for rank 2 vector bundles on regular projective arithmetic varieties.
• The construction provides a consistent framework for arithmetic intersection theory in the presence of singular or degenerate Green currents.
• The paper establishes that degeneracy in Green currents does not obstruct the validity of fundamental arithmetic duality and index theorems.
• The results are valid for arbitrary regular projective arithmetic varieties, not restricted to surfaces or specific base schemes.