[Paper Review] Canonical heights, invariant currents, and dynamical systems of morphisms associated with line bundles

Shu Kawaguchi|ArXiv.org|May 1, 2004

Algebraic Geometry and Number Theory18 references7 citations

TL;DR

This paper constructs canonical heights for subvarieties in dynamical systems of multiple morphisms associated with line bundles over number fields, generalizing Call–Silverman and Zhang’s work on single-morphism systems. It establishes a unique height function satisfying a degree-d scaling property and proves finiteness of small-height points under mild conditions, extending results to K3 surfaces and projective spaces.

ABSTRACT

We construct canonical heights of subvarieties for dynamical system of several morphisms associated with line bundles defined over a number field, and study some of their properties. We also construct invariant currents for such systems over $\mathbb{C}$.

Motivation & Objective

• To generalize canonical height constructions from single morphisms to systems of k morphisms on projective varieties over number fields.
• To extend the theory of canonical heights to subvarieties of K3 surfaces, including Wheler’s K3 surfaces and other special families.
• To establish the existence and uniqueness of a canonical height function satisfying a scaling identity under the action of k morphisms.
• To prove finiteness of periodic points of bounded degree and small height in such systems.
• To construct invariant currents for these systems over complex projective varieties, extending the theory of Green currents.

Proposed method

• Define a dynamical system of k morphisms over a number field K associated with a line bundle L of degree d > k via the condition ⨂_{i=1}^k f_i^*(L) ≅ L^{ ensor d}.
• Construct a canonical height function ĥ_{L,{f₁,…,fₖ}}: X(overline{K}) → ℝ satisfying ĥ(f_i(x)) sum to d·ĥ(x) and ĥ = h_L + O(1).
• Prove uniqueness of the canonical height via a linear algebra argument using non-negative matrices and Perron–Frobenius-type eigenvalue theory.
• Establish non-negativity and vanishing condition (zero iff orbit is finite) when L is ample and morphisms are surjective.
• Extend the height to subvarieties via specialization and use Northcott’s finiteness theorem to bound points of small height.
• Construct invariant (1,1)-currents over ℂ using the dynamics of morphisms and the theory of positive currents, generalizing Green currents.

Experimental results

Research questions

• RQ1 Can canonical heights for subvarieties be constructed in dynamical systems involving multiple morphisms rather than a single endomorphism?
• RQ2 Does the canonical height function satisfy a scaling identity under the joint action of k morphisms, and is it unique?
• RQ3 Under what conditions is the canonical height non-negative, and when does it vanish (indicating finite orbit)?
• RQ4 Can the theory be extended to K3 surfaces with multiple symmetries, such as Wheler’s K3 surfaces?
• RQ5 Do invariant currents exist for such multi-morphism systems over ℂ, and how do they relate to Green currents?

Key findings

• A unique canonical height function exists for dynamical systems of k morphisms associated with a line bundle L of degree d > k over a number field.
• The canonical height satisfies ∑_{i=1}^k ĥ(f_i(x)) = d·ĥ(x) for all x ∈ X(overline{K}), and ĥ = h_L + O(1).
• When L is ample, ĥ(x) ≥ 0 for all x, and ĥ(x) = 0 if and only if the forward orbit of x under {f₁,…,fₖ} is finite.
• For normal X with ample L and surjective morphisms, a canonical height ĥ(Y) ≥ 0 is defined for any subvariety Y ⊂ X_{overline{K}}.
• The canonical height on Wheler’s K3 surface (S;σ₁,σ₂) coincides with Silverman’s canonical height up to a scalar multiple (1+√3).
• The number of K-rational points of bounded degree with small canonical height is finite, as shown via Northcott’s theorem and a matrix eigenvalue argument.