[Paper Review] On $3$-dimensional foliated dynamical systems and Hilbert type reciprocity law

Kim Junhyeong, Masanori Morishita+2 more|arXiv (Cornell University)|Jun 6, 2019

Geometry and complex manifolds13 references4 citations

TL;DR

This paper establishes foundational results for 3-dimensional foliated dynamical systems (FDS 3), introducing a decomposition theorem that classifies FDS 3 structures via transverse and non-transverse compact leaves. It constructs explicit examples for each class, including countably infinite closed orbits, and develops geometric local symbols using smooth Deligne cohomology to prove a Hilbert-type reciprocity law: the sum of local symbols over all closed orbits and non-transverse leaves vanishes modulo the period group, generalizing class field theory analogies in arithmetic topology.

ABSTRACT

We show some fundamental results concerning $3$-dimensional foliated dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give a decomposition theorem for an FDS$^3$, which yields a classification of FDS$^3$'s. Secondly, for each type of the classification, we construct concrete examples of FDS$^3$'s. Finally, by using the integration theory for smooth Deligne cohomology, we introduce geometric analogues of local symbols and show a Hilbert type reciprocity law for an FDS$^3$. Our results answer the question posed by Deninger.

Motivation & Objective

• To resolve a question posed by Deninger on geometric analogues of Hilbert symbols and reciprocity laws in the context of 3-dimensional foliated dynamical systems (FDS 3).
• To classify FDS 3 structures via a decomposition theorem based on transverse and non-transverse compact leaves.
• To construct explicit examples of FDS 3 for each classification type, including those with countably infinite closed orbits.
• To develop a cohomological framework using smooth Deligne cohomology to define geometric local symbols and prove a reciprocity law.

Proposed method

• A decomposition theorem is proven for FDS 3 by cutting along non-transverse compact leaves, yielding well-structured components that classify the system.
• Examples are constructed using open book decompositions, showing every closed 3-manifold admits an FDS 3 structure.
• Local symbols ⟨f,g⟩γ are defined via integration of Deligne cohomology classes, using holomorphic data and foliated local coordinates.
• The integration theory of smooth Deligne cohomology is applied to compute local symbols explicitly for closed orbits.
• The reciprocity law is proven by showing the sum of local symbols over all γ ∈ P̄S vanishes modulo the period group ΛS(3), using Stokes' theorem and closedness of ωS.
• The method generalizes Deligne-Bloch-Beilinson’s interpretation of the tame symbol on Riemann surfaces to 3D FDS 3 via higher cohomology and iterated integrals.

Experimental results

Research questions

• RQ1 Can a classification of 3-dimensional foliated dynamical systems be achieved via a decomposition theorem based on compact leaves?
• RQ2 Can explicit examples of FDS 3 be constructed for each class in the classification, including those with countably infinite closed orbits?
• RQ3 How can geometric analogues of Hilbert symbols be defined for meromorphic functions on FDS 3?
• RQ4 Does a Hilbert-type reciprocity law hold for these geometric local symbols, analogous to class field theory?
• RQ5 Can the integration theory of smooth Deligne cohomology be used to define and compute local symbols in a geometric setting?

Key findings

• A decomposition theorem is established that cuts a 3-manifold along non-transverse compact leaves, yielding components with well-defined foliated dynamical structure, enabling classification of FDS 3 by the types of such leaves.
• Every closed smooth 3-manifold admits an FDS 3 structure via open book decomposition, demonstrating a unique property of the 3-dimensional case.
• For each classification type, concrete examples of FDS 3 are constructed, including systems with countably infinitely many closed orbits, analogous to arithmetic curves with infinitely many finite primes.
• An explicit integral formula is derived for the local symbol ⟨f,g⟩γ along a closed orbit γ ∈ P_S, using foliated local coordinates and holomorphic logarithmic differentials.
• A Hilbert-type reciprocity law is proven: ∑γ∈P̄S ⟨f,g⟩γ ≡ 0 mod (2πi)^3ΛS, where ΛS is the period group of the FDS 3.
• The proof relies on the closedness of the form ωS and the exactness of d(log f ∧ dlog g ∧ ωS), showing the total integral over ∂Z vanishes, thus confirming the reciprocity law.