Bibliographic Information: Deaconu, S. (2024). On units with Galois complex conjugates of equal absolute value. arXiv preprint arXiv:2310.18958v2.
Research Objective: This paper aims to provide a simplified and unified proof for the relationship between the existence of specific units in a number field and the presence of locally conformally Kähler (l.c.K) or pluriclosed metrics on Oeljeklaus-Toma (OT) manifolds.
Methodology: The author utilizes concepts from algebraic number theory, including Galois theory, heights on number fields, the Northcott-Weil theorem, and Dirichlet's unit theorem, to analyze the properties of units in number fields. These properties are then linked to the geometric structures of OT manifolds.
Key Findings: The paper demonstrates that if a number field K with s real embeddings and 2t complex embeddings has a subgroup of units U where all elements have complex Galois conjugates of equal absolute value, then t must equal 1. This finding implies that OT manifolds X(K, U) admitting l.c.K metrics correspond precisely to cases where U is a congruence subgroup of the units in K. Additionally, the paper proves that for OT manifolds with s ≥ 1 admitting pluriclosed metrics, the number of real and complex embeddings of the associated number field must be equal (s = t).
Main Conclusions: The results provide a concrete connection between the algebraic properties of number fields and the geometric structures possible on OT manifolds. The existence of l.c.K or pluriclosed metrics on these manifolds can be directly determined by examining the Galois conjugates of units within the associated number field.
Significance: This research contributes significantly to the understanding of the interplay between number theory and complex geometry. It provides a powerful tool for studying the geometry of OT manifolds by leveraging the algebraic structure of their underlying number fields.
Limitations and Future Research: The paper focuses specifically on l.c.K and pluriclosed metrics. Exploring the existence of other geometric structures on OT manifolds and their connection to number field properties could be a potential avenue for future research. Additionally, investigating the implications of these findings for related classes of complex manifolds would be of interest.
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by Stefan Deaco... at arxiv.org 10-15-2024
https://arxiv.org/pdf/2310.18958.pdfDeeper Inquiries