Continuum envelops on Fargues-Fontaine curves and elliptic curves

2024/04/06 by Heng Du, Du, Heng, Qingyuan Jiang +3 · 1 citation
Mathematics · Social Sciences · #11H99 #14G45 #14H52 #14H60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2404.04551

openalex publication_date 2024/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we apply the theory of Bridgeland stability conditions, which was originated from string theory, to study the derived category of coherent sheaves on Fargues--Fontaine curves. This leads us to consider the quasi-coherent sheaves O(θ^±) via the convergents of an irrational number θ. We define the continuum envelop QCoh_ℝ(X_FF) to be the smallest abelian subcategory in QCoh(X_FF) containing Coh(X_FF) and O(θ^±). We study the homological algebra of QCoh_ℝ(X_FF) via Farey diagrams. We show that the homological property of O(θ^±) depends heavily on the arithmetic property of θ. The Fargues--Fontaine curves present strong similarity with complex elliptic curves in this point of view.

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• Stability conditions on irreducible projective curves

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