Monodromy Representations: Decomposing Rank-Three Bundles over the Projective Line with Three Marked Points

2025/07/09 by Yépez, Diego
#14F06 #32L10 #53B15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.07080

Abstract

Given a monodromy representation ρ of the projective line minus m points, one can extend the resulting vector bundle with connection map canonically to a vector bundle with logarithmic connection map over all of the projective line. Now, since vector bundles split as twisting sheaves over the projective line, the focus of this work regards knowing the exact decomposition; i.e., computing the roots. Particularly, we use the monodromy derivative to compute the roots for all three-dimensional ρ when m = 3.

Citations

• Moduli space of rank three logarithmic connections on the projective line with three poles
• Moduli space of irregular rank two parabolic bundles over the Riemann sphere and its compactification
• Parabolic Hecke eigensheaves
• Indecomposable vector-valued modular forms and periods of modular curves
• Vector valued modular forms and the modular orbifold of elliptic curves

Related

• Computing the Roots of Twisting Sheaves over the Projective Line arising from Monodromy Representations
• Monodromy, vanishing cycles, knots and the adjoint quotient
• Monodromy representations of meromorphic projective structures
• Monodromy of triple point line arrangements
• Connections on trivial vector bundles over projective schemes