Computing the Roots of Twisting Sheaves over the Projective Line arising from Monodromy Representations
Diego Yépez
TL;DR
The work analyzes how monodromy representations of the punctured projective line give rise to vector bundles with logarithmic connections on $\mathbb{P}^1$ and aims to determine their Birkhoff–Grothendieck decompositions into twisting sheaves, i.e., the roots. It develops an algebraic framework based on Ohtsuki's formula for the first Chern class and introduces a Serre-like derivative via an auxiliary connection, called the monodromy derivative, to handle irreducible representations and compute explicit root data. The main contributions provide explicit root decompositions for all finite-dimensional representations when $m=2$ and for representations with $\dim\rho<3$ when $m=3$, including a two-dimensional irreducible case that yields precise splitting patterns; the modular-group example demonstrates a concrete outcome with $c_1(\mathcal{V}_{Log(\rho)})=-2$ and $\mathcal{V}_{Log(\rho)}\cong \mathcal{O}(-1)^{\oplus 2}$. By connecting monodromy representations, parabolic-bundle perspectives, and explicit construction of the monodromy derivative, the paper provides concrete, computable decompositions of these logarithmic connection-induced bundles into twisting components, advancing the explicit understanding of roots in this geometric setting.
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 compute the roots for all finite-dimensional $ρ$ when $m = 2$ and for all $ρ$ of dimension less than $3$ when $m = 3$.