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Cohomology of vector bundles on the moduli space of parabolic connections on $\mathbb{P}^1$ minus $5$ points

Yuki Matsubara

TL;DR

The paper analyzes the moduli space $\mathcal{M}$ of rank-two parabolic connections on ${\mathbb P}^1$ with five punctures, constructing a universal twisted $D$-module $\xi_{\boldsymbol{\nu}}$ and proving a generic orthogonality property for $n=5$. It develops a robust algebro-geometric framework using relative twisted compactified Jacobians, Fourier–Mukai transforms, and the Hitchin system to compute cohomology on both the boundary $\mathcal{M}_H$ and the compactification $\overline{\mathcal{M}}$, culminating in explicit results for $H^i(\mathcal{M}, \mathcal{O}_{\mathcal{M}})$ and related cohomologies of products of $\xi_x$-bundles. The work also provides a detailed geometric description of the smooth compactification and connects these cohomological computations to a conjectural Geometric Langlands correspondence, including a Radon-transform perspective and implications for tamely ramified Langlands-type equivalences. Overall, the paper extends Arinkin’s results to the $n=5$ case, offering new insights into the interplay between parabolic connections, spectral data, and derived-equivalence phenomena in the Langlands program.

Abstract

We study the moduli space of parabolic connections of rank two on the complex projective line $\mathbb{P}^1$ minus five points with fixed spectral data. This paper aims to compute the cohomology of the structure sheaf and a certain vector bundle on this space. We use this computation to extend the results of Arinkin, which proved a specific Geometric Langlands Correspondence to the case where these connections have five simple poles on $\mathbb{P}^1$. Moreover, we give an explicit geometric description of the compactification of this moduli space.

Cohomology of vector bundles on the moduli space of parabolic connections on $\mathbb{P}^1$ minus $5$ points

TL;DR

The paper analyzes the moduli space of rank-two parabolic connections on with five punctures, constructing a universal twisted -module and proving a generic orthogonality property for . It develops a robust algebro-geometric framework using relative twisted compactified Jacobians, Fourier–Mukai transforms, and the Hitchin system to compute cohomology on both the boundary and the compactification , culminating in explicit results for and related cohomologies of products of -bundles. The work also provides a detailed geometric description of the smooth compactification and connects these cohomological computations to a conjectural Geometric Langlands correspondence, including a Radon-transform perspective and implications for tamely ramified Langlands-type equivalences. Overall, the paper extends Arinkin’s results to the case, offering new insights into the interplay between parabolic connections, spectral data, and derived-equivalence phenomena in the Langlands program.

Abstract

We study the moduli space of parabolic connections of rank two on the complex projective line minus five points with fixed spectral data. This paper aims to compute the cohomology of the structure sheaf and a certain vector bundle on this space. We use this computation to extend the results of Arinkin, which proved a specific Geometric Langlands Correspondence to the case where these connections have five simple poles on . Moreover, we give an explicit geometric description of the compactification of this moduli space.
Paper Structure (32 sections, 47 theorems, 151 equations, 2 tables)

This paper contains 32 sections, 47 theorems, 151 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Summary of main results
  4. Proof of Theorem main
  5. Outline of the paper
  6. Acknowledgements
  7. Cohomology of line bundles on relative twisted compactified Jacobians
  8. Relative compactified Jacobians
  9. Autoduality of compactified Jacobians
  10. Relative twisted compactified Jacobians
  11. Trivialization along a multi-section
  12. Étale local descriptions
  13. Cohomology of line bundles on relative twisted compactified Jacobians
  14. Bundles $\xi_x$ on $\mathcal{M}_H$
  15. Relation between ${\mathcal{M}}_H$ and ${\mathfrak{s}l_2}$-Higgs bundles
  16. ...and 17 more sections

Key Result

Theorem 1.2

Suppose $x_1,\dots, x_4 \in {\mathbb P}^1$ and $x_i \neq x_j$ for $i \neq j$. Then for any $i \geq 0$.

Theorems & Definitions (87)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorem generic_orthogonal_property
  • Theorem 1.4: D. Arinkin AF, Corollary proof_of_main2
  • Definition 1.5
  • Proposition 1.6: Proposition computation_along_boundary
  • Proposition 1.7: Proposition computation_compactified
  • Lemma 1.8
  • proof : Proof of Theorem main
  • Theorem 2.1: A11
  • ...and 77 more