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Deformation theory of parabolic representation pairs

Zhi Hu, Pengfei Huang, Wanmin Yan, Runhong Zong

Abstract

In this paper, we introduce the notions of parabolic representation pair variety and relative representation variety of a given parabolic type. We investigate the local behavior of these varieties. The Zariski tangent space and the tangent quadratic cones are described. By the Riemann--Hilbert--Deligne correspondence, we pro-represent the analytic germs of these varieties by functors related to certain groupoids of parabolic logarithmic flat bundles. Under suitable assumptions, we prove that the differential graded Lie algebra (DGLA) controlling the deformation of parabolic logarithmic flat bundle is mixedly formal. Finally, we construct the moduli space of weighted parabolic representation pairs, and, by means of quiver representation theory, we establish the Kobayashi--Hitchin-type theorem for polystable parabolic representation pairs.

Deformation theory of parabolic representation pairs

Abstract

In this paper, we introduce the notions of parabolic representation pair variety and relative representation variety of a given parabolic type. We investigate the local behavior of these varieties. The Zariski tangent space and the tangent quadratic cones are described. By the Riemann--Hilbert--Deligne correspondence, we pro-represent the analytic germs of these varieties by functors related to certain groupoids of parabolic logarithmic flat bundles. Under suitable assumptions, we prove that the differential graded Lie algebra (DGLA) controlling the deformation of parabolic logarithmic flat bundle is mixedly formal. Finally, we construct the moduli space of weighted parabolic representation pairs, and, by means of quiver representation theory, we establish the Kobayashi--Hitchin-type theorem for polystable parabolic representation pairs.

Paper Structure

This paper contains 12 sections, 26 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Deformations of parabolic representation pairs
  3. Parabolic representation pairs
  4. Zariski tangent spaces and tangent quadratic cones
  5. Deformations of parabolic logarithmic flat bundles
  6. Parabolic-groupoid version of the Riemann--Hilbert--Deligne correspondence
  7. Zariski tangent spaces and tangent quadratic cones
  8. DGLA-descriptions of deformation functors
  9. Review of Deligne--Goldman--Millson deformation theory
  10. DGLAs associated to parabolic logarithmic flat bundles
  11. Mixed formality
  12. Moduli spaces of weighted parabolic representation pairs

Key Result

Theorem 1.1

For a representation $\rho: \Xi\rightarrow G$ of a group $\Xi$ into $G$, we introduce the following notations:

Figures (1)

  • Figure :

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Definition 2.4: sitm2
  • ...and 44 more