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On the Existence of Gr-semistable Filtrations of Orthogonal/Symplectic $λ$-connections

Mao Sheng, Hao Sun, Jianping Wang

Abstract

In this paper, we study the existence of gr-semistable filtrations of orthogonal/symplectic $λ$-connections. It is known that gr-semistable filtrations always exist for flat bundles in arbitrary characteristic. However, we found a counterexample of orthogonal flat bundles of rank 5 in positive characteristic. The central new idea in this example is the notion of quasi gr-semistability for orthogonal/symplectic $λ$-connections. We establish the equivalence between gr-semistability and quasi gr-semistablity for an orthogonal/symplectic $λ$-connection. This provides a way to determine whether an orthogonal/symplectic $λ$-connection is gr-semistable. As an application, we obtain a characterization of gr-semistable orthogonal $λ$-connections of rank $\leq 6$.

On the Existence of Gr-semistable Filtrations of Orthogonal/Symplectic $λ$-connections

Abstract

In this paper, we study the existence of gr-semistable filtrations of orthogonal/symplectic -connections. It is known that gr-semistable filtrations always exist for flat bundles in arbitrary characteristic. However, we found a counterexample of orthogonal flat bundles of rank 5 in positive characteristic. The central new idea in this example is the notion of quasi gr-semistability for orthogonal/symplectic -connections. We establish the equivalence between gr-semistability and quasi gr-semistablity for an orthogonal/symplectic -connection. This provides a way to determine whether an orthogonal/symplectic -connection is gr-semistable. As an application, we obtain a characterization of gr-semistable orthogonal -connections of rank .
Paper Structure (13 sections, 34 theorems, 141 equations)

This paper contains 13 sections, 34 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rees Construction
  4. Logarithmic Orthogonal/Symplectic $\lambda$-connections
  5. Stability Condition and Harder--Narasimhan Filtration
  6. Quasi Gr-semistable Filtration: Odd Rank
  7. Definition and Basic Properties
  8. The Existence of Gr-semistable Filtration
  9. Quasi Gr-semistable Filtration: Even Rank
  10. Classification of Quasi Gr-semistable Orthogonal $\lambda$-Connections in Small Ranks
  11. Case I: $\boldsymbol{\mathrm{rk}(V) \leq 4}$
  12. Case II: $\boldsymbol{\mathrm{rk}(V) = 5}$
  13. Case III: $\boldsymbol{\mathrm{rk}(V) = 6}$

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field and $X$ a smooth projective variety over $k$, equipped with an ample line bundle $L$. Let $(V,\nabla)$ be an integrable $\lambda$-connection over $X$. Then $(V,\nabla)$ is $\mu_L$-semistable if and only if it is $\mu_L$-gr-semistable.

Theorems & Definitions (77)

  • Theorem 1.1: Simpson (Theorem 2.5 S10), Lan-Sheng-Yang-Zuo (Theorem A.4 LSZ19), Langer (Theorem 5.5 L13)
  • Example 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3: Proposition 2.2.7 in Lo17b
  • Lemma 2.4: §3 in Lo17a
  • Definition 2.5
  • ...and 67 more