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An obstruction to the local lifting problem

Aristides Kontogeorgis, Alexios Terezakis

TL;DR

The paper addresses the local lifting problem for actions by metacyclic groups $G = C_q ⋊ C_m$ in characteristic $p$, using the Harbater-Katz-Gabber framework to translate local liftability into the problem of lifting HKG-covers and their canonical ideals. It develops a new obstruction based on the $G$-module structure of holomorphic differentials and the associated canonical ideal, deriving a practical lift criterion that can be checked via decompositions into indecomposable summands $U_{ℓ,μ}$ and their pairings. The authors provide an algorithm (implemented in Sage) to decide liftability for given ramification data, and they exhibit explicit non-lift examples for dihedral groups (notably $D_{125}$ with jumps $(1,21,521)$ and $(1,5,25)$, and also $(9,189,4689)$), demonstrating obstructions beyond the KGB obstruction and yielding a counterexample to a generalized Oort conjecture. The work connects local lifting to deformation theory through the canonical ideal, Petri-type liftability criteria, and a detailed analysis of the Galois module structure on holomorphic differentials, offering both theoretical insight and computational tools for assessing liftability.

Abstract

We are investigating the lifting problem for local actions involving semidirect products of a cyclic $p$-group with a cyclic group prime to $p$, where $p$ represents the characteristic of the special fiber. We establish a criterion based on the Harbater-Katz-Gabber compactification of local actions, enabling us to determine whether a given local action can be lifted or not. Specifically, in the case of the dihedral group, we present an example of a local dihedral action that cannot be lifted. This instance provides a more potent obstruction than the KGB obstruction.

An obstruction to the local lifting problem

TL;DR

The paper addresses the local lifting problem for actions by metacyclic groups in characteristic , using the Harbater-Katz-Gabber framework to translate local liftability into the problem of lifting HKG-covers and their canonical ideals. It develops a new obstruction based on the -module structure of holomorphic differentials and the associated canonical ideal, deriving a practical lift criterion that can be checked via decompositions into indecomposable summands and their pairings. The authors provide an algorithm (implemented in Sage) to decide liftability for given ramification data, and they exhibit explicit non-lift examples for dihedral groups (notably with jumps and , and also ), demonstrating obstructions beyond the KGB obstruction and yielding a counterexample to a generalized Oort conjecture. The work connects local lifting to deformation theory through the canonical ideal, Petri-type liftability criteria, and a detailed analysis of the Galois module structure on holomorphic differentials, offering both theoretical insight and computational tools for assessing liftability.

Abstract

We are investigating the lifting problem for local actions involving semidirect products of a cyclic -group with a cyclic group prime to , where represents the characteristic of the special fiber. We establish a criterion based on the Harbater-Katz-Gabber compactification of local actions, enabling us to determine whether a given local action can be lifted or not. Specifically, in the case of the dihedral group, we present an example of a local dihedral action that cannot be lifted. This instance provides a more potent obstruction than the KGB obstruction.
Paper Structure (6 sections, 12 theorems, 49 equations)

This paper contains 6 sections, 12 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. HKG-covers and their canonical ideal
  4. Invariant subspaces of vector spaces
  5. Galois module structure of holomorphic differentials, special fibre
  6. Examples

Key Result

Proposition 1

Let $f_1,\ldots,f_r \in S:=\mathop{\mathrm{Sym}}\nolimits H^0(X,\Omega_X)=k[\omega_1,\ldots,\omega_g]$ be quadratic polynomials which generate the canonical ideal $I_{X}$ of a curve $X$ defined over an algebraic closed field $k$. Any deformation $\mathcal{X}_A$ is given by quadratic polynomials $\ti

Theorems & Definitions (29)

  • Proposition 1
  • Theorem 2
  • Proposition 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Lemma 7
  • proof
  • Remark 8
  • ...and 19 more