Obus conjecture on the sufficiency of the KGB obstruction

Determine whether, for every faithful local action of the metacyclic group G=C_{p^s}\rtimes_{\chi}C_m on k[[y]], the KGB obstruction is the only obstruction to lifting the action to characteristic zero.

Background

The paper studies lifting representations and actions associated with the metacyclic groups G=C_{ps}\rtimes_{\chi}C_m in characteristic p. For a faithful local action of G on k[[y]], the Harbater–Katz–Gabber construction produces a global action on an HKG cover of the projective line. The KGB obstruction is a numerical obstruction defined through the genera of quotient curves and is known to be necessary for a characteristic-zero lift.

The authors present module-theoretic lifting results that remove a previously proposed obstruction in several settings and describe their results as additional evidence for the conjecture. However, the conjecture is stated for general faithful local actions and is not proved in the paper.

References

Obus Conjecture (see , Conj.\ 1.9) predicts that if $\Gamma=G$, then for each $\phi$ the KGB obstruction defined in , Sect.\ 1, p.\ 539 is the only obstruction to the existence of a lift of $\phi$ to characteristic $0$.

— On lifting representations and actions on curves of the metacyclic groups $C_{p^s}\rtimes C_m$  (2609.03191 - Dang et al., 2 Sep 2026) in Section 1, Introduction, paragraph beginning “Obus Conjecture”

Moreover, on the positive side we add that Theorem \ref{T1} and several other results of ours could be viewed as additional evidence for the likelihood of Obus Conjecture to be true in many cases of general nature and interest, on the negative side we add that we introduce a new obstruction, and on the neutral side we add that we do not know when the new obstruction vanishes and in particular there exists the possibility that it vanishes in all cases in which the KGB obstruction vanishes.

— On lifting representations and actions on curves of the metacyclic groups $C_{p^s}\rtimes C_m$  (2609.03191 - Dang et al., 2 Sep 2026) in Section 1, Introduction, paragraph immediately following Theorem 1