Non-invertible quasihomogeneous singularities and their Landau-Ginzburg orbifolds
Anton Rarovskii
Journal of Singularities
volume 28 (2025), 217-233
Received: 28 May 2024. In revised form: 10 December 2025.
Abstract:
Based on the classification of quasihomogeneous singularities, any polynomial $f$ defining such a singularity can be decomposed as f = f_\kappa + f_{add}. The polynomial f_\kappa takes a specific form, whereas f_{add} is constrained only by the requirement that the singularity of f should be isolated. The polynomial f_{add} is zero if and only if f is invertible; otherwise, in the non-invertible case, f_{add} may be arbitrarily complicated. This paper investigates all possible polynomials f_{add} for a given non-invertible f. For a fixed f_\kappa, we introduce a specific, small collection of monomials that constitute f_{add}, which guarantees that the polynomial f = f_\kappa + f_{add} defines an isolated quasihomogeneous singularity. Furthermore, if (f, Z/2Z) is a Landau-Ginzburg orbifold with such a non-invertible polynomial f, we provide a quasihomogeneous polynomial \bar{f} that satisfies the orbifold equivalence (f, Z/2Z) ~ (\bar{f}, {id}). We also present an explicit isomorphism between the corresponding Frobenius algebras.
Author(s) information:
Anton Rarovskii
Faculty of Mathematics, National Research University Higher School of Economics, Usacheva str., 6, 119048 Moscow, Russian Federation
and
Skolkovo Institute of Science and Technology, Nobelya str., 3, 121205 Moscow, Russian Federation
email: aararovskiy@edu.hse.ru