A Bishop surface with a vanishing Bishop invariant

TL;DR

This paper develops a complete set of invariants for formal Bishop surfaces with vanishing Bishop invariant, revealing their infinite-dimensional moduli space and showing that finite Taylor expansions cannot classify their equivalence classes.

Contribution

It introduces a full set of invariants for Bishop surfaces with vanishing Bishop invariant and demonstrates the infinite-dimensionality of their moduli space.

Findings

Moduli space of such surfaces is infinite-dimensional.

Finite Taylor expansions do not determine equivalence classes.

Answers a long-standing problem posed by J. Moser in 1985.

Abstract

We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant $s<\infty$ is of infinite dimension. We also prove that the equivalence class of the germ of a generic real analytic Bishop surface near a complex tangent with a vanishing Bishop invariant can not be determined by a finite part of the Taylor expansion of its defining equation. This answers, in the negative, a problem raised by J. Moser in 1985 after his joint work with Webster in 1983 and his own work in 1985.