Non-embeddable 1-convex manifolds

TL;DR

This paper demonstrates that certain small resolutions of three-dimensional hypersurface singularities can produce non-embeddable 1-convex manifolds, providing explicit examples and studying specific singularities.

Contribution

It introduces the first explicit examples of non-embeddable 1-convex manifolds arising from small resolutions of hypersurface singularities.

Findings

Existence of non-embeddable 1-convex manifolds from small resolutions

Explicit example with an irreducible rational curve with (1,-3) normal bundle

Analysis of small resolutions of cD_4-singularities

Abstract

We show that every small resolution of a three-dimensional terminal hypersurface singularity can occur on a non-embeddable 1-convex manifold. We give an explicit example of a non-embeddable manifold containing an irreducible exceptional rational curve with normal bundle of type (1,-3). To this end we study small resolutions of cD_4-singularities.