Chern-Moser-Weyl Tensor and Embeddings into Hyperquadrics

TL;DR

This paper surveys the role of the Chern-Moser-Weyl tensor in the embeddability of real hypersurfaces into hyperquadrics and provides a negative answer to a longstanding conjecture about embedding compact algebraic hypersurfaces into high-dimensional spheres.

Contribution

It offers a comprehensive survey of the Chern-Moser-Weyl tensor's application and disproves a folklore conjecture on embeddability into spheres.

Findings

Negative answer to the folklore conjecture.

Clarification of conditions for embeddability into hyperquadrics.

Insights into the structure of real algebraic hypersurfaces.

Abstract

The first part of the article surveys some work on the Chern-Moser-Weyl tensor and its application in the embeddability problem into hyperquadrics. In the last section, we give a negative answer to a folklore conjecture concerning the embeddability of compact, strongly pseudoconvex, real algebraic hypersurfaces into a sphere of sufficiently high dimension.