# Totally real embeddings with prescribed polynomial hulls

## TL;DR

This paper demonstrates that any smooth compact manifold can be embedded into complex Euclidean space as a totally real submanifold with a specified polynomial hull, expanding understanding of complex embeddings.

## Contribution

It constructs totally real embeddings of smooth manifolds with prescribed polynomial hulls, showing such embeddings exist in certain complex dimensions without requiring complex structure.

## Key findings

- Existence of totally real embeddings with prescribed polynomial hulls
- Embeddings in complex dimension loor(3d/2) for manifolds of dimension d
- Manifolds can have non-trivial polynomial hulls without complex structure

## Abstract

We embed compact $C^\infty$ manifolds into $\mathbb C^n$ as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact $C^\infty$ manifold of dimension $d$ admits a totally real embedding into $\mathbb C^{\lfloor \frac{3d}{2}\rfloor}$ with non-trivial polynomial hull without complex structure.