Cayley deformations of compact complex surfaces

TL;DR

This paper investigates Cayley deformations of compact complex surfaces within Calabi--Yau four-folds, demonstrating that complex and Cayley deformations coincide and that their moduli space is smooth.

Contribution

It proves that the moduli space of complex deformations of compact complex submanifolds in Calabi--Yau manifolds is smooth, linking complex and Cayley deformations.

Findings

Complex and Cayley deformations of compact complex surfaces are equivalent.

The moduli space of complex deformations is smooth.

Provides a geometric understanding of deformations in Calabi--Yau contexts.

Abstract

In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi--Yau manifold is a smooth manifold.