Deformations of conically singular Cayley submanifolds

TL;DR

This paper investigates the deformation theory of conically singular Cayley submanifolds, establishing expected dimensions of their moduli spaces and analyzing smoothness in special cases, with explicit calculations for examples.

Contribution

It provides new results on the expected dimension and smoothness of moduli spaces of Cayley deformations, especially for complex submanifolds in Calabi--Yau four-folds.

Findings

Expected dimension of Cayley deformation moduli space established

Smoothness of moduli space shown for certain complex submanifolds

Explicit calculations performed for specific examples

Abstract

In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a Calabi--Yau four-fold we show by comparing Cayley and complex deformations that in this special case the moduli space is a smooth manifold. We also perform calculations of some of the quantities discussed for some examples.