Deformations of geometric structures on a subclass of LVM threefolds

TL;DR

This paper investigates how certain geometric structures deform on specific complex three-dimensional LVM manifolds, linking these deformations to complex structure variations and describing their moduli space.

Contribution

It provides a detailed description of the Kuranishi family for these LVM manifolds and constructs a comprehensive family encompassing all such structures.

Findings

Description of the Kuranishi family for LVM threefolds

Construction of a universal family of geometric structures

Analysis of the link between geometric and complex structure deformations

Abstract

We study deformations of geometric structures on some LVM manifolds of complex dimension $3$. More precisely, we study resonant structures, a particular type of $(G,X)$-structures, via the Ehresmann-Thurston principle, and their link with the deformation of the complex structure of the LVM manifold in the sense of Kodaira, Spencer and Kuranishi. We describe the Kuranishi family of these manifolds and construct a family containing all of them and complete at every point.