Overview
Research published on arXiv details the construction of the first compact examples of type-changing generalized Calabi--Yau manifolds. This work presents a structural form that transitions from a generic type 1 to type 3. The construction utilizes Fei's methodology, which involves non-Kähler Calabi--Yau manifolds on branched covers of twistor spaces.
Approach
The described generalized Calabi--Yau structures are explicitly built on a specific product manifold: $\Sigma_3 \times \mathbb{T}^4$. This manifold consists of a hyperelliptic surface of genus three ($\Sigma_3$) and a four-torus ($\mathbb{T}^4$). The starting point for this construction was a pre-existing non-Kähler Calabi--Yau structure. The methodology employed is based on Fei's construction as outlined in reference \cite{Fei16}, which pertains to non-Kähler Calabi--Yau manifolds derived from branched covers of twistor spaces.
Findings
The primary finding is the provision of the first compact examples of type-changing generalized Calabi--Yau manifolds. These constructed examples exhibit a specific type change in their structure. They are generically of type 1 and transition to type 3. The structures were observed on the manifold formed by the product of a hyperelliptic surface of genus three and a four-torus ($\Sigma_3 \times \mathbb{T}^4$). This construction initiated with a non-Kähler Calabi--Yau structure.