Classification and uniqueness conjecture for exact Calabi–Yau manifolds with tangent cone $\mathbf{C}\times A_1$

Let $n>4$. A $\partial\bar\partial$-exact Calabi–Yau manifold is a Calabi–Yau manifold of complex dimension $n$ whose Calabi–Yau metric is $\partial\bar\partial$-exact. Suppose its tangent cone is $\mathbf{C}\times A_1$. Metrics are considered up to scaling and isometry. Classification and uniqueness conjecture. The only $\partial\bar\partial$-exact Calabi–Yau manifolds of dimension $n$ with tangent cone $\mathbf{C}\times A_1$ are $\mathbf{C}\times Q^{n-1}$, $\mathbf{C}^n$ and $Q^n$. Moreover up to scaling and isometry each of these manifolds admits a unique such Calabi–Yau metric. This conjecture proposes both a classification of the possible manifolds and uniqueness of the corresponding exact Calabi–Yau metrics. The source motivates it by the expected existence of a Calabi–Yau metric on the smoothing $Q^n$ of the $n$-dimensional $A_1$ singularity and by the methods used for the paper’s uniqueness theorem.