ScalingStacks

Example 7.7 . [02ZY]

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Example 7.7.

Consider the polytope Δ\Delta of Example 3.2. The dual polytope ∇\nabla is the convex hull of the points (−1,−1,−1,−1),(1,0,0,0),…,(0,0,0,1)(-1,-1,-1,-1),(1,0,0,0),\ldots,(0,0,0,1). The corresponding projective toric variety ℙ∇\mathbb{P}_{\nabla} has a crepant resolution XΣ→ℙ∇X_{\Sigma}\rightarrow\mathbb{P}_{\nabla} where Σ\Sigma is the fan consisting of cones over all elements of the decomposition 𝒫\mathscr{P} of ∂Δ\partial\Delta as described in Example 3.2. Consider in ℙ∇×𝔸1\mathbb{P}_{\nabla}\times\mathbb{A}^{1} the degenerating family 𝒳→𝔸1\mathcal{X}\rightarrow\mathbb{A}^{1} of Calabi-Yau manifolds given by

s0+t​∑m∈∇∩ℤ4cm​sm=0s_{0}+t\sum_{m\in\nabla\cap\mathbb{Z}^{4}}c_{m}s_{m}=0

where sms_{m} is the section of 𝒪ℙ∇​(1)\mathcal{O}_{\mathbb{P}_{\nabla}}(1) corresponding to m∈∇∩ℤ4m\in\nabla\cap\mathbb{Z}^{4}. Let 𝒳~\widetilde{\mathcal{X}} be the proper transform of 𝒳\mathcal{X} in XΣ×𝔸1X_{\Sigma}\times\mathbb{A}^{1}. Then the family 𝒳~→𝔸1\widetilde{\mathcal{X}}\rightarrow\mathbb{A}^{1} is a toric degeneration with general fibre the mirror quintic, and its dual intersection complex is the affine manifold BB constructed in Example 3.2.

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