ScalingStacks

Example 2.2 . [03GE]

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Example 2.2.

Let U=S1×ℝ2U=S^{1}\times\mathbb{R}^{2} and let VV be a Green’s function with exactly one pole on S1×{0}S^{1}\times\{0\}. In [GW00] VV is constructed by passing to the universal cover U~=ℝ3\widetilde{U}=\mathbb{R}^{3}, where the lifted function V~\widetilde{V} is a periodic Green’s function constructed using a Weierstrass series. VV is only positive in a certain bounded open set in UU. The corresponding hyperkähler metric on this bounded open set is called the Ooguri-Vafa metric. With one particular choice of a compatible complex structure, 𝔐\mathfrak{M} becomes a holomorphic elliptic fibration over a disc 𝔻⊂ℝ2=ℂ\mathbb{D}\subset\mathbb{R}^{2}=\mathbb{C}, and the singular fiber has monodromy of type I1I_{1}. The Ooguri-Vafa metric plays a crucial role in the work of Gross-Wilson [GW00] on collapsing Calabi-Yau metrics on elliptic K3⁡3\K 3 surfaces with exactly 24 singular fibers of type I1I_{1}.

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