Example 2.2 . [03GE]
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Example 2.2.
Let and let be a Green’s function with exactly one pole on . In [GW00] is constructed by passing to the universal cover , where the lifted function is a periodic Green’s function constructed using a Weierstrass series. is only positive in a certain bounded open set in . The corresponding hyperkähler metric on this bounded open set is called the Ooguri-Vafa metric. With one particular choice of a compatible complex structure, becomes a holomorphic elliptic fibration over a disc , and the singular fiber has monodromy of type . The Ooguri-Vafa metric plays a crucial role in the work of Gross-Wilson [GW00] on collapsing Calabi-Yau metrics on elliptic surfaces with exactly 24 singular fibers of type .