A Gluing Construction for Calabi-Yau Metrics of Kummer Surfaces

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Ebrahimi, Mohammad Amin | 2020

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 52865 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Esfahanizadeh, Mostafa; Seyyed Ali, Reza
Abstract:
let T4 = C2/ be a complex torus with induced flat Kähler metic. The quotient X = T4/Z2 (where the generator of Z2 acts by multiplication by −1) is an orbifold with 24 singularities, called a singular Kummer Surface. Locally any of the singularities are modelled on the singularity of C2/Z2. We can describe the minimal resolution X of X by resolving all of the singularities as : TCP1 ! C2/Z2.There is a well known Calabi–Yau metric on TCP1 named the Eguchi–Hanson metric, with the important property that far from the exceptional divisor −1(0) the metric is asymptotically locally Euclidean (ALE). We identify a small annulus around any singular point of X/Z2 with a large annulus over TCP1 far from the exceptional divisor, interpolating between the scaled-down Eguchi-Hanson metric and the flat one over the annulus via a partition of unity.The newly constructed metric is an approximate Calabi–Yau metric. We can find the exact Calabi–Yau metric with a purturbation argument
Keywords:
Calabi-Yau Manifold ; Kahler Manifold ; Eguchi-Hanson Metric ; Connected Sum Custuction ; Gluing Custuction ; Kummer Surfaces