UC Berkeley

In this thesis, we study constant scalar curvature Kähler metrics of Poincaré type. We give constructions of new examples in the following setting.• Consider a compact Kähler manifold admitting a constant scalar curvature Kähler metric and with no nontrivial holomorphic vector fields. After blowing up the manifold at finitely many points, we prove the existence of constant scalar curvature Kähler metrics on the complement of the exceptional divisors, with Poincaré-type singularities along them.• Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat Kähler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact Kähler surfaces. We also provide an alternative construction of toric scalar-flat Kähler metrics with conical singularity along the toric divisor, following the approach of Abreu and Sena-Dias.We also make progress on the construction of a degenerate scalar-flat Kähler neck region in complex dimension two, motivated by degeneration problems in Calabi-Yau and negative Kähler-Einstein metrics. The neck region exhibits Poincaré-type singularities at both ends and may serve as a candidate local model for interpolating between appropriately chosen geometric ends.