Some Remarks on Conic Degeneration and Bending of Poincaré-einstein Metrics

Let (M, g) be a compact Kähler-Einstein manifold with c1 > 0. Denote by K → M the canonical line-bundle, with total space X, and X0 the singular space obtained by blowing down X along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincaré–Einstein metrics on X. One 1-parameter subfamily {gt}t>0 has the property that as t ց 0, gt converges to a PE metric g0 on X0 with conic singularity, while t −1gt converges to a complete Ricci-flat Kähler metric ĝ0 on X. Another 1-parameter subfamily has an edge singularity along the zero section of X, with cone angle depending on the parameter, but has constant conformal infinity. These illustrate some unexpected features of the Poincaré-Einstein moduli space.