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  1. Abstract

    We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity.Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry.

     
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  2. Abstract

    Let$$X\rightarrow {{\mathbb {P}}}^1$$XP1be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$ωiof Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$ωi. Given the corresponding sequence$$\Xi _i$$Ξiof Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$Ξi|Econverges to a flat connection$$A_0$$A0. Furthermore, if the restriction$$V|_E$$V|Eis of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$j=1nOE(qj-0)forndistinct points$$q_j\in E$$qjE, then these points uniquely determine$$A_0$$A0.

     
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