Projective embedding of degenerating family of K\"ahler-Einstein manifolds of negative curvature

TL;DR

This paper investigates how Bergman embeddings of degenerating Kähler-Einstein manifolds with negative curvature behave, showing convergence to the limit space's embedding and bubbles in a special case.

Contribution

It demonstrates the convergence of Bergman embeddings for degenerating families of Kähler-Einstein manifolds, including the construction of orthonormal bases in a specific scenario.

Findings

Bergman embeddings converge to the limit space's embedding.

Construction of orthonormal bases in a special case.

Convergence includes bubbles in the limit.

Abstract

We study the Bergman embeddings of degenerating families of K\"{a}hler-Einsten manifolds of negative curvature. In one special case, we show that we can construct orthonormal bases so that the induced Bergman embeddings converge to the Bergman embedding of the limit space together with bubbles.