Projective embedding of pairs and logarithmic K-stability

TL;DR

This paper investigates the projective embedding and K-stability of certain complex manifolds with line bundles, showing conditions under which the embedding is almost balanced and the manifold is K-semistable.

Contribution

It demonstrates that under specific geometric conditions, the Kodaira embedding becomes almost balanced and the manifold exhibits K-semistability, linking metric properties to stability.

Findings

Kodaira embedding is almost balanced under given conditions.

The manifold $(uildrelrown elax L, D, cA, 0)$ is K-semistable.

Constructs complete CSCK metrics on $uildrelrown elax L ackslash D$.

Abstract

Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSCK metric on $D$, we show that the Kodaira embedding induced by orthonormal basis of the Bergman space of $kA$ is almost balanced. As a corollary, $(\hat{L},D,cA,0)$ is K-semistable.