Constant scalar curvature Kaehler metrics and the stabilty of polarized algebraic manifolds in the geometric invariant theory

我们研究了几何不变量理论中极化代数流形的稳定性等价于该流形上存在常数量曲率Kaehler度量的猜想，即“极化代数流形上的Kobayashi-Hitchin对应”，特别是从K-能量的角度观察到了Kaehler-Ricci孤子，它是Einstein-Kaehler度量的推广，在Einstein-Kaehler度规的研究中起着重要的作用，我们可以将Kaehler-Ricci孤子推广到一般Kaehler类的情形。此外，我们还研究了这种推广的Kaehler-Ricci孤子的非平凡例子的构造。然后，我们证明了在复射影直线上的全纯线丛的紧化下，对于纤维体积比O截面足够小的Kaehler类，广义Kaehler-Ricci孤子的存在性。

We have studied the conjecture which states that the stability of polarized algebraic manifolds in the geometric invariant theory is equivalent to the existence of constant scalar curvature Kaehler metrics on that manifolds, namely "Kobayashi-Hitchin correspondence for polarized algebraic manifolds".In particular, we observed Kaehler-Ricci solitons, which are an generalization of Einstein-Kaehler metrics, from the view points of K-energy, which plays an important role in the research of Einstein-Kaehler metrics, and we could generalize Kaehler-Ricci solitons to the case of general Kaehler classes.Moreover, we have studied the constraction of non-trival examples of this generalized Kaehler-Ricci solitons. Then, we could prove the existence of generalized Kaehler-Ricci solitons for the Kaehler classes, for which the volume of fibers is sufficiently small compare to that of O-section, on the compactification of the holomorphic line bundle over the complex projective line.