極化流形的Gieseker-Mumford穩定性之幾何準則報告

Abstract

本篇論文主要是研讀羅華章博士在1997年的論文，對其內容與證明進行詳細的研讀，並且提出四個陳述，對其做出證明。 首先，羅介紹了極化流形以及它的希爾伯特點。藉著幾何不變理論中的對於希爾伯特點的穩定性定義了極化流形在幾何不變理論下的穩定性。藉此他給出穩定性的性質。我們在這邊證明了我們第一個陳述。 再來，使用微分幾何的方法，羅改進了上面的性質。藉由格林流的定義，給出了更廣的Gieseker-Mumford穩定性的性質。在這邊我們證明了兩個陳述。 最後，透過上述的分析，羅證明了他最後一個定理。在這邊我們證明了最後一個陳述以及對於羅的定理我們做了一點修改得到一個關於Gieseker-Mumford穩定性的幾何準則。

This paper is to study Luo’s paper in 1997. We give four statements with their proofs. Firstly, Luo introduce the polarized manifold and its Hilbert point. By the stability of Hilbert points in the Geometric Invariant Theory, he defined the stability of polarized manifolds in the Geometric Invariant Theory; hence he gave the proposition for the stability. We prove our first statement. Secondly, Luo use the differential geometric method to reduce the proposition. By the definition of Green current, it gave the extended proposition for the Gieseker-Mumford stability, which is the first main theorem . Here we prove two statements. Finally, use the above analysis, Luo proved the last theorem. We prove our final statement and do a slight improvement to give the geometric criterion for the Gieseker-Mumford stability.