Analytical Approach To Multiple Ideal Sheaves

Multiple ideal sheaf is a very powerful tool in mathematics. It can be s-tudied from either algebraic view point, or analytical one. Especially, with the remarkable work of Siu, J-P.Demaiily and so forth, multiple ideal sheaf has thrown strong light on problems in complex geometry, particularly on the ampleness of some specific line bundles.This survey focus on the analytical approach to the basic concepts and theorems on multiple ideal sheaves, and we use two different methods to prove the high-order vanishing theorem of its cohomology groups.In the third chapter, in terms of the tools that we have developed in the first two chapters, we give three applications:1.a criterion to the existence of Kahler-Einstein metric on some Fano varieties,2.a relatively concise proof to Ko-daira Embedding Theorem,3.following Siu’s paper, we prove a theorem related to Fujita Conjecture.And in the last Chapter, we discuss Openness Conjecture as an example of recent work on multiple ideal sheaf and application of L2estimate.This survey is a summary and collection of what the author has learned dur-ing his master’s study.