Vanishing Theorems In Complex Geometry

The vanishing theorems for holomorphic vector bundles on complex manifolds play a very important role in complex and algebraic geometric.The vanishing theorem can be traced back to the Kodaira vanishing theorem in 1953,but its root cause can be attributed to the result of Riemann and Roch on the curve or Picard's case on the surface.After Kodaira,many mathematicians made various extensions to the Kodaira vanishing theorem.In complex geometry,there are mainly three ways to generalize the vanishing theorem.The first aspect is the difference of manifolds,the second is the positive condition of vector bundles,and the third is the difference of vector bundles.This article summarizes the results and progress of the vanishing theorem.The first chapter of this paper is an introduction to some of the basic knowledge in complex geometry and algebraic geometry.The vanishing theorem is introduced in three parts.The first part describes the classical vanishing theorem in chronological order.In this section we also introduce the Bochner-Kodaira-Nakano identities and use it to prove some of the vanishing theorems.The second part is to study the vanishing theorem on vector bundles,which is also the main part of this paper.In the first part we introduce several kinds of definitions of positive conditions and their relationships.Then we introduce the vanishing theorems of vector bundles under different positive conditions under three different underlying manifolds.For the case of special vector bundles,we study two cases.One is the adjiont vector bundles,that is,the tensor product and the symmetric power of vector bundles.The other is the case of bounded vector bundles.In the third part we introduce the vanishing theorem of the sections of vector bundles.In this section we study two situations,one is the case of compact Kahler manifolds,and the other is the case of Higgs bundles.In this paper,we study the vanishing theorem under various positive conditions on different manifolds and combine the recent results to get a more comprehensive summary.