BBD:perverse Sheaf And Research Of Semi-small Maps Between Quiver Varieties

One of the three important parts of BBD is about the classification of simple perverse sheaf in the framework of category and in the framework of seperated scheme.In this master’s thesis,we have systematically discussed this part of BBD’s article,carefully stated the classification of simple per-verse sheaf,the theorem of construction of simple perverse sheaf of Deligne,and completed some proofs which are not obvious and also some details are added.In order to prove fundamental lemma and to give a geometric proof of decomposition theorem,Ng?.B.C and de Cataldo.M.A.A give some new characterizations of Beilinson,Bernstein,Deligne and Gabber[BBD]’s de-composition theorem.We use thoes results,by constructing some small maps,to give support of simple perverse sheaves corresponding to the canon-ical elements of power of root vectors in[6],which is πv!((?))[dim(?)v]=IC-(?)nP[dim(?)np],where P is a indecomposable representation of symmetric quiver of finite type Q。Furthermore,we use decomposition theorem by constructing certain sequence of dimension vector to give support of canon-ical bases of positive part of symmetric quantum group of finite type.