L~2 Extension Theorem For Jets With Variable Denominators And An Analytic Proof For A Case Of The Unobstructed Lograthmic Deformations

L2 extension theorems are very important and have many applications in several complex variables,algebraic geometry and complex geometry.One interesting problem is to study the L2 extension theorem for jets.By studying the variable denominators introduced by X.Zhou-L.Zhu,we gener-alize the results of D.Popovici for the L2 extension theorem for jets[Po05]in the first part of this thesis.As a direct corollary,we also give a generalization of T.Ohsawa-K.Takegoshi’s classical extension theorem[OT87]to a jet version.On projective manifolds,L.Katzarkov-M.Kontsevich-T.Pantev[KKP08,Section 4.3.3(ⅰ),(ⅱ)]proved the unobstructedness of logarithmic deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by using a purely algebraic method.We can regard[KKP08,Section 4.3.3(ⅲ)]as a natural common generalization of the above two cases.In the second part of this thesis,we will give an analytic proof for the unobstructed logarithmic deformations in the case of[KKP08,Section 4.3.3(ⅲ)]for compact Kahler manifolds.