Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are as the supports of quotient sheaves multiplier ideal quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to question posed by Cao-Demailly-Matsumura unifies few well-known injectivity theorems. second generalizes optimizes general L 2 L^2 theorem obtained Demailly.