Abstract: Kodaira maps provide a natural sequence of embeddings of a given complex projective manifold X into projective spaces associated with the dual of the space of holomorphic sections of Lk, H0(X; Lk), k 2 N. A theorem of Tian states that for any positive metric on the line bundle these embeddings become almost isometries (after a proper rescaling) when X is endowed with the Kähler metric induced by the metric on L and the projective spaces are endowed with Fubini-Study metrics associated with the induced L2-norms on H0(X; Lk). This theorem is related with the asymptotic expansion of Bergman kernel, which has been studied by Yau, Bouche, Ruan, Catlin, Zelditch, Lu, Wang, Dai-Liu-Ma and many others. In this lecture, we explain the semiclassical extension theorem, generalizing Tian’s result. More precisely, Ohsawa-Takegoshi gave a condition under which a holomorphic section of a vector bundle on a submanifold extends to a holomorphic section over an ambient manifold with a reasonable bound on the L2-norm of the extension in terms of the L2-norm of the section. For a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal L2-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic formula for this operator.