In this paper, we obtain optimal L 2 extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular .where c is a nonnegative number, g j ∈ O X (U ) and u ∈ C ∞ (U ).Definition 1.2. If ψ is a quasi-plurisubharmonic function on a complex manifold X, the multiplier ideal sheaf I(ψ) is the coherent analytic subsheaf of O X defined by+∞) and R(+∞) := lim t→+∞ R(t) ∈ (0, +∞] in the above definition of R α0,α1 .Remark 1.3. The number α 0 , α 1 and the function R(t) are equal to the number A, 1 δ and the function 1 cA(−t)e t defined just before the main theorems in [14]. If α 0 = +∞ and R is decreasing on (−∞, α 0 ], the longest inequality in the definition of R α0,α1 holds for all t ∈ (−∞, α 0 ). If α 0 = +∞, the longest inequality in the definition of R α0,α1 implies that +∞ t α1 R(+∞) dt 2 < +∞ for all t ∈ (−∞, +∞). Therefore, α1 R(+∞) = 0, i.e., α 1 = 0 or R(+∞) = +∞.
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