Extensions of Multiply Twisted Pluri-Canonical Forms

Given a projective variety X, a smooth divisor D, and semipositive line bundles (L1;h1);:::; (Lm;hm), we consider the \multiply twisted pluricanonical bundle" F := Om i=1 (KX +D +Li) on X and FD := Om i=1 (KD +LijD). Let Ij be the multiplier ideal sheaves associated to hjjD, j = 1;:::;m. We show that, under a certain conditions on curvature, H 0 (D;FD I1I2 Im) lies in the image of the restriction map H 0 (X;F )! H 0 (D;FD). Our result is inspired by Siu's proof of invariance of plurigenera and its simplication made by P aun. We emphasize its nature as a multiple version of extension theorem of Ohsawa-Takegoshi type (the case with m = 1), whose proof is also given here in detail.