Extensions of multiply twisted pluri-canonical forms

TL;DR

This paper extends the theory of pluricanonical forms on projective varieties by establishing conditions under which sections on a divisor extend to the ambient variety, using advanced extension theorems and multiplier ideal sheaves.

Contribution

It introduces a new extension theorem for multiply twisted pluricanonical bundles, generalizing previous results and incorporating a modified induction construction and a detailed Ohsawa-Takegoshi type extension proof.

Findings

Sections on divisor D extend to X under curvature conditions.

Multiplier ideal sheaves are shown to contain the product of individual ideals.

The method generalizes previous invariance of plurigenera results.

Abstract

Given a projective variety X, a smooth divisor D, and semipositive line bundles (L_1,h_1),,...,(L_m,h_m), we consider the "multiply twisted pluricanonical bundle" F:=m(K_X+D)+L_1+...+L_m on X and F_D:=mK_D+(L_1+...+L_m)|_D. Let I_j be the multiplier ideal sheaves associated to h_j, j=1,...,m. We show that, under a certain conditions on curvature, H^0(D,F_D\otimes I_1I_2...I_m) lies in the image of the restriction map H^0(X,F)->H^0(D,F_D). The format of our result is inspired both by Paun's simplification of Siu's proof of invariance of plurigenera and an earlier similar result due to Demailly. The main ingredient is a modification of Siu-Paun's induction construction and an extension theorem of Ohsawa-Takegoshi type (O-T). We also include a detail proof of O-T. The key feature is that the ideal sheaf we use is the product of the multiplier ideals associated to the singular metrics h_1,...,h_m, which contains the multiplier ideal sheaf of the product of the metrics h_1\otimes...\otimes h_m.