Siu’s lemma, optimal $$L^2$$ extension and applications to twisted pluricanonical sheaves

A brief introduction of several complex variables

“…利用我们解 决强开性猜想的思想与工作, 朱朗峰和作者 [50] 发现了广义的萧荫堂引理. 基于弱拟凸 Kähler 流形上具奇异度量线丛的最优 L 2 延拓定理、广义的萧荫堂引理和乘子理想 层的强开性等, 朱朗峰和作者 [51] 得到了 Kähler 纤维化时扭化相对多典则丛 (twisted relative pluricanonical bundles) 及其直接像 (direct images) 的正性, 该结果在代数纤维空间情形由文献 [52,53] 获 得. 朱朗峰和作者 [51] 还解决了文献 [52,53] 两次提及的一个猜想.…”

“…基于弱拟凸 Kähler 流形上具奇异度量线丛的最优 L 2 延拓定理、广义的萧荫堂引理和乘子理想 层的强开性等, 朱朗峰和作者 [51] 得到了 Kähler 纤维化时扭化相对多典则丛 (twisted relative pluricanonical bundles) 及其直接像 (direct images) 的正性, 该结果在代数纤维空间情形由文献 [52,53] 获 得. 朱朗峰和作者 [51] 还解决了文献 [52,53] 两次提及的一个猜想. 在文献 [54,55] 中, 我们结合 L 2 方法, 在几何不变量理论框架下建立了比 Kiselman-Berndtsson (均为瑞典皇家科学院院士) 更广的关于多次调和函数的极小原理.…”

L2 Extensions with Singular Metrics on Kähler Manifolds

“…Our proof of the generalized Siu's lemma is based on the strong openness property of multiplier ideal sheaves (conjectured by Demailly in [12] and [13] and by many others, and solved in [21]). The reader is referred to [43] for further studies on the generalized Siu's lemma. These studies on the generalized Siu's lemma constitute a key element of the proofs of our main results outlined in the present paper, being combined with many known ideas, results and techniques.…”

“…In [4] (see also [5]), Berndtsson and Pȃun established an iteration method to obtain L q extension theorems (0 < q < 2) from L 2 extension theorems, where the constants in the final estimates are consistent. One can use this iteration method to obtain optimal L q extension theorems (0 < q < 2) with singular metrics in the Kähler case from optimal L 2 extension theorems with singular metrics in the Kähler case (the reader is referred to [20] for projective cases, and to [7,41,43] for general Kähler cases).…”

“…The positivity of the direct images of twisted relative canonical bundles with multiplier ideal sheaves for projective fibrations was proved in [23] by using the optimal L 2 extension theorem and its geometric meaning. The positivity of twisted relative pluricanonical bundles and their direct images with singular metrics in the Kähler case was discussed in [7,41] and [43] with the applications of optimal L q (0 < q ≤ 2) extension theorems with singular metrics in the Kähler case.…”