论文标题
$ l^p $ - 条件的$ \ bar \ partial $的阳性矢量捆绑包的积极性
Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$
论文作者
论文摘要
我们根据$ \ bar \ partial $和$ l^p $ extensions of Holomorphic对象的$ l^p $估计，研究了Hermitian（甚至Finsler）Holomorthic Vector束的积极性。为此，我们介绍了四个条件，称为最佳$ l^p $ - estimate条件，多个粗糙的$ l^p $ - estimate条件，最佳$ l^p $ - extension条件和多个粗$ l^p $ - extension条件，用于Hermitian（或Finsler）矢量bundle $（e，h）$。本文的主要结果是通过最佳$ l^2 $ estimate条件对$（e，h）$的nakano积极性进行表征。我们还表明，如果$（e，h）$满足某些$ p> 1 $的多个粗$ l^p $ esstimate条件，则是Griffiths正面的，最佳$ l^p $ - 扩展条件或多个粗$ l^p $ - extension条件对于某些$ p> 0 $。可以将这些结果粗略地视为Hörmander的$ l^2 $ - bar \ bar \ partial $的对话 和ohsawa-takegoshi型扩展定理。作为主要结果的应用，我们获得了一种完全不同的方法，可以通过与复杂歧管的霍明型家族相关的扭曲的相对规范束的直接图像滑轮阳性。
We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condition. We also show that $(E,h)$ is Griffiths positive if it satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, the optimal $L^p$-extension condition, or the multiple coarse $L^p$-extension condition for some $p>0$. These results can be roughly viewed as converses of Hörmander's $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type extension theorems. As an application of the main result, we get a totally different method to Nakano positivity of direct image sheaves of twisted relative canonical bundles associated to holomorphic families of complex manifolds.
