Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $X' \longrightarrow X$, \'{e}tale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that the standard tangent bundle of such a cover is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the quotient of the ball we consider.

中文翻译：

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具有尖点的复双曲流形上的对称微分

令 $(X, D)$ 为对数对，令 $h$ 为切线丛上的奇异度量，在 $X$ 的开部分上平滑。我们给出了关于 $h$ 曲率的充分条件，以使对数丛和标准余切丛变大。作为一个应用程序，我们给出了一个有界对称域的任何环形紧化上的对数余切丛大小的度量证明。然后，我们使用这种奇异度量方法来研究在更具体的球情况下标准切丛的大小和净度。我们获得了边界外的封面 $X' \longrightarrow X$, \'{e}tale 的有效分支命令，使其所有子变种都具有大余切束。我们还证明，如果分枝足够高，这种覆盖的标准切丛是 nef。而且，