We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kov\'acs-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the "local vanishing conjecture" proposed by Musta\c{t}\u{a}, Olano, and Popa.

中文翻译：

---

将全纯形式从复杂空间的规则轨迹扩展到奇点的解析

我们研究在什么条件下，定义在简化复空间的规则轨迹上的全纯形式扩展到奇点分辨率上的全纯（或对数）形式。我们为此给出了一个简单的充要条件，其证明依赖于分解定理和斋藤混合霍奇模理论。我们用它来将 Greb-Kebekus-Kov\'acs-Peternell 定理推广到具有有理奇点的复空间，并证明这些空间上自反微分的函子回拉的存在。我们还使用我们的方法解决了 Musta\c{t}\u{a}、Olano 和 Popa 提出的“局部消失猜想”。