We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

中文翻译：

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平滑环形交叉空间

我们在温和的假设下证明了环形交叉空间的平滑存在。通过将原木结构与无限小的变形联系起来，结果获得了一个非常紧凑的正常交叉空间形式。主要方法是研究在余维数为 2 的子空间上不连贯的对数结构，并证明此类对数空间的 Hodge-de Rham 退化定理也解决了 Danilov 的猜想。我们证明了 Maurer-Cartan 解和变形之间的同伦等价性结合 Batalin-Vilkovisky 理论可用于获得平滑。在模空间上构建新的 Calabi-Yau 和 Fano 流形以及 Frobenius 流形结构提供了潜在的应用。