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Smoothing toroidal crossing spaces. (English) Zbl 1483.14067

The paper proves the existence of smoothing for a proper normal crossing space with effective anticanonical class, globally generated \(\mathcal{T}^1_X := \mathrm{Ext}^1(\Omega_X,\mathcal{O}_X)\), and projective singular part \(X_{\mathrm{sing}}\) (Theorem 1.1). It allows non-toric components in the central fiber, compared tot he Gross-Siebert program of toric degenerations.
The main tool of the paper is log structure. It equips \(X\) with log structure, and employs elementary log toroidal local models to show the Hodge-de Rham degeneration, which is a key ingredient for the smoothing of \(X\). In showing unobstructedness of log deformations, the paper uses recent advancements of Bogomolov-Tian-Todorov theory for degenerate Calabi-Yau varieties motivated from mirror symmetry.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G05 Deformations of complex structures
14J45 Fano varieties
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