Log Toroidal Families
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Abstract
We study families of logarithmic varieties with mild singularities, the log toroidal families. They generalize and unify various classes of spaces with controlled singularities, including toroidal varieties, toroidal embeddings, semistable degenerations, log smooth morphisms, and toric log Calabi–Yau spaces. Starting from Kato’s toroidal characterization of log smoothness and Gross–Siebert’s local models for the singularities of a toric log Calabi–
Yau space, we construct elementary log toroidal families from combinatorial data as étale local models for the singularities which we allow in a log toroidal family. We study the reflexive de Rham complex W*_{X/S} of a log toroidal family and prove the Hodge–de Rham degeneration for proper log toroidal families over a log point S = Spec (Q → k). This in particular settles a conjecture of Danilov on the cohomology of toroidal pairs (X, D). This
thesis is an expanded version of the article "Smoothing Toroidal Crossing Spaces" (attached in the file), where we prove the Hodge–de Rham degen-
eration and apply it to obtain a smoothing of a normal crossing space as well as a toroidal crossing space.