A proof that toroidal crossing spaces with a simple logarithmic section and a transverse anticanonical divisor admit smoothings, together with a proof of Danilov's Hodge-de Rham degeneration conjecture.
Local uniqueness of approximations and finite determinacy of log morphisms
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abstract
I prove a local finite determinacy result of singular fibres in families. As an application, I generalize finite determinacy results on the fibres of log morphisms and nearby and vanishing cycles sheaves by Illusie and Kisin.
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Smoothing toroidal crossing spaces
A proof that toroidal crossing spaces with a simple logarithmic section and a transverse anticanonical divisor admit smoothings, together with a proof of Danilov's Hodge-de Rham degeneration conjecture.