Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties

Mirror Symmetry via Logarithmic Degeneration Data I

“…Again, we hope in the simple case that a log Calabi-Yau space is always the central fibre of a toric degeneration. The Bogomolov-Tian-Todorov type results of [29] give some moral support that such should be true. However, the presence of the singular set means we cannot apply known results off the shelf.…”

“…B, P, s)). One can show [12], [29] that LS X 0 (B,P,s) is the O × D -torsor associated to the line bundle Ext P,s) ), the local T 1 -sheaf. There exists a log smooth structure on X 0 (B, P, s) if and only if this is the trivial line bundle.…”

“…, [29], [27]). So the existence of a log smooth structure restricts not only the type of singularities of X, but poses also some more subtle global analytical conditions (here the triviality of a locally free sheaf over X sing ).…”

“…(2) To turn these ideas into a genuine mirror symmetry construction, we need to study the log deformation theory of log Calabi-Yau spaces,à la Kato [27] and KawamataNamikawa [29]. Again, we hope in the simple case that a log Calabi-Yau space is always the central fibre of a toric degeneration.…”

Weak Coupling, Degeneration and Log Calabi-Yau Spaces

“…Finally when Y 0 has a log structure and satisfies other mild hypotheses, we can deform Y 0 to a smooth global model Y t , where t ∈ C is a parameter, by the fundamental smoothing theorem of Kawamata and Namikawa [8]. Obviously this construction comes with a small parameter and a built-in degeneration limit.…”