Log smooth deformation theory

Abstract

This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.

Mirror Symmetry via Logarithmic Degeneration Data I

“…The definition runs analogously to formal smoothness for schemes. For our purposes it is more instructive to use the characterization of log smooth morphisms due to K. Kato [28] Theorem 3.5, see also [27] Theorem 4.1. A model log smooth map is given by the map of log schemes induced by a morphism of monoids Q → P with finite kernel, with some subtlety in non-zero characteristic.…”

“…, [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.…”

“…The fundamental notions of log geometry are due to Fontaine, Illusie and K. Kato. Standard references are [28], [27].…”

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

“…(see in particular the remark on deformation theory and log schemes preceding the proof), we will check that our stack is log smooth, and smoothness in the (2-)category of stacks will follow formally. For the definition and properties of log deformations, see [10], and in particular Def. 8.1, Prop.…”