Log structure

In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others.
Motivation.
The idea is to study some algebraic variety (or scheme) "U" which is smooth but not necessarily proper by embedding it into "X", which is proper, and then looking at certain sheaves on "X". The problem is that the subsheaf of formula_1 consisting of functions whose restriction to "U" is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of formula_2, multiplicatively. Remembering this additional structure on "X" corresponds to somehow remembering the inclusion formula_3, which likens "X" with this extra structure to a variety with boundary (corresponding to formula_4).
Definition.
Let "X" be a scheme. A pre-log structure on "X" consists of a sheaf of (commutative) monoids formula_5 on "X" together with a homomorphism of monoids formula_6, where formula_2 is considered as a monoid under multiplication of functions.
A pre-log structure formula_8 is a log structure if in addition formula_9 induces an isomorphism formula_10.
A morphism of (pre-)log structures consists in a homomorphism of sheaves of monoids commuting with the associated homomorphisms into formula_11.
A log scheme is simply a scheme furnished with a log structure.
Applications.
One application of log structures is the ability to define logarithmic forms on any log scheme. From this, one can for instance define corresponding notions of log-smoothness and log-étaleness which parallel the usual definitions for schemes. This then allows the study of deformation theory.
In addition, log structures serve to define the mixed Hodge structure on any smooth variety "X", by taking a compactification with boundary a normal crossings divisor "D", and writing down the Hodge–De Rham complex associated to "X" with the standard log structure defined by "D".
Log objects also naturally appear as the objects at the boundary of moduli spaces, i.e. from degenerations.
Log geometry also allows the definition of log-crystalline cohomology, an analogue of crystalline cohomology which has good behaviour for varieties that are not necessarily smooth, only log smooth. This then has application to the theory of Galois representations, and particularly semistable Galois representations.