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.
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.
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
, which likens X with this extra structure to a variety with boundary (corresponding to
A pre-log structure on X consists of a sheaf of (commutative) monoids
is considered as a monoid under multiplication of functions.
A morphism of (pre-)log structures consists in a homomorphism of sheaves of monoids commuting with the associated homomorphisms into
One application of log structures is the ability to define logarithmic forms (also called differential forms with log poles) on any log scheme.
From this, one can for instance define log-smoothness and log-étaleness, generalizing the notions of smooth morphisms and étale morphisms.
In addition, log structures serve to define the mixed Hodge structure on any smooth complex variety X, by taking a compactification with boundary a normal crossings divisor D, and writing down the corresponding logarithmic de Rham complex.
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.