The logarithmic Kodaira dimension 
of a smooth quasiprojective
variety 
over a field of characteristic
zero is the Iitaka dimension of 
on a smooth projective compactification 
with reduced simple normal crossing boundary 
. It is independent of such a compactification. The boundary
allows logarithmic poles in pluricanonical forms. For example, 
, while 
and 
.
For a projective surjective morphism 
with connected fibers between smooth connected complex
quasiprojective varieties, let 
be the smooth projective geometric generic fiber. The variation
is the least transcendence degree over 
of an algebraically closed subfield of 
over which 
has a birational model. It records the whole fiber function
field, rather than only its canonical model.
