Given an -dimensional smooth
projective variety , we can study the canonical line bundle
of holomorphic -forms. The dimensions of the spaces of global
sections of
are useful birational invariants of
which aid in the classification of varieties (birational means
they only depend on a Zariski-open subset of ). As
, these numbers either behave asymptotically like
for a unique integer or are eventually zero. We define the
Kodaira dimension to be this integer in the first case and
in the second case.
Another intepretation is as follows. For each , we have a rational
map of into projective space given by

where
are the global sections of
.
The Kodaira dimension is the supremum, as
, of the
dimension of the image of under these maps. Hence the Kodaira
dimension of takes values in
.

Jeffrey Herschel Giansiracusa
2005-06-27