Kodaira dimension

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