Kodaira dimension

Given an $ n$-dimensional smooth projective variety $ X$, we can study the canonical line bundle $ K_X$ of holomorphic $ n$-forms. The dimensions of the spaces of global sections of $ K_{X}^{\otimes r}$ are useful birational invariants of $ X$ which aid in the classification of varieties (birational means they only depend on a Zariski-open subset of $ X$). As $ r \rightarrow
\infty$, these numbers either behave asymptotically like $ C\cdot
r^{k}$ for a unique integer $ k$ or are eventually zero. We define the Kodaira dimension to be this integer $ k$ in the first case and $ -\infty$ in the second case.

Another intepretation is as follows. For each $ r$, we have a rational map of $ X$ into projective space given by

$\displaystyle [s_0, \ldots, s_m]: X \dashrightarrow \mathbb{P}^n

where $ s_0,
\ldots, s_n$ are the global sections of $ K_X^{\otimes r}$. The Kodaira dimension is the supremum, as $ r \to \infty$, of the dimension of the image of $ X$ under these maps. Hence the Kodaira dimension of $ X$ takes values in $ \{-\infty,0,\dots,n\}$.

Jeffrey Herschel Giansiracusa 2005-06-27