= Kodaira dimension
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{title2=$\kappa(X)$}
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For a smooth <projective variety> $X$ over the <complex numbers>, its <plurigenera> are $P_m(X)=\dim H^0(X,K_X^{\otimes m})$, where $K_X$ is the <canonical line bundle of a smooth variety>. If every $P_m$ vanishes for $m>0$, define $\kappa(X)=-\infty$. Otherwise, $\kappa(X)$ is the largest <dimension> of the <image> of a <rational map> defined by a <complete linear system of a divisor>. Equivalently, there are positive constants $a,b$ such that
$$
a m^{\kappa(X)}\leq P_m(X)\leq b m^{\kappa(X)}
$$
for all sufficiently large, sufficiently divisible $m$. Restricting to divisible $m$ matters because some <plurigenera> can vanish periodically.<Kodaira dimension of algebraic varieties by Mihnea Popa>[Sections 1.2–1.3, Definition 1.2.1, Proposition 1.2.4 and Definition 1.3.1]{ref}
For a <smooth projective curve>, <genus> zero, one and at least two give $\kappa=-\infty,0,1$, respectively. For <smooth projective surfaces>, $\mathbb P^2$, a product of two <elliptic curves>, the product of an <elliptic curve> with a <curve> of <genus> at least two, and the product of two such higher-<genus> <curves> realize $-\infty,0,1,2$. The <plurigenera>, hence $\kappa$, are invariant under <birational> between smooth <projective varieties>, making this growth rate useful for classification.<Kodaira dimension of algebraic varieties by Mihnea Popa>[Section 1.1, birational invariance of plurigenera; Section 1.3, Examples 1.3.2–1.3.7]{ref}
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