In Kollár and Mori's Birational Geometry of Algebraic Varieties, the authors say a Cartier divisor is big iff its birational pullback is big (Definition 2.59 below). But I can't understand. Maybe the varieties are proper by definition of big divisor.
And I know when the varieties are integral normal and proper, this can be done by Zariski main theorem and projectiive formula. Are conditions 'integral' and 'normal' necessary?
Thank you for any answer or comments.
In this book Kollar and Mori define big divisors only for proper (irreducible) varieties, so when they say that for a birational morphism $f:X\to Y$, the pullback $f^*D$ of a divisor $D$ on $Y$ is big if and only if $D$ is big, they probably mean $X$ and $Y$ are proper.
To see that $D$ is big if and only if $f^*D$ is, you can use Lemma 2.60. This says in particular that a Cartier divisor $E$ on a variety $Z$ is big if, and only if, the rational map $\phi_{kE}$ defined by some multiple of it has image of dimension equal to $\dim Z$. The point is that $X$ and $Y$ have isomorphic dense open subsets and the maps $\phi_{kD}$ and $\phi_{kf^*D}$ coincide on them.
I guess volume can also be defined on a compact kahler manifold according to Lazarsfelds book 2.2.53. On the other hand, if the manifold is not compact, then it might not be possible to always get an actual number for the volume (although I don't know any examples off the top of my head) (and D is big iff volume > 0, so you want the volume to be well-defined first).
