Volume of a divisor?

Question

One can define the volume of a divisor $D$ by the formula $\displaystyle \limsup_{m \to \infty} \frac{h^0(X, O(mD))}{m^n/n!}$, where $n$ is the dimension of $X$.

For example, see the definition here (a paper that I randomly chose from a google search): http://www.math.northwestern.edu/~mpopa/papers/RVLS.pdf

Does anyone know if there is a geometric meaning to this definition? ("Why volume?") The author says that this is the "top self-intersection" number, but I don't really know what this means, or what it has to do with the concept of volume (on a complex algebraic manifold).

I'm just curious to know.

Edit: probably this contains an answer to my question: http://www.ms.uky.edu/~corso/Purdue_2011/posters/fulger-poster.pdf

Answer 1

To expand a bit on msteves answer, depending on how familiar you are with intersection theory, remember $L$ is a (something like) a hypersurface, and if we intersect $n$ hypersurfaces $L_1\cdot L_2\cdots L_n$, the dimension drops one at a time (unless we're very unlucky) and the result is a set of points - we can forget their locations and think of this as a number.

Intersection theory let's us circumnavigate the above "unluckiness." Even though the set-theoretic intersection $L^n = L\cdot L\cdots L$ with itself $n$ times is just $L$, we can still interpret this as a zero-dimensional thing (a number), philosophically by intersection $n$ different slightly perturbered copies of $l$ that are not so unlucky.

In the case that $L$ is nef, the volume as you defined turns out to agree with this number. The question remains: why should $L^n$ be related with volume?

One explanation is that a dimension $k$ subvariety can be thought of as a singular $(n-k)$-cycle, and by Poincare duality + de Rham theory these are related to differential $(n-k)$-forms. Keep intersecting your divisor and you go from codimension 1 right down to codimension n, and hence from 1-forms right up to $n$-forms. Another name for an $n$-form is a "volume form" since one integrates it over a volume.

In other words, $L$ can be thought of (under this intersection->singular->de Rham analogy/correspondence) as related to some $1$-form, and $\text{vol}(L)$ is the $n$-form that we get by wedging with itself $n$ times.

Answer 2

If $L$ is a nef line bundle on a smooth projective complex variety $X$ of dimension $n$, then the volume equals the top self-intersection $(L^n)$, which in turn is the integral of the volume form $c_1(L)^n$, hence the name.