# Multiplication maps for big line bundles

In Birational Geometry of Algebraic Varieties, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line bundle $$L$$ on a projective variety $$X$$ of dimension $$d$$ is big if

$$\limsup_{n \to \infty } \dfrac{H^0(X,L^n)}{n^d} \neq 0.$$

In other words, the rate of growth of the spaces of global sections is as big as possible. Big line bundles tend to exhibit behavior analogous to ample line bundles. I will give a couple of examples. In what follows, let $$X$$ be a variety over the complex numbers and let $$L$$ be a line bundle on $$X$$.

1. Suppose $$X$$ is normal. If $$L$$ is ample, some power of $$L$$ defines an embedding in a projective space. Analogously, if $$L$$ is big, some power of $$L$$ defines a map

$$\varphi_m: X \dashrightarrow H^0(X,L^m)$$

that is birational onto its image (Positivity in Algebraic Geometry I, page 139).

1. If $$L$$ is ample, some power of $$L$$ is globally generated. On the other hand, if $$L$$ is big, some positive power of $$L$$ is generically globally generated; that is, the natural map

$$H^0(X,L^m) \otimes \mathcal{O}_{X} \rightarrow L^m$$

is generically surjective (Positivity in Algebraic Geometry I, page 141).

Now, to get to my question, recall that if $$L$$ is ample, there exists a natural number $$m$$ such that the multiplication maps

$$H^0(X,L^a) \otimes H^0(X,L^b) \rightarrow H^0(X,L^{a+b})$$

are surjective for $$a,b \geq m$$ (Positivity in Algebraic Geometry I, page 32).

Question: Do big line bundles have a property analogous to the surjectivity of multiplications maps?

It is not clear to me what this property should be, but I would hope that these multiplication maps have eventually high rank in some suitable sense.

If $$R(L)=\oplus H^0(mL)$$ is not finitely generated, the above surjectivity will fail, however it will hold "asymptotically" for any big line bundle $$L$$. In fact, by Fujita's approximation of big classes (see eg. Lazarsfeld's Positivity book Theorem 11.4.4), for any $$\epsilon >0$$ there is a birational modification $$f:X'\to X$$ such that $$f^*L=A+E$$ where $$A$$ is an ample $$\mathbb Q$$-divisor and $$E$$ is an effective $$\mathbb Q$$-divisor such that $${\rm vol}(A)>{\rm vol}(L)-\epsilon$$. Thus, by the ample case, there is an $$m>0$$ such that $$H^0(aA)\otimes H^0(bA)\to H^0((a+b)A)$$ is surjective for all $$a,b\geq m$$ sufficiently divisible (so that $$aA$$ and $$bA$$ are Cartier). Since $$f_*\mathcal O _{X'}(aA)\subset \mathcal O _X(aL)$$, we see that if $$V_{a,b}={\rm Im} \left( H^0(aL)\otimes H^0(bL)\to H^0((a+b)L))\right),$$ then $$\dim V_{a,b}/h^0((a+b)L)>(1-\epsilon)$$ for $$m\gg 0$$.
• Thank you. This is exactly the sort of statement I had hoped would be true. I have a couple of quick questions. 1. Since $a$ and $b$ should clear the denominators of $A$ for the argument to work, we could still expect the dimension of $V_{a,b}$ to be quite small even for arbitrarily big $a$ and $b$. Is that correct? 2. Does your argument use the fact that the volume is a limit rather than just a limsup? Thanks again. Aug 24, 2020 at 23:39
• The statement is true for all $a,b$ sufficiently big, but more annoying to state/prove. Essentially, we have proven the statement for ll $a,b$ divisible by $m$. Since $L$ is big, we can assume that $H^0(cL)>0$ for all $c\geq m$. So in general we write $a=km+a'$ with $m\leq a'<2m$, $b=jm+b'$ for $m\leq b'<2m$ and then $V_{a,b}\supset V_{km,jm}$ and we still have $\dim V_{km,jm}/h^0((a+b)L)>(1-\epsilon)$ for $j,m\gg 0$. Aug 25, 2020 at 4:27