When is a nef line bundle big

Question

Suppose $M^n$ is a smooth projective variety. A line bundle $L$ on $M$ is nef (numerically effective) if on any complete curve $C$ in $M$, $L$ has positive degree, i.e.
$$
L\cdot C=\int_{C}R_h\geq 0.
$$
where $R_h$ is the curvature form of any Hermitian metric $h$ on $L$.

A line bundle $L$ on $M$ is big if there exist constants $m_0$ and $c>0$ such that
$$
h^0(M,L^m)=\operatorname{dim} H^0(M,L^m)\geq cm^n.
$$
for all $m\geq m_0$.

I read a result which states that by Riemann-Roch theorem that a nef line bundle is big if and only if $\int_Mc_1(L)^n>0$. The reference given by the author is Hartshorne's Algebraic Geometry, which is written in the language of algebraic geometry. However, I'm not familiar with it. Can anyone explain the result and Riemann-Roch theorem? Thanks in advance.

Answer 1

The right reference for this material is probably Lazarsfeld's book Positivity in Algebraic Geometry, Volume $1$, and this is a combination of results. (I'd argue it does not follow immediately from Riemann-Roch.) This book is also written in the language of algebraic geometry, but it at least contains this result. See the end of section 1.4 for example.

For a vector bundle $E$ we can define the Euler characteristic $\chi(M, E) = \sum_i (-1)^i h^i(M, E)$. Riemann-Roch theorems seek to compute this using the intersection theory of $M$ and $E$. For our purpose, we only need the following version of the asymptotic Riemann-Roch theorem.

Theorem: Let $L$ be a line bundle on (a smooth projective variety) $M$. Then, $$\chi(M, L^m) = Cm^n + O(m^{n - 1})$$ is a polynomial in $m$ of degree $\leq n$, where $C = \int_M c_1(L)^n$.

When $L$ is nef we also have the following fact which controls the asymptotic growth of the higher cohomology.

Theorem: Let $L$ be a nef line bundle on $M$. Then $$h^i(M, L^m) = O(m^{n - i})$$ for $m$ sufficiently large and $i \geq 0$.

Together, these theorems imply that for a nef line bundle, we can describe the asymptotic growth $h^0(M, L^m)$ to be $$h^0(M, L^m) = \left (\int_M c_1(L)^n\right)m^n + O(m^{n - 1})$$ for $m$ sufficiently large. In particular, $L$ is big if and only if this first coefficient is positive.

Both these theorems are nontrivial and require a significant amount of algebraic geometry to prove. The proof of the second fact is given under theorem 1.4.40 in Lazarsfeld's book, and uses vanishing theorems. The asymptotic Riemann-Roch theorem is stated in Lazarsfeld's book but a proof can only be found elsewhere. (Olivier Debarre's Higher Dimensional Algebraic Geometry, for example.) Both these books assume and use a lot of the language algebraic geometry, though.