General type implies ample canonical class?

Question

Let $X$ be a complex algebraic surface of general type. Does it then follow that $\omega_{X}$ is ample? I have been told that it does follow, but I am not sure if it does. I know that the pluricanonical maps $\Phi_{|nK_{X}|}$ are birational for $n\ge 5$, but this doesn't seem to entail ampleness, since it's not an embedding.

If this is not the case, what is a counterexample?

Answer 1

Upshot: The short answer is that $X$ is of general type if and only if $\omega_X$ is big. In particular, $\omega_X$ may neither be ample nor nef, contrary to the claim in Mohan's comment.

For simplicity, let us work with (connected) smooth projective varieties $X$, defined over the field of complex numbers. Given a line bundle $L$ on $X$, the Iitaka dimension $\kappa(X,L)$ is the maximum of the dimensions of the images of $\Phi_{|mL|}$ if $H^0(X,L^{\otimes m}) \not= 0$ for some $m > 0$, and $\kappa(X,L) = -\infty$ otherwise. In particular, the Kodaira dimension $\kappa(X)$ of $X$ is the Iitaka dimension of $L = \omega_X$.

Usually, a variety $X$ as above is called of general type if it has maximal Kodaira dimension $\kappa(X) = \mathrm{dim}(X)$. Occasionally, the term is used with a different meaning, for example for varieties with $\kappa(X) \geq 1$, mimicking the curve case, or for varieties with ample canonical bundle $\omega_X$. Under this latter definition, of course $\omega_X$ is ample as asked for, but for the remainder, we stick to the standard definition.

A line bundle $L$ is called big if it has maximal Iitaka dimension $\kappa(X,L) = \mathrm{dim}(X)$. Thus, the claim that $X$ is of general type if and only if $\omega_X$ is big is a tautology, but it is nonetheless meaningful because bigness is a well-established property of line bundles.

Given a line bundle $L$ on $X$ and a reduced and irreducible (but possibly singular) curve $C \subset X$, we can define the intersection number $(C \cdot L) \in \mathbb{Z}$. As a weakening of the intersection properties of ample line bundles (see the Nakai-Moishezon criterion), a line bundle $L$ is called nef if $(C \cdot L) \geq 0$ for all reduced and irreducible curves $C \subset X$. Loosely speaking, a nef line bundle is one which is a limit of ample ones, and there is a precise meaning of this in the sense of $\mathbb{R}$-divisor classes.

Every ample line bundle is nef and big. Thus, if $\omega_X$ is ample, then $X$ is of general type.

Now let us consider a few examples.

• Let $X \subset \mathbb{P}^3$ be a smooth quintic (hyper-)surface. Then $\omega_X \cong \mathcal{O}_X(1)$, so $\omega_X$ is ample, hence $X$ is of general type. Since $X$ has degree $5$, the self-intersection of $\omega_X$ is $(\omega_X \cdot \omega_X) = 5$.
• Assume that $Y \subset \mathbb{P}^3$ is a (hyper-)surface of degree $d \geq 5$, and assume that $Y$ has only ordinary double points as singularities, and at least one of them. Let $Z \rightarrow Y$ be a minimal resolution of singularities. Then $\omega_Z$ is nef and big but not ample.
• Let $S \subset \mathbb{P}^3$ be a smooth quartic (hyper-)surface. Then $\omega_S \cong \mathcal{O}_S$, so $\omega_S$ is nef but not big.
• Let $B \rightarrow X$ be the blow-up of the quintic surface in a point. Then $\omega_B$ is big but not nef. Namely, the Kodaira dimension is a birational invariant, so it must be big, but $\omega_B$ is not nef because $B$ is not a minimal surface.
• For $F = \mathbb{P}^2$, the canonical bundle $\omega_F$ is neither nef nor big.

In short, the properties of being nef and being big are independent.

Two additional points are worth mentioning:

• A nef line bundle $L$ is big if and only if $(D^{\mathrm{dim}(X)}) > 0$ for the top self-intersection.
• A line bundle $L$ is semi-ample if some positive power is globally generated. Every semi-ample line bundle $L$ is also nef.

Ample material on this topic can be found in Lazarsfeld's book "Positivity in Algebraic Geometry I".