Minimal complex surfaces with pseudo-effective canonical bundles

Question

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of currents. That is, if we locally represent $h$ as $h = e^{\varphi}$, where $\varphi$ is an integrable real function, then $\Theta(h) = i \bar{\partial}\partial \varphi \geq 0$ (in the sense of currents).

All possible smooth compact complex minimal surfaces are listed in the famous Enriques-Kodaira classification. Denote by $K_X$ the canonical bundle of a complex surface $X$. I wonder about the following question:

• Which minimal complex surfaces $X$ have $K_X$, $K^{-1}_X$ or both pseudo-effective?

I am sure this is well-known by experts and, in fact, one can go case by case through Enriques-Kodaira's list to get the answer. However, some of the cases in the list are riddled with subtleties, and I seem to fail to obtain a satisfactory answer. A comprehensive list answering this question would be very much appreciated, and hopefully, it would be of use to other people.

Thanks.

Answer 1

Edit: My answer deals with the projective case.

First of all, for any projective manifold, $K_X$ is pseudoeffective if and only if $X$ is not uniruled (Boucksom-Demailly-Paun-Peternell). In the case of surfaces, this is also known to be equivalent to $\kappa(X)\ge 0$.

Next, $K_X$ and $-K_X$ are both pseudoeffective if and only if $c_1(K_X)=0$. In the case of surfaces (then, automatically minimal), you recover abelian surfaces, bi-elliptic surfaces, K3 surfaces and Enriques surfaces.

Finally, in the uniruled case ($K_X$ not pseudoeffective), $X$ is birational to $\mathbb P^1\times C$ for some curve $C$. Then $-K_X$ is psef if and only if $C=\mathbb P^1$ or $C=E$ is an elliptic curve.