Is the canonical bundle of a projective complex surface an ample bundle if its self-intersection is positive?

Let $$X$$ be a projective complex surface ($$\dim_\mathbb C X = 2$$). If $$(K_X, K_X) > 0$$, is it true that $$K_X$$ is an ample line bundle?

I am trying to the affirmative answer using Nakai's criterion, which says that a line bundle $$L$$ on a compact surface is ample iff $$(L,L)>0$$ and $$(L,D) > 0$$ for every effective divisor $$D$$ on $$X$$.

By Riemann-Roch we have that $$\dim H^0(X, O_X(D)) + \dim H^0(X, O_X(-D) \otimes K_X) - \dim H^1 (X, O_X(D) ) = \chi(X) + 1/2( (D, D) - (D,K_X) )$$

I'm not sure where to go from here, or if RR helps at all.

• As in your last question, you need more assumptions. Try the case of $\Bbb P^2$ first and see what happens. (This is the same example as last time.) Commented Jan 9, 2023 at 16:20
• @KReiser the self-intersection is negative then though, right? Commented Jan 9, 2023 at 16:34
• The intersection product on $\Bbb P^2$ is $\mathcal{O}(a).\mathcal{O}(b)=ab$, right? Or am I misremembering? Commented Jan 9, 2023 at 16:40

The answer is no. For example, the canonical divisor of $$\mathbb{P}^2$$ is $$K=\mathcal{O}(-3)$$, and so $$K^2=9$$. However, Nakai-Moishezon's criterion says that if $$D$$ is a divisor on a smooth surface such that $$D^2>0$$ and for any curve $$C$$, the intersection $$DC$$ is also positive, then $$D$$ is ample. See, for instance, Hartshorne's Book.