Cohomology of structure sheaf on surfaces of general type with $p_g=0$ Let $S$ be a surface of general type (the canonical divisor $K_{S}$ is big and nef). 
Then I have read that $\chi(\mathcal{O}_{S}) \geq 1$.
Why?
If this requires vanishing theorems, then assume that they hold.
 A: This result can be found in [Beauville, Complex Algeberaic Surfaces, Chapter X], see in particular Theorem X.4.
For the reader's convenience, let me give a short account of the proof.
Since $K_S$ is big and nef, we have $K_S^2 >0$. By Noether formula $$\chi(\mathcal{O}_S) = \frac{1}{12}(K_S^2+c_2(S))$$ it is therefore enough to show that $c_2(S) \geq 0$. 
By contradiction, assume $c_2(S) <0$. Then it is possible to show that $S$ has an étale cover $S' \to S$ such that $c_2(S') <0$ and $p_g(S') \leq 2q(S')-4$, and the second inequality implies that there are two linearly independent $1$-forms $\omega_1, \omega_2 \in H^0(\Omega^1_{S'})$ such that $\omega_1 \wedge \omega_2 \equiv 0$.
By Castelnuovo-De Franchis theorem this yields the existence of a connected morphism $p \colon S' \to B$, where $B$ is a curve with $g(B)\geq 2$. If $S'$ is ruled then so is $S$, which is excluded by hypothesis. Then the general fibre $F$ of $p$ satisfies $g(F) \geq 1$, so the Zeuthen-Segre formula gives $$c_2(S') \geq (2g(B)-2)(2g(F)-2) \geq 0,$$ again a contradiction.
Then $c_2(S)  \geq 0$ and we are done.
