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.



Cohomology of structure sheaf on surfaces of general type with $p_g=0$
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.


