### Sheaves associated to holomorphic first integrals

Let $\mathcal{F}:L\to TS$ be a foliation on a complex, smooth and irreducible projective surface $S$, assume $\mathcal{F}$ admits a holomorphic first integral $f:S\to {\mathbb{P}}^{1}$. If ${h}^{0}(S,{\mathcal{O}}_{S}(-n{\mathcal{K}}_{S}\left)\right)\>0$ for some $n\ge 1$ we prove the inequality: $(2n-1)(g-1)\le {h}^{1}(S,{\mathcal{L}}^{\text{'}-1}(-(n-1){K}_{S}\left)\right)+{h}^{0}(S,{\mathcal{L}}^{\text{'}})+1$. If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of $\mathcal{F}$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.