The Poincaré Problem for a foliated surface

Let $\mathcal{F}$ be a foliation on a smooth projective surface $S$ over the complex number $\mathbb{C}$. We introduce three birational non-negative invariants $c_1^2(\mathcal{F})$, $c_2(\mathcal{F})$ and $\chi (\mathcal{F})$, called the Chern numbers. If the foliation $\mathcal{F}$ is not of general type, the first Chern number $c_1^2(\mathcal{F})=0$, and $c_2(\mathcal{F})=\chi (\mathcal{F})=0$ except when $\mathcal{F}$ is induced by a non-isotrivial fibration of genus $g=1$. If $\mathcal{F}$ is of general type, we obtain a slope inequality when $\mathcal{F}$ is algebraically integral. As a corollary, $\mathcal{F}$ is always transcendental if the slope is less than $2$. On the other hand, we also prove three sharp Noether type inequalities if $\mathcal{F}$ is of general type. As applications, we obtain a criterion for foliations to be transcendental using Noether type inequalities, and we also give a partial positive answer to the question on the lower bound on the volume of a foliation of general type.