The title is changed and one more author is added. In this new version, we provide two approaches to study the Poincare's problem on the algebraic integrability of foliations on smooth surfaces: the first one is to use the Chern numbers and slope inequalities; the other one is to use the Noether type inequalities obtained in the earlier version.of this paper. Any comment is warmly welcome
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 $χ(\mathcal F)$, 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 $c_2(\mathcal F)=χ(\mathcal F)=0$ 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.
Canonical Models of Adjoint Foliated Structures on Surfaces
School of Mathematical Sciences, Key Laboratory of MEA(Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University（华东师范大学数学科学学院，教育部MEA重点实验室及上海PMMP上海市重点实验室）；School of Mathematics and Systems Science, and Center for Mathematical Sciences, Wuhan University of Science and Technology（武汉科技大学数学与系统科学学院暨数学科学中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
