On the slope of non-algebraic holomorphic foliations

Let (Y, {\mathcal {G}}) be a Riccati foliation on Y and let \pi :(X,{\mathcal {F}}){\rightarrow } (Y,{\mathcal {G}}) be a double cover ramified over some normal-crossing curves. We will determine the minimal model of {\mathcal {F}} and compute its Chern numbers c_1^2({\mathcal {F}}), c_2({\mathcal {F}}), and \chi ({\mathcal {F}})=(c_1^2({\mathcal {F}})+ c_2({\mathcal {F}}))/12. We will prove that the slope \lambda ({\mathcal {F}})=c_1^2({\mathcal {F}})/\chi ({\mathcal {F}}) satisfies 4\leq \lambda ({\mathcal {F}})