Geography On Non-algebraic Holomorphic Foliations

Holomorphic foliation on an algebraic surface is a generalization of fibration.An algebraic foliation is birationally equivalent to a fibration.Some birational invariants of fibrations can be generalized to those of foliations,e.g.,Kodaira dimension and modular invariants can be generalized as Kodaira dimension and Chern numbers respectively.However,the genus of a fibration is a birational invariant which cannot be generalized to non-algebraic foliations.According to Kodaira dimensions,foliations of non-general type can be classified into eight classes,and four of them are non-algebraic.Foliations of general type can be divided into two classes,algebraic foliations(non-isotrivial fibrations of genus 2)? 2)and non-algebraic foliations.We know very little about non-algebraic foliations of general type.The main problem for foliations of general type is the classification problem according to their Chern numbers and some other invariants.The first fundamental problem is the Problem of Geography: which numbers can be their Chern numbers?What we know is that the first Chern numbers and the Euler characteristic numbers are positive rational numbers,and the second Chern numbers are non-negative rational numbers.The Noether equality holds true.One can define the slope of a foliation as the ratio of the First Chern number to the Euler number,which is a positive rational number less than or equal to twelve.The main purpose of the doctoral dissertation is to study non-algebraic foliations by using the method of double covers which is used in the study of hyperelliptic fibrations,especially the Problem of Geography.In this case,the base foliation is a Riccati foliation whose Kodaira dimension is 0 or 1.We have no fibers or genus,and the minimal model of the double cover is not easy to determine(the algebraic case of this problem is solved by Gang Xiao).Thus the calculation of double Riccati foliations is much more complicated than the hyperelliptic case.In order that the double cover method works,we impose some general conditions on the singularities of the branch locus.Then we give the formulas for the Chern numbers,which generalize Kodaira's formulas for the -invariants and Horikawa-Xiao's formulas for the Chern numbers in the fibration case.As the main result,we give formulas for the Chern numbers in terms of some local indexes for non-algebraic double Riccati foliations,which is similar to Xiao's singularity indexes.We find some new phenomenon for non-algebraic double Riccati foliations,the slopes cannot reach the maximal value twelve,similar phenomenon for hyperelliptic fibrations found by Gang Xiao.In our examples,the slopes are between four and twelve.We give some characterizations and examples of nonalgebraic double Riccati foliations whose slope equals to four.