The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry. It is mainly focused on natural questions in the birational geometry of complex projective varieties such as the study of the pluricanonical maps of varieties of general type and understanding how rational curves and holomorphic 1-forms influence the geometry of these varieties. Some of the main problems to be investigated are: For what values of m is the m-th pluricanonical map of a variety of general type birational? Are the fibers of a resolution of a variety with mild singularities always rationally chain connected? Is it the case that on a smooth projective variety of general type, holomorphic 1-forms always have non-empty zero set?
The classification of complex algebraic surfaces was initiated by the Italian school at the beginning of the twentieth century. Its main features have been long understood. In Higher Dimensional Complex Algebraic Geometry, one hopes to extend the theory surfaces to dimensions grater or equal to three. The minimal model program, for example aims to show, as in the case of surfaces, that any variety is either covered by rational curves or birationally equivalent to a minimal model. The minimal model program is complete only in dimension 3, but it has inspired many important advances in algebraic geometry. This project hopes to answer some of the questions and conjectures that naturally arise in this context.
