Birational geometry of complex projective varieties

所提出的研究涉及高维复代数几何的主题。它主要研究复射影簇的双有理几何中的一些自然问题，如研究一般类型簇的多重正则映射，理解有理曲线和全纯1-形式如何影响这些簇的几何，其中一些主要问题是：对于m的什么值，一般类型簇的m次多重正则映射是双有理的？具有温和奇点的簇的分解的纤维总是理性链连接的吗？在一般型光滑射影簇上，全纯1-形式是否总有非空零集？复代数曲面的分类是由意大利学派在世纪初提出的。它的主要特征早已为人们所了解。在高维复代数几何中，人们希望将理论曲面扩展到大于或等于3维。例如，极小模型程序的目的是表明，与曲面的情况一样，任何簇要么被有理曲线覆盖，要么与极小模型双有理等价。极小模型程序只在3维中是完备的，但它激发了代数几何学的许多重要进展。2这个项目希望回答在这种背景下自然出现的一些问题和假设。

The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry. It is mainly focused on natural questions inthe birational geometry of complex projective varietiessuch as the study of the pluricanonical maps of varieties of general typeand 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 theItalian school at the beginning of the twentieth century. Its main featureshave 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 ofsurfaces, that any variety is either covered by rational curves orbirationally 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.