Local and global problems on singularities for higher dimensional algebraic varieties

拟议的研究是在代数几何的一般领域。所提出的项目从局部和全局两个角度研究了代数簇奇点的复杂性。关于局部奇点理论，PI建议研究奇点的不同方法之间的关系：奇点的分解，Hodge理论和D-模，以及JET方案。第一个项目考虑了霍奇理论对乘子理想的限制。利用D-模和组合技术，PI计划研究称为伯恩斯坦-佐藤多项式的奇点不变量的微妙性质。在第三个项目中，PI寻找D-模块和JET方案之间的联系。其方法是在D-模的派生范畴中构建与值的动机整合的概念。关于整体理论，PI和L.Ein计划根据乘子理想找到几何稳定性的标准。这些准则对于Kahler几何中的极值度量的研究将是重要的，并且将涉及奇点关于乘子理想的整体上同调不变量。在提出的最后一个项目中，PI研究了这些全局上同调不变量在局部系统中的自然设置。代数几何是研究代数方程的解。它是古希腊几何的现代继承者之一，因为几何形状已经被定义几何形状的方程所取代。这些几何形状中最丰富、最复杂的结构出现在某些称为奇点的地方。对奇点的局部研究大致就像把它们放在显微镜下，然后对它们进行放大。一项全球性的研究是了解奇点对几何形状相对于其他几何形状的行为施加的限制。

The proposed research is in the general area of algebraic geometry. The proposed projects investigate the complexity of singularities of algebraic varieties from both the local and the global point of view. Concerning the local theory of singularities, the PI proposes to investigate relations among different approaches to singularities: resolutions of singularities, Hodge theory and D-modules, and jet schemes. The first project considers restrictions on multiplier ideals induced by Hodge theory. Employing the use of D-modules and combinatorial techniques, the PI plans to study the subtle nature of an invariant of singularities called the Bernstein-Sato polynomial. In a third project, the PI looks for the connection between D-modules and jet schemes. The approach is to construct a notion of motivic integration with values in a derived category of D-modules. Concerning the global theory, the PI together with L. Ein plans to find criteria for geometric stability in terms of multiplier ideals. Such criteria would be important for the study of extremal metrics in Kahler geometry and would involve global cohomological invariants of singularities in terms of multiplier ideals. In the last project proposed, the PI studies a natural setting for these global cohomological invariants in terms of local systems.Algebraic geometry is the study of solutions of algebraic equations.It is one of the modern successors of ancient Greek geometry in the sense that geometrical shapes have been replaced by the equations defining them. The richest and most complicated structure of these geometrical shapes appears at certain places called singularities. A local study of singularities is roughly like placing them under a microscope and zoomming on them. A global study is to understand the restrictions imposed by the singularities on the behavior of the geometrical shape with respect to other geometrical shapes.