Local and global problems on singularities for higher dimensional algebraic varieties

提出的研究是在代数几何的一般领域。提出的项目从局部和全局的角度研究代数变种奇点的复杂性。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：奇点的解析，Hodge理论和d -模，以及射流方案。第一个项目考虑了霍奇理论对乘数理想的限制。利用d模和组合技术，PI计划研究被称为Bernstein-Sato多项式的奇点不变量的微妙本质。在第三个项目中，PI寻找d模块和喷气计划之间的联系。该方法是构造一个与d模派生范畴中的值的动机积分的概念。关于全局理论，PI和L. Ein计划从乘数理想的角度寻找几何稳定性的准则。这样的准则对于研究Kahler几何中的极值度量是很重要的，并且涉及乘子理想中奇点的全局上同不变量。在最后提出的项目中，PI研究了局部系统中这些全局上同不变量的自然设置。代数几何是研究代数方程解的学科。它是古希腊几何学的现代继承者之一，因为几何形状已经被定义几何形状的方程所取代。这些几何形状的最丰富和最复杂的结构出现在被称为奇点的某些地方。对奇点的局部研究大致就像把它们放在显微镜下放大。一项全面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equations.It is one of the modern successors of ancient Greek geometry in the sense that geometrical shapes have been replaced by the equations defining them.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.