Local and global problems on singularities for higher dimensional algebraic varieties

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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equations.It is one of the modern successors of ancient Greek geometry in the sense that geometrical shapes have been replaced by the equations defining them.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.