Complex Manifold Theory and Kaehler Geometry

首席研究员萧雨棠将继续他在复流形理论和Kaehler几何方面的研究，一方面将最新的超越方法应用于复代数几何，另一方面将代数几何方法应用于偏微分方程组。一些激发他工作的问题将是复代数几何中的丰度猜想，通过奇点放大的复Monge-Ampere方程构造Fano流形中的有理曲线，关于一般类型流形中的全纯曲线的Green-Griffiths猜想，正则极化流形的模空间的值分布理论的第二主要定理，紧致Kaehler流形的多属元的形变不变性，不可约紧致Hermite对称流形的整体不可变形性，以及复Neumann问题的一般正则性问题。最近的超越方法涉及乘子理想层技术，一方面已经解决了复代数几何中一些长期悬而未决的问题，如代数流形的多亏格的变形不变性和正则环的有限生成；另一方面，在另一个方向上，开辟了将代数几何方法应用于偏微分方程组的新途径，解决了复杂Neumann问题的一些正则性问题。这样的界面一直在带来两个领域的高水平交叉受精。一个领域的专家一直在研究和应用另一个领域的方法。这导致了一些长期悬而未决的问题的解决，这些问题只有一个领域的技术无法解决。在界面的一侧，代数几何研究代数结构、它们的分类和关系。解析方法通过使用分析的极限、估计和最优化技术，使得在代数几何问题中构造和处理代数对象成为可能。在界面的另一边，分析处理偏微分方程式和估计。界面中的一些最新技术涉及乘数理想滑轮。乘子理想层确定了分析中估计的失败位置和失败顺序，并使利用代数技巧来表示偏微分方程解和正则性的整体条件成为可能。除了为分析中的问题开辟了一种新的代数方法之外，它们还提供了强大的新工具，可用于从任何科学领域提出的全球问题研究偏微分方程。国际和平研究所将继续与研究生和初级研究人员合作。

The principal investigator, Yum-Tong Siu, will continue his research in complex manifold theory and Kaehler geometry by applying recent transcendental methods to complex algebraic geometry on the one hand and applying algebraic-geometric methods to partial differential equations on the other. Some of the motivating problems for his work will be the abundance conjecture in complex algebraic geometry, the construction of rational curves in Fano manifolds by singularity-magnifying complex Monge-Ampere equations, the Green-Griffiths conjecture concerning entire holomorphic curves in manifolds of general type, the second main theorem of value distribution theory for moduli spaces of canonically polarized manifolds, the deformational invariance of plurigenera for compact Kaehler manifolds, the global nondeformability of irreducible compact Hermitian symmetric manifolds, and general regularity questions of the complex Neumann problem. The recent transcendental methods involve techniques of multiplier ideal sheaves which, in one direction, has already led to the solution of a number of longstanding open problems in complex algebraic geometry such as the deformational invariance of plurigenera for algebraic manifolds and the finite generation of canonical rings, and in the other direction, opened up a new way of applying algebraic-geometry methods to partial differential equations and settled some regularity questions of the complex Neumann problem.The proposed research is in the interface of algebraic geometry and analysis. Such an interface has been bringing about a high level of cross-fertilization of both fields. Experts in one field have been investigating and applying the methods of the other. This has led to the solution of some longstanding open problems which have been inaccessible to the techniques of only one field. On the one side of the interface, algebraic geometry studies algebraic structures, their classifications and relations. Analytic methods make it possible to construct and work with algebraic objects in algebraic-geometry problems by using limits, estimates, and optimization techniques of analysis. On the other side of the interface, analysis deals with partial differential equations and estimates. Some of the recent techniques in the interface involve multiplier ideal sheaves. Multiplier ideal sheaves identify the location and the order of failure of estimates in analysis and make it possible to formulate global conditions for the solvability and regularity of partial differential equations by using algebraic techniques. Besides opening up a new algebraic approach to problems in analysis, they provide powerful new tools useful for the investigation of partial differential equations from global problems posed by any scientific field. The PI will continue work with graduate students and junior researchers.