Complex Manifold Theory and Kaehler Geometry

首席研究员萧耀东将继续研究复杂流形理论和凯勒几何，一方面将最新的超越方法应用于复杂代数几何，另一方面将代数-几何方法应用于偏微分方程。他的工作的一些激励问题将是复杂代数几何中的丰度猜想，用奇异放大的复Monge-Ampere方程构造法诺流形中的有理曲线，关于一般型流形中全纯曲线的Green-Griffiths猜想，正则极化流形模空间的值分布理论的第二个主要定理，紧致Kaehler流形的多生变形不变性，不可约紧致hermite对称流形的全局不变形性，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束的技术，在一个方向上，它已经导致了复杂代数几何中许多长期开放问题的解决，例如代数流形的多生的变形不变性和正则环的有限生成；开辟了将代数-几何方法应用于偏微分方程的新途径，解决了复诺伊曼问题中的一些正则性问题。本研究是在代数几何与分析学的交叉领域进行的。这样的一个界面已经带来了两个领域的高水平的交叉施肥。一个领域的专家一直在研究和应用另一个领域的方法。这导致解决了一些长期存在的开放性问题，而这些问题仅是一个领域的技术无法解决的。在界面的一边，代数几何研究代数结构、代数结构的分类和代数结构之间的关系。解析方法通过使用分析的极限、估计和优化技术，使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和估计。最近在界面中的一些技术涉及到倍增器理想轴。乘数理想轴确定了分析中估计失效的位置和顺序，并使利用代数技术表述偏微分方程的可解性和正则性的全局条件成为可能。除了为分析问题开辟了一种新的代数方法外，它们还提供了强大的新工具，用于从任何科学领域提出的全球性问题中研究偏微分方程。PI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.