Complex Manifold Theory and Kaehler Geometry

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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.