The project is to investigate the following problems in complex manifold theory. (1) Invariance of plurigenera for manifolds not of general type. (2) Fujita conjecture type problems and the sharpening of known bounds. (3) Finite generation of canonical rings for manifolds of general type. (4) Global regularity of the complex Neumann problem and nonexistence problem for Levi-flat sets with small singularities. (5) Hyperbolicity of generic high-degree hypersurfaces in the complex projective space and their complements. The investigation will use and further develop the method of multiplier ideal sheaf which has already produced very good results.
Many problems in geometry and related fields, such as mathematical physics, are reduced to questions about the existence and the properties, such as regularity, of global solutions of partial differential equations. A priori estimates have for a long time been the dominant tool for globally solving partial differential equations. Such a priori estimates are usually derived from pointwise properties of the partial differential equations. In many important global geometric problems, some of which are listed above, pointwise arguments are insufficient. To solve such problems, this project uses and further develops the method of "mulitplier ideal sheaf". What is to be estimated is multiplied by a "multiplier" before estimation so that the a priori estimates hold. The set of all such multipliers forms the "multiplier ideal sheaf". Global properties of the "multiplier ideal sheaf", such as closedness under certain kinds of differentiation, for certain problems force to be a multiplier the function which is identically 1, thus giving the desired global solutions of the partial differential equations. Some long outstanding problems in algebraic geometry have already been solved by this method. With further development this new method of using the multiplier ideal sheaf to solve global partial differential equations should be a very powerful tool with broad applications and deep impact.
