Complex Manifold Theory and Kaehler Geometry

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ABSTRACTThe project develops transcendental methods, such as multiplier ideal sheaves, to apply to problems in algebraic geometry and complex geometry, such as the finite generation of canonical rings, the Zariski nondensity of the union of all entire holomorphic curves in a compact complex manifold with positive canonical line bundle, the deformational invariance of plurigenera for compact Kaehler manifolds, and the relation between finite type condition of weakly pseudoconvex domains and subelliptic estimates from the viewpoint of complex Frobenius integrability theorem over Artinian subschemes.The investigations in this project are in the interface between severalcomplex variables, complex algebraic geometry, complex differential geometry, and partial differential equations. Besides applications to problems in algebraic geometry and complex geometry, a good understanding of multiplier ideal sheaves, especially those defined by differentiation, will introduce to partial differential equations new powerful tools. Multiplier ideal sheaves identify the higher-order directions of differentiation where estimates fail and then glue them together in global geometric entities with rich properties. They are able to provide global conditions for the solvability of partial differential equations instead of local conditions and will be especially useful for partial differential equations from global problems posed by any scientific field.