Siu

Multiplier Ideals -- A New Technique Linking Analysis and Algebraic Geometry

A multiplier is a function such that local a priori estimates for partial differential equations hold only after the test function is multiplied by it. The ideal sheaf consisting of multipliers identifies the location and the jet orders where local a priori estimates fail to hold. Solvability of a partial differential equation is reduced to algebraic conditions which force the multiplier ideal sheaf to be the structure sheaf. On the side of analysis, the method of multiplier ideal sheaves has been applied to problems such as the global regularity problem of the complex Neumann equation on pseudoconvex domains and the existence of Kaehler-Einstein metrics of Fano manifolds. On the side of algebraic geometry, the method of multiplier ideal sheaves has been successfully applied to solving, or making substantial progress towards solving, a number of long outstanding problems in algebraic geometry such as the Fujita conjecture, the effective Matsusaka big theorem, the deformational invariance of plurigenera, and the finite generation of canonical rings.