《《Multiplier ideal sheaves in complex and al-》.pdf

Science in China Ser. A Mathematics 2005 Vol. 48 Supp. 1—31 1 Multiplier ideal sheaves in complex and al- gebraic geometry Yum-Tong Siu Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: siu@) Received January 27, 2005 Abstract The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invariance of plurigenera for general compact algebraic manifolds is presented and discussed. Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed. Keywords: algebraic geometry, multiplier ideal sheaf, effective problem, invariance of plurigenera, canonical ring, minimal model, complex Monge-Amp`ere equation, complex Neumann estimate, weakly pseudoconvex domain. DOI: 10.1360/05za0001 There are two parts in this article. The first part, which is the main part of the article, discusses the application, by the method of multiplier ideal sheaves, of analysis to complex algebraic geometry. The second part discusses the other direction which is the application of complex algebraic geometry to analysis, ¯ mainly to problems of estimates and subellipticity for the ∂ operator. 1 Application of analysis to algebraic geometry For the application of analysis to algebraic geometry, we will start out with the general technique of reducing problems in algebraic geometry to problems in n 2 ¯ ¯ ∂ estimates for Stein domains spread over C . The L