Person:

Siu, Yum-Tong

This article is written for the Proceedings of the Conference on Current Developments in Mathematics in Harvard University, November 16-17, 2007. It is an exposition of the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type. It lists and discusses the main techniques and explains how they are put together in the proof. Of the various main techniques some special attention is given to (i) the technique of discrepancy subspaces and (ii) the technique of subspaces of minimum additional vanishing.

We sketch a proof of the abundance conjecture that the Kodaira dimension of a compact complex algebraic manifold equals its numerical Kodaira dimension. The proof consists of the following three parts: (i) the case of numerical Kodaira dimension zero, (ii) the general case under the assumption of the coincidence of the numerically trivial foliation and fibration for the canonical bundle, and (iii) the verification of the coincidence of the numerically trivial foliation and fibration for the canonical bundle. Besides the use of standard techniques such as the L2 estimates of d-bar, the first part uses Simpson's method of replacing the flat line bundle in a nontrivial flatly twisted canonical section by a torsion flat line bundle. Simpson's method relies on the technique of Gelfond-Schneider for the solution of the seventh problem of Hilbert. The second part uses the semi-positivity of the direct image of a relative pluricanonical bundle. The third part uses the technique of the First Main Theorem of Nevanlinna theory and its use is related to the technique of Gelfond-Schneider in the first part.

This article discusses the recent transcendental techniques used in the proofs of the following three conjectures. (1) The plurigenera of a compact projective algebraic manifold are invariant under holomorphic deformation. (2) There exists no smooth Leviflat hypersurface in the complex projective plan. (3) A generic hypersurface of sufficiently high degree in the complex projective space is hyperbolic in the sense that there is no nonconstant holomorphic map from the complex Euclidean line to it.

The purpose of this paper is threefold. (i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context of an example of Catlin-D’Angelo concerning the ineffectiveness of the latter. (ii) To extend the techniques of multiplier ideal sheaves for the complex Neumann problem to general systems of partial differential equations. (iii) To present a new procedure of generation of multipliers in the complex Neumann problem as a special case of the multiplier ideal sheaves techniques for general systems of partial differential equations.

The deformational invariance of plurigenera proved recently by the author is a special case of the torsion-free property of the first direct image sheaf of the pluricanonical line bundle when the target space of the proper surjective holomorphic map is the open unit 1-disk with nonsingular fibers. This article discusses the adaptation of the techniques used in the proof of the deformational invariance of plurigenera to the general problem of proving the torsion-free property of the first direct image sheaf of the pluricanonical line bundle. The discussion covers also the more general case of the torsion-free property of the first direct image of the pluricanonical line bundle after twisting by a Hermitian holomorphic line bundle with semipositive curvature current and by its multiplier ideal sheaf. A number of results are obtained for the torsion-free problems by the adaptation of the techniques of the proof of the deformational invariance of plurigenera.