# Abundance Conjecture

 dc.contributor.author Siu, Yum-Tong dc.date.accessioned 2011-11-09T15:37:25Z dc.date.issued 2010 dc.identifier.citation Siu, Yum-Tong. 2010. Abundance conjecture. In Geometry and analysis, no. 2, edited by Lizhen Ji, 271-317. Advanced Lectures in Mathematics. Boston: International Press. en_US dc.identifier.isbn 978-1-57146-225-1 en_US dc.identifier.issn 0932-7134 en_US dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:5343170 dc.description.abstract 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. en_US dc.description.sponsorship Mathematics en_US dc.language.iso en_US en_US dc.publisher International Press en_US dash.license OAP dc.subject algebraic geometry en_US dc.title Abundance Conjecture en_US dc.type Monograph or Book en_US dc.description.version Accepted Manuscript en_US dc.relation.journal Advanced Lectures in Mathematics en_US dash.depositing.author Siu, Yum-Tong dc.date.available 2011-11-09T15:37:25Z

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