2012/11/30 by Tsz On Mario Chan · 2 citations
Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Atiyah–Singer index theorem #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hermitian matrix #Holomorphic function #Isomorphism (crystallography) #Line bundle #Mathematics #Pure mathematics #Sesquilinear form #Torus #math.AG #math.CV #math.FA #msc:14F17 #msc:32J25 #msc:32T27 #msc:46C05
paper · pdf · doi:10.1016/j.matpur.2013.01.019
published in Journal de Mathématiques Pures et Appliquées 100(5), 719-747 (Elsevier BV) · 44 pages, author's PhD thesis
openalex publication_date 2013/01/30 · arxiv created 2013/03/04 · arxiv updated 2013/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert-Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L2-methods coupled with the Kazama-Dolbeault isomorphism and Bochner-Kodaira formulas.