On surfaces of class VII₀⁺ with numerically anticanonical divisor

2004/06/19 by Georges Dloussky, G. Dloussky, Dloussky, G.
Mathematics · #32J15 #53C21 #53C55 #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV #math.DG #msc:32J15 #msc:53C21 #msc:53C55

paper · pdf · doi:10.48550/arxiv.math/0406387

31 pages, revised version, statement of thm 3.44 corrected, proof not changed. Accepted in Am. J. of Math

openalex publication_date 2004/06/19 · arxiv created 2005/04/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider minimal compact complex surfaces S with Betti numbers b₁=1 and n=b₂>0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m>0 and a flat line bundle F such that -mK⊗ F has nontrivial sections, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces.

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