A characterization of surfaces whose universal cover is the bidisk

2008/03/20 by Fabrizio Catanese, Catanese, Fabrizio, Marco Franciosi +1 · 1 citation
Mathematics · #14J25 #14J29 #32 Q 30 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:14J25 #msc:14J29 #msc:30 #msc:32

paper · pdf · doi:10.48550/arxiv.0803.3008

12 pages, references added

arxiv created 2008/03/26 · arxiv updated 2009/12/01

Abstract

We show that the universal cover of a compact complex surface X is the bidisk \HH × \HH, or X is biholomorphic to \PP¹ × \PP¹, if and only if K_X² > 0 and there exists an invertible sheaf η such that η²≅ \hol_X and H⁰(X, S²Ω¹_X (-K_X) ⊗ η) ≠ 0. The two cases are distinguished by the second plurigenus, P₂(X)≥ 2 in the former case, P₂(X)= 0 in the latter. We also discuss related questions.

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