On varieties whose universal cover is a product of curves

2008/12/23 by Fabrizio Catanese, Catanese, Fabrizio, Marco Franciosi +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.0812.4317

Abstract

We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product Cⁿ of a curve C = \PP¹ or \HH: the existence of a semispecial tensor ω. A semispecial tensor is a non zero section 0 ≠ ω∈ H⁰(X, SⁿΩ¹_X (-K_X) ⊗ η) ), where η is an invertible sheaf of 2-torsion (i.e., η²≅ \hol_X). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension n= 2 or n= 3; but it is not sufficient alone, even in dimension 2. In the case of Kähler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases.

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