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Regular version of the site

Article

Closed symmetric 2-differentials of the 1st kind

Pure and Applied Mathematics Quarterly. 2013. Vol. 9. No. 4. P. 613-642.
Fedor Bogomolov, De Oliveira B.
A closed symmetric differential of the 1st kind is a differential that locally is the product of closed holomorphic 1-forms. We show that closed symmetric 2-differentials of the 1st kind on a projective manifold X come from maps of X to cyclic or dihedral quotients of Abelian varieties and that their presence implies that the fundamental group of X (case of rank 2) or of the complement X∖E of a divisor E with negative properties (case of rank 1) contains subgroup of finite index with infinite abelianization. Other results include: i) the identification of the differential operator characterizing closed symmetric 2-differentials on surfaces (using a natural connection to flat Riemannian metrics) and ii) projective manifolds X having symmetric 2-differentials w that are the product of two closed meromorphic 1-forms are irregular, in fact if w is not of the 1st kind (which can happen), then X has a fibration f:X→C over a curve of genus ≥1.