## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 613-642 |

Number of pages | 30 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

## Keywords

- Albanese
- Closed meromorphic 1-forms
- Symmetric differentials
- Webs

## ASJC Scopus subject areas

- Mathematics(all)