### 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) |
---|---|

Pages (from-to) | 613-642 |

Number of pages | 30 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pure and Applied Mathematics Quarterly*,

*9*(4), 613-642. https://doi.org/10.4310/PAMQ.2013.v9.n4.a2

**Closed symmetric 2-differentials of the 1st kind.** / Bogomolov, Fedor; De Oliveira, Bruno.

Research output: Contribution to journal › Article

*Pure and Applied Mathematics Quarterly*, vol. 9, no. 4, pp. 613-642. https://doi.org/10.4310/PAMQ.2013.v9.n4.a2

}

TY - JOUR

T1 - Closed symmetric 2-differentials of the 1st kind

AU - Bogomolov, Fedor

AU - De Oliveira, Bruno

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Albanese

KW - Closed meromorphic 1-forms

KW - Symmetric differentials

KW - Webs

UR - http://www.scopus.com/inward/record.url?scp=84907022489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907022489&partnerID=8YFLogxK

U2 - 10.4310/PAMQ.2013.v9.n4.a2

DO - 10.4310/PAMQ.2013.v9.n4.a2

M3 - Article

AN - SCOPUS:84907022489

VL - 9

SP - 613

EP - 642

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

SN - 1558-8599

IS - 4

ER -