Closed symmetric 2-differentials of the 1st kind

Fedor Bogomolov, Bruno De Oliveira

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)613-642
Number of pages30
JournalPure and Applied Mathematics Quarterly
Volume9
Issue number4
DOIs
StatePublished - 2013

Fingerprint

Closed
Abelian Variety
Meromorphic
Fibration
Riemannian Metric
Fundamental Group
Divisor
Differential operator
Irregular
Genus
Quotient
Complement
Subgroup
Imply
Curve
Form

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Pure and Applied Mathematics Quarterly, Vol. 9, No. 4, 2013, p. 613-642.

Research output: Contribution to journalArticle

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