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# Pure and Applied Mathematics Quarterly

## Volume 9 (2013)

### Number 4

### Special Issue: In Memory of Andrey Todorov, Part 1 of 3

### Closed Symmetric 2-Differentials of the 1st Kind

Pages: 613 – 642

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n4.a2

#### Authors

#### 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 $X \backslash E$ 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 $f : X \to C$ over a curve of genus $\geq 1$.

#### Keywords

symmetric differentials, Albanese, webs, closed meromorphic 1-forms