Symmetric Differentials of Rank 1 and Holomorphic Maps

Let X be a projective manifold of dimension n, a symmetric differential of degree m is a section of the m-th symmetric power of the sheaf of holomorphic 1-forms, SΩX . A symmetric differential w of degree m is of rank 1 if it can be locally written in the form w|U = fμ, where μ ∈ H(U,ΩX) and f ∈ O(U). Symmetric differentials of degree 1, i.e. holomorphic 1-forms, are trivially of rank 1. Holomorphic 1-forms μ on compact projective manifolds X have the following properties: i) μ ∈ H(X, ΩX) are closed (and locally exact). ii) The presence of a nontrivial holomorphic 1-form μ implies the existence of a holomorphic map to an abelian variety A(X), f : X → A(X) and μ = f∗u with u ∈ H(A(X), ΩA(X)). iii) The presence of a nontrivial holomorphic 1-form μ imply that the abelianization of π1(X) is infinite.