The presence of holomorphic 1-forms on a compact kahler manifold $X$ implies topological properties of $X$. Moreover, from their presence also follows the existence of a holomorphic map from $X$ into a complex torus from which all the holomorphic $1$-forms of $X$ are induced from. The talk gives a complete extension of this result to symmetric differentials of rank $1$. This result belongs to the program whose aim is to understand the class of symmetric differentials that have a close to topological nature (symmetric differentials of rank $1$ will be shown to be closed symmetric differentials). We will also discuss these results for degree $2$.