Abelian differential

Summary

An Abelian differential is a meromorphic $1$-form on a Riemann surface, written locally as $f(z)\,dz$ with $f$ meromorphic. The holomorphic ones are the differentials of the first kind. Their integrals along paths give the abelian integrals, the periods, and the Abel–Jacobi map. On a compact Riemann surface of genus $g$, the holomorphic Abelian differentials form a complex vector space of dimension $g$.