Algebraic dimension of complex nilmanifolds

Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that a(M)⩽k, where a(M) is the algebraic dimension a(M) (i.e. the transcendence degree of the field of meromorphic functions) and k is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kähler manifolds.