Algebraic dimension of complex nilmanifolds

Abstract: 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 $a(M)\leq k$, 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 Kahler manifolds.