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 $a(M)\le k$, 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.

20 pages, v. 2.0, minor corrections, some reference added

References in corpus (1)

• Geometry of nilmanifolds with left-invariant complex structure and deformations in the large

Cited by in corpus (2)

• Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds
• The Lee--Gauduchon cone on complex manifolds