Algebraic dimension of complex nilmanifolds

TL;DR

This paper investigates the algebraic dimension of complex nilmanifolds, establishing an upper bound based on the dimension of holomorphic differentials, and extends results to meromorphic maps to Kähler manifolds.

Contribution

It proves that the algebraic dimension of a complex nilmanifold is bounded above by the dimension of its space of holomorphic differentials, and extends this to meromorphic maps to Kähler manifolds.

Findings

Algebraic dimension is at most the dimension of holomorphic differentials.

Established bounds for meromorphic maps to Kähler manifolds.

Provides new insights into the structure 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.