This is a basic question but I have found it annoyingly hard to find an answer (which means its incredibly trivial or non-trivial!)
Are there examples of compact complex manifolds (of dimension at least 2) which do not admit any global, non-constant, meromorphic functions?
Of course this question can be phrased in a number of ways. For one, is there an example of a compact complex manifold which does not admit a rational map into a projective space? Equivalently, is there an example of a compact complex manifold which does not admit a linear system with a base locus of at least codimension 2?
Yes.
The generic Hopf surfaces (surfaces diffeomorphic to $S^3\times S^1$, or equivalently $\mathbb{C}^2-0$ quotient by an infinite cyclic group $\Gamma\subset GL(2,\mathbb{C})$) are such an example. Basically if $\Gamma$ is generated by a matrix $A$ with distinct eigenvalues (WLOG $A=\operatorname{\mathrm{diag}}(\lambda_1,\lambda_2)$), then the existence of non-constant meromorphic function $f/g$ gives two holomorphic functions $f,g$ on $\mathbb{C}^2$ by Hartog's and hence using $\Gamma$-invariance and considering lowest order terms when you clear denominators, you get $\lambda_1^m=\lambda_2^n$ for some $m,n$.