Siegel's theorem, best possible bound? I am studying classic Siegel's theorem, and I was wondering if that result is the best possible bound, meaning: is there a compact connected complex manifold $X$ of dimension $n$ in which $\operatorname{trdeg}_{\mathbb{C}} K(X) = n$ ?
Also, is there one in which it is $0$? (I was thinking about the manifold made of a single point, but is there another less trivial one?) And more in general, how can I compute this value for the classic ones? (projective complex space, torus, etc)

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*Any quasiprojective algebraic manifold (or indeed any quasiprojective algebraic variety at all) of dimension $n$ always has $\operatorname{trdeg}_{\mathbf C} K(X) = n$. (Indeed, this is a popular way to define the dimension of an algebraic variety!) This is very clear for projective space: picking an affine patch $\mathbf C^n$, the $n$ coordinate functions are an algebraically independent set which generate the field of meromorphic functions. 

*Yes, there are complex manifolds with no nonconstant meromorphic functions. The most common examples are complex tori $X=\mathbf C^n/\Lambda$: for a general choice of the lattice $\Lambda$, the only meromorphic functions on $X$ will be the constants.
A reference for these things is Chapter VIII of Basic Algebraic Geometry by Shafarevich.
