The theorem that inkspot refers to in his answer is originally from Inoue's paper *New Surfaces with No Meromorphic Functions, II* which seems like a more complete reference for this question. In particular, Inoue gives an explicit example of a compact complex surface which has an analytic subvariety but no compact complex submanifolds.

If $x$ is a real quadratic irrationality (i.e. a real irrational solution of a real quadratic equation), denote it's conjugate by $x'$. Let $M(x)$ be the free $\mathbb{Z}$-module generated by $1$ and $x$, then set $U(x) = \{\alpha \in \mathbb{Q}(x) \mid \alpha > 0, \alpha\cdot M(x) = M(x)\}$ and $U^+(x) = \{\alpha \in U(x) \mid \alpha\cdot\alpha' > 0\}$. Both $U(x)$ and $U^+(x)$ are infinite cyclic groups and $[U(x) : U^+(x)] = 1$ or $2$.

If $\omega$ is a real quadratic irrationality such that $\omega > 1 > \omega' > 0$, then $\omega$ is a purely periodic modified continued fraction; that is, $\omega = [[\overline{n_0, n_1, \dots, n_{r-1}}]]$ where $n_i \geq 2$ for all $i$, $n_j \geq 3$ for at least one $j$, and $r$ is the smallest period. For every such $\omega$, Inoue constructs a compact complex surface $S_{\omega}$ which is now known as an *Inoue-Hirzebruch surface*.

There are compact subvarieties $C$ and $D$ of $S_{\omega}$ with irreducible components $C_0, \dots, C_{r-1}$ and $D_0, \dots, D_{s-1}$ respectively; here $s$ is the smallest period of the modified continued fraction expansion of another element $\omega^*$ related to $\omega$ (alternatively, $s$ can be determined from the modified continued fraction expansion of $\frac{1}{\omega}$). When $r \geq 2$, $C$ is a cycle of non-singular rational curves, and when $r = 1$, $C$ is a rational curve with one ordinary double point. Proposition $5.4$ shows that $C_0, \dots, C_{r-1}, D_0, \dots, D_{s-1}$ are the only irreducible curves in $S_{\omega}$.

In the case where $[U(\omega) : U^+(\omega)] = 2$, we have $r = s$. Furthermore, there is an involution $\iota$ such that $\iota(C_i) = D_i$ for $i = 0, \dots, r - 1$. The quotient of $S_{\omega}$ by $\iota$ is denoted $\hat{S}_{\omega}$ and is now known as a *half Inoue surface*. Note that the images of $C_0, \dots, C_{r-1}$ are the only irreducible curves in $\hat{S}_{\omega}$.

If we can find a real quadratic irrationality $\omega$ such that $\omega > 1 > \omega' > 0$, $r = 1$, and $[U(\omega) : U^+(\omega)] = 2$, then $\hat{S}_{\omega}$ is a compact complex surface containing a unique curve, namely a rational curve with one ordinary double point. In particular, it provides an example of a compact complex manifold with a subvariety but no compact complex submanifolds. One such $\omega$ was given in the paper (end of section 6).

**Example.** Take $\omega = (3 + \sqrt{5})/2$. Then $[U(\omega) : U^+(\omega)] = 2$ and $\alpha_0 =\ \text{a generator of}\ U(\omega) = (1 + \sqrt{5})/2$, $\alpha = \alpha_0^2 = (3 + \sqrt{5})/2$, $\omega = [[\overline{3}]]$, $r = 1$.

In this case, $b_2(\hat{S}_{\omega}) = 1$ and $\hat{S}_{\omega}$ contains exactly one curve $\hat{C}$. Moreover, $\hat{C}$ is a rational curve with one ordinary double point and $(\hat{C})^2 = -1$.

For those interested in the details, in addition to Inoue's paper, it may also be worth reading the earlier paper *Hilbert modular surfaces* by Hirzebruch. As mentioned in his paper, Inoue used some methods from Hirzebruch's paper (which gives some indication of why the resulting surfaces are jointly named).