  Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper
is devoted to various questions concerning the analytic K-homology of $(X,h)$.
In the fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or
$\mathrm{dim}(X)=2$, we show that the rolled-up operator of the minimal
$L^2$-$\overline{\partial}$ complex, denoted here
$\overline{\eth}_{\mathrm{rel}}$, induces a class in $K_0 (X)\equiv
KK_0(C(X),\mathbb{C})$. A similar result, assuming
$\mathrm{dim}(\mathrm{sing}(X))=0$, is proved also for
$\overline{\eth}_{\mathrm{abs}}$, the rolled-up operator of the maximal
$L^2$-$\overline{\partial}$ complex. We then show that when
$\mathrm{dim}(\mathrm{sing}(X))=0$ we have
$[\overline{\eth}_{\mathrm{rel}}]=\pi_*[\overline{\eth}_M]$ with
$\pi:M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth}_M]\in
K_0 (M)$ the analytic K-homology class induced by
$\overline{\partial}+\overline{\partial}^t$ on $M$. In the second part of the
paper we focus on complex projective varieties $(V,h)$ endowed with the
Fubini-Study metric. First, assuming $\dim(V)\leq 2$, we compare the
Baum-Fulton-MacPherson K-homology class of $V$ with the class defined
analytically through the rolled-up operator of any $L^2$-$\overline{\partial}$
complex. We show that there is no $L^2$-$\overline{\partial}$ complex on
$(\mathrm{reg}(V),h)$ whose rolled-up operator induces a K-homology class that
equals the Baum-Fulton-MacPherson class. Finally in the last part of the paper
we prove that under suitable assumptions on $V$ the push-forward of
$[\overline{\eth}_{\mathrm{rel}}]$ in the K-homology of the classifying space
of the fundamental group of $V$ is a birational invariant.
