Be $T$ a complex torus (which is not necessarily am abelian variety) of complex dimension $n \geq 2$. On $T$ we have an involution corresponding to the $(-1)$ application (i.e. passing to the inverse for the group law on $T$). Quotient $T$ by the action of such involution: we have a complex analytic space $\tilde{T}$ which is singular exactly at those points where the action of the involution is not free, i.e. $2-$torsion points. If we blow-up $\tilde{T}$ at these points we obtain again a complex manifolds and the exceptional divisors dominating the $2-$torsion points are all $\mathbf{P}^{n-1}$. Now is such a desingularization of the quotient of $T$ again Kaehler? It is clear to me that on the non-singular points of $\tilde{T}$ there are Kaehler forms coming from $T$, since invariant closed $2-$forms on $T$ are fixed for the $(-1)$ involution.

More generally, given a Kaehler manifold $X$, given a group $G$ acting on $X$ (not necessarily in a free, proper discountinous way), under what restrictions on $G$ and its action we will have that the desingularization of $X/G$ is still Kaehler?