There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and don't yet have the understanding of even the basic notions like schemes. So I am asking the question again.
In the context of a ramified covering map for Riemann surfaces, the formula relates the Euler characteristics of two surfaces. More precisely if $\pi : X \to Y$ is a complex analytic covering map between two Riemann surfaces, and if the degree of $\pi$ is $N$, then we have
$$ 2-2g_X = N(2-2g_Y) - R $$
where
$g_X, g_Y$ are the genus of $X,Y$ respectively,
$R=\sum_{p\in Y}(e_p-1)$ is a finite sum over the points of $X$ at which $\pi$ is ramified and $e_p$ denotes the ramification index.
Now to my actual questions:
$1)$ What if $X,Y$ are connected complex manifolds of dimension greater than $1$? How does the formula read? (An explanation as elementary as possible will be greatly appreciated.)
$2)$ Why can't the same formula as above be used for manifolds of higher dimension?