Let $f: X \to Y$ be a proper surjective morphism of complex irreducible varieties such that general fibre of $f$ is connected and $Y$ integrally closed\normal. Say, we even assume wlog $Y=\text{Spec}(A)$ local affine.
Q: Is it possible to show with "relatively elementary methods" from complex geometry that then then every fibre must be actually connected? (Maybe I should add also flatness assumption; don't know, see considerations below)
The motivation is that there is well known Zariski's connectedness theorem vastly generalizing this beyond complex setting but whose proof uses quite heavy machinery and provides not an "obvious" geometrical insight.
But it is also said that Zariski's result is motivated by Enriques' "principle of degeneration" which was stated only in complex setting so far I know, which seems to base on rather "plausible geometric considerations". So I was hopping that in context of complex algebraic geometry the proof of this statement simplifies making the statement "intuitively/geometrically plausible" and making more "transparent" why the assumptions need to be posed in that way they are posed, especially the normality of $Y$
(recall, in modern terms normality assumption is made plausible to assure $f_*O_X =O_Y$ invoking Zariski's Main Thm), or, why general fibre need to be irreducible and not just connected.
But I would like to see a more "intuitive/geometric" picture of the connectedness principle which modern version not provides and maybe if we specialize to complex geometry setting becomes more "apparent".
I hoped that it should we work in similar vein as Ehresmann lemma where one essentially constructed - via constructing flow from appropr vector field solving certain dgl - even a diffeo between neighboured fibres.
But here we are no more in $\mathcal{C}^{\infty}$ realm, so flow & vector field techniques are a priori not available.
Considerations: Essentially, we want to show that $f^{-1}(y)$ ($y$ unique closed point of $Y$) is connected. Assume $f^{-1}(y)$ is not connected, so there exist idempotent $r \in \Gamma(f^{-1}(y), O_{f^{-1}(y)})$ with $r \neq 1,0$. Can we lift it somehow sophisticately exploiting normality of $A$ to obtain a contradition, eg non irreducibility of generic fibre?
Also, it seems that we implicitly need some kind of flatness (as we want to regard the fibres as a family). As we are in complex setting we have definitely generic flatness of proper open set on the base. But does it suffice? It seems that above we may try to use flat base change techniques to recornize the idempotent $r$ to live in $\Gamma(X) \otimes_A A/m_y$, but not sure if Enriques tacitly assumed the family of fibres to "vary continuously" - as at that time presumably notion of flatness wasn't introduced.
Sorry, if the nature of the question is a bit too vague, the point is that I'm not sure what assumptions actually Enriques tacitly took into account for his "principle of degeneration"(...compare the mentioned issue on flatness above). But it seemed that in the way he stated it the picture "what actually happens geometrically" was rather "plausible", and that's what I would like to understand.
