Reference/Known results on the singular behaviour of the fibres of a holomorphic map between compact Kähler manifolds

I have been interested in the following situation of late: Let $$X$$ and $$Y$$ be compact Kähler manifolds with $$\dim_{\mathbb{C}}(Y) < \dim_{\mathbb{C}}(X)$$ and let $$f : X \to Y$$ be a surjective holomorphic map with connected fibres. Let $$S = \{ s_1, …, s_k \}$$ denote the critical values of $$f$$, which is a subvariety of $$Y$$.

I cannot find a detailed account of how bad the singular behaviour of the fibres of $$f$$ can be. For example, do the fibres contain $$(-1)$$ curves (i.e., curves with self-intersection number $$-1$$) or $$(-2)$$ curves?

If anyone can provide references where I can get a better understanding of this, that would be tremendously appreciated.

Module of Kahler differentials for manifolds

Let $$A$$ be a $$k$$-algebra and let $$\mathcal{M}_A$$ be the set of all $$A$$-modules. In $$\mathcal{M}_A$$, there exists a universal object $$\Omega_{A/k}$$, called the module of Kahler differentials, and a $$k$$-derivation $$d: A \to \Omega_{A/k}$$ such that for any $$k$$-derivation $$D: A \to M$$ there exists a map $$f: \Omega_{A/k} \to M$$ such that $$f\circ d = D$$.

I’m trying to understand what this object has to be when I look at manifolds that are also smooth affine varities. I think that the answer should be the space of sections of the cotangent bundle over the ring of $$C^\infty$$ functions on the manifold. I’m unable to figure out how to show this.

Any help would be appreciated.