Tag: Kahler

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.