It is always a pain to move back and forth between definitions in algebraic geometry and complex analytic geometry. Dictionary is much easier when are working with (family of) smooth varieties but the pain grows exponentially when we include singular varieties.
Here is some of that:
Suppose $f: X\rightarrow Y$ is a map of possibly singular complex analytic varieties, then is there a simpler definition of flatness for $f$ ? this one should be hard to answer but how about following,
—Let $f:X\rightarrow Y$ be a family of curves. there might be multiple fibers, non-reduced fibers, nodal curves, cusp curves,… in the family, but fibers are connected.
—When this fibration is flat? What kind of bad fibers mentioned above are allowed in a flat family?
—Same question for higher dimensional family of analytic varieties?
For simplicity you may assume that the base of fibration is smooth.
Best Answer
Instead of trying to say what flatness in analytic geometry means I'll give you some street-fighting tricks for recognizing whether a morphism of analytic spaces ( not necessarily reduced) $f:X\to Y $ is , or has a chance to be, flat.
a) A flat map is always open. . Hence, contraspositely, the embedding $\lbrace 0\rbrace \hookrightarrow \mathbb C$ is not flat. More generally, the embedding of a closed (and not open !) subspace $X \hookrightarrow Y$ is never flat.
b) An open map need not be flat: think of the open map $ Spec(\mathbb C) \to Spec(\mathbb C[\epsilon ]) $ from the reduced point to the double point, which is open but not flat.
[It is not flat because the $\mathbb C[\epsilon]$- algebra $\mathbb C=\mathbb C[\epsilon]/(\epsilon)$ is not flat : recall that a quotient ring $A/I$ can only be flat over $A$ if $I=I^2$ and here $I=(\epsilon) \neq I^2=(\epsilon)^2=(0)$]
An example with both spaces reduced is the normalization [see g) below] $f:X^{nor} \to X $ of the cusp $X\subset \mathbb C^2$ given by the equation $y^2=x^3$ .That normalization is a homeomorphism and so certainly open, but it is not flat : this results either from g) or from h) below.
c) Given the morphism $f:X\to Y $ , consider the following property:
$\forall x \in X, \quad dim_x(X)=dim_{f(x)} (Y) +dim_x(f^{-1}(f(x)))$ $\quad (DIM) $
We then have: $ f \; \text {flat} \Rightarrow f \;\text { satisfies } (DIM)$
For example a (non-trivial) blowup is not flat.
d) For a morphism $f:X\to Y $ between connected holomorphic manifolds we have:
$$f \text { is flat} \iff f \text { is open} \quad \iff (DIM) \;\text {holds}$$ For example a submersion is flat, since it is open.
e) Given two complex spaces $X,Y$ the projection $X\times Y\to X$ is flat ( Not trivial: recall that open doesn't imply flat!).
f) flatness is preserved by base change.
g) The normalization $f:X^{nor} \to X $ of a non-normal space is never flat.
For example if $X\subset \mathbb C^2$ is the cusp $y^2=x^3$, the normalization morphism $\mathbb C\to X: t\mapsto (t^2,t^3)$ is not flat.
h) Given a finite morphism $f:X\to Y $, each $y\in Y$ has a fiber $X(y) \subset X$ and for $x\in X$ we can define $\mu (x)= dim_{\mathbb C} (\mathbb C \otimes_{\mathcal O_{Y,y}} \mathcal O_{X,x})$ . Now for $ y\in Y$ we put $\; \nu (y)= \Sigma_{x\in X(y)} \mu(x)$ and we obtain :
$$f \; \text{ flat} \iff \nu :Y\to \mathbb N \text { locally constant}$$
[Of course for connected $Y$, locally constant = constant]
Bibliography:
A. Douady, Flatness and Privilege, L'Enseignement Mathématique, Vol.14 (1968)
G.Fischer, Complex Analytic Geometry, Springer LNM 538, 1976