I am struggling with the following problem: given $f : M \rightarrow B$ a proper smooth family of compact complex manifolds I want to show that the normal bundle of the fiber $M_b = f^{-1}(b)$ is trivial. First of all let me write down the definition: a proper smooth family of compact complex manifolds is a proper and holomorphic map $f: M \rightarrow B$ between two complex manifolds such that:
- M is nonempty and B is connected
- the differential of $f$ is surjective at every point
From this definition follows that $M_b = f^{-1}(b)$ is a compact complex submanifold of $M$. As I said I want to show that its normal bundle in $M$ is trivial.
My idea is: by implicit function theorem we know that the local coordinates are given by the projections. Let's suppose $dimM =m$, $dimB=k$. Then I define: $$ s_i : M_b \rightarrow N_{M_b|M}, s_i(p) = \frac{\partial}{\partial z_{j^p_i,p}} $$ Where $J(p) = \{j^p_1, \dots, j^p_{k}\} \subset \{1, \dots, m\}$ is the subset of indexes whose local coordinates are not needed to describe $M_b$ and the local coordinate around $p$ in $M$ are $z_{1,p}, \dots, z_{m,p}$. If I show that these are sections of the normal bundle then I've done because by definition they are independent at every point. What I have to show is only that these applications are holomorphic. However the sets $J(p)$ are locally constants because they are made by the sets of indexes such that $$ det(\frac{\partial f}{\partial z_{i,p}}) \neq 0 $$ for $i \in J(p)$. Being $f$ holomorphic it is continuous, therefore $det(\partial f/\partial z_{i,*})$ is continuous. Consequently for each point $p \in M_b$ there exists a neighborhood such that $J(p) = J(q)$ for all $q \in U$. Then in this neighborhood the applications $s_i$ are simply those that associate to every point the derivation $\partial/\partial z_{j^p_i,*}$. Being these holomorphic maps I've done.
Am I right? Thank you in advance!
EDIT Thinking again about this problem I came out with the following example that seems to contradict the statement I was trying to prove. Let us take: $$ f: \mathbb{P}^1\mathbb{C} \times \mathbb{P}^1\mathbb{C} \rightarrow \mathbb{P}^1\mathbb{C} $$ the projection. Then $f$ is a proper smooth family. The normal bundle of the fiber $M_b$ is simply the tangent bundle to $\mathbb{P}^1\mathbb{C}$ that is not trivial. I would be very grateful if anyone could tell if this example is correct or whether I missed something. If it's wrong in which direction should I go to prove the statement? Thank you!