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Let $B=B(0,1)$ be the open unit disc in $\mathbf C$. Let $X\to B$ be a smooth projective morphism of complex manifolds with $X$ connected. What can we say about $X$? For instance, if $X\to B$ is of relative dimension zero, we have that $X\cong B$. Does a similar type of "triviality" hold in relative dimension $>0$? That is, do we have that $X \cong B\times V$ for some manifold $V$? |
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Consider $X\to B$ the elliptic fibration given by an affine equation $y^2=x^3+tx+1$, where $t$ is the coordinate function of $B$. Computing the $j$-invariants of the fibers we see that the family is not constant. If $X$ was isomorphic to some $B\times V$, then one can show that each fiber $X_t$, $t\in B$, is isomorphic to $V$ (because $X_t$ is a projective complex manifold and its image in $B\times V$ is a closed analytic subset and compact). Contradiction. |
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