3
$\begingroup$

Disclaimer. I don't know much about the things I'm asking. This is why my other question pencils on varieties of general type was a bit unclear. I believe the following question makes up for this.

Let $X$ be a smooth projective complex algebraic variety with $K_X$ ample. When does there exist a flat projective (non-isotrivial) morphism $X\to \mathbf{P}^1$ with geometrically connected fibres which are not of general type?

The answer is never when $\dim X \leq 2$. Is the answer also never when $\dim X =3$?

If you assume $\Omega^1_X$ to be ample, then the answer is also never. In fact, in this case, every subvariety of $X$ is of general type.

What if we also put a restriction on the Kodaira dimension of the fibration $X\to \mathbf P^1$, say, the Kodaira dimension is not positive.

What if we replace $\mathbf P^1$ by a smooth projective curve $C$ of positive genus?(This is not the same question, because I want the fibres of my fibration to be connected.) Of course, this is asking for much more. For instance, the Albanese of such an $X$ is going to be non-zero.

Here's a slightly different question: are there $X$ as above with infinitely many distinct abelian varieties $A_1,A_2,\ldots$ mapping non-constantly to $X$? (This is related to "hyperbolicity" properties of $X$.)

$\endgroup$
0

1 Answer 1

10
$\begingroup$

Let $F$ be a general fiber of a fibration $X\to B$, where $B$ is a smooth variety. The normal bundle to $F$ in $X$ is trivial, so $K_F=K_X|_F$. Thus if $K_X$ is ample then $K_F$ is also ample and $F$ is of general type. The same argument shows that $F$ is of general type also when $K_X$ is just big.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.