# A criterion for isotriviality of families of varieties of general type

Let $$f:X\rightarrow Y$$ be a surjective map between smooth varieties with connected fibers. Assume that the generic fiber of $$f$$ is of general type, and everything is over $$\mathbb{C}$$. $$f$$ is said to be isotrivial if all its smooth fibers are isomorphic.

When $$\mathrm{dim}X=2$$ and $$\mathrm{dim}Y=1$$, if $$\mathrm{deg}(f_*\omega_{X/Y})=0$$, then $$f$$ is isotrivial, in BPV,Chapter III. Theorem 17.3.

My question is that does this true for $$\mathrm{dim}X\geq 3$$ and $$\mathrm{dim}Y=1$$?

The tools to prove this are using the period map and Torelli's theorem for curves.

In case $$\mathrm{dim}X=2$$ and $$\mathrm{dim}Y=1$$: If $$\mathrm{deg}(f_*\omega_{X/Y})=0$$, then the period map $$\mathcal{P}:Y\rightarrow \overline{\Gamma/D}$$ is constant, where $$D$$ is the period domain and $$\Gamma$$ is the monodromy group. Thus $$\mathcal{P}(Y^0)$$ is a point in $$\Gamma/D$$, where $$Y^0\subseteq Y$$ such that $$f^0:X^0=f^{-1}(Y^0)\rightarrow Y^0$$ is smooth. By Torelli's theorem, all nonsingular fibers will be isomorphic.

In higher dimensional cases, it seems that Torelli's theorem is not true in general. Does this true that $$\mathrm{deg}(f_*\omega_{X/Y})=0$$ implies the associated period map is constant?

• The title of the question has general type in it, but not the question itself! Without general type assumption, e.g. for del Pezzo surfaces, the statement will be definitely false because the canonical class of the fibers will be negative, so the pushforward will be zero, but del Pezzo surfaces have moduli. Commented Jun 16, 2020 at 16:19
• Thank you for your remind! The assumption of the generic fiber of $f$ being of general type is necessary for my question. Commented Jun 17, 2020 at 7:42
• Couldn't this sheaf be zero? Commented Jun 17, 2020 at 9:28
• A stupid counterexample would be to blow up a moving family of curves inside a product of two fake projective planes, in which case $f_* \omega_{X/Y} = 0$. I guess that a corrected conjecture would be that if $\deg f_* \omega_{X/Y}^\nu = 0$ for all $\nu>0$ then the family is birationally isotrivial. Commented Jun 17, 2020 at 9:51
• I agree with Piotr; in fact the "corrected conjecture" seems to be a theorem in great generality (log case; any dimension fiber/base) see Theorem 1.3 arxiv.org/pdf/1503.02952.pdf. For surfaces, I would expect $\nu =5$ or $4$ should suffice (as $5K_X$ recovers the canonical model and $4K_X$ gives a birational map if I remember correctly). Commented Jun 18, 2020 at 17:14