Consider a surjective holomorphic map between two complex projective manifolds $\pi :X \rightarrow Y$. Iitaka conjectured the subadditivity of Kodaira dimensions: $\kappa(X)\geqslant\kappa(Y)+\kappa(X_y)$ where $X_y$ is a generic fibre. We know already this holds when $dim(X)=dim(Y)+1$ and when $\pi$ is a fibre bundle. In particular, the conjecture holds for surfaces fibred over curves.

I want to know if there are some conditions to garantee the equality in the conjecture. The equality holds for a product, how about fibre bundles? In dimension two, if we have a fibred surface with neither singular nor multiple fibre, does the equality hold?


There is another inequality which says the following:

Easy addition (Using the same notation): $$ \kappa(X)\leqslant \kappa(X_y) + \dim Y $$

Consequently, if $Y$ is of general type, i.e., $\kappa(Y)=\dim Y$, then the subadditivity conjecture is equivalent with equality instead of inequality.

Subadditivity is also known in the case $X_y$ is of general type (due to Kollár) and of course the inequality is trivially an equality (without easy addition) if both $X_y$ and $Y$ are of general type.

A problematic case is when $\kappa(Y)<0$. Then subadditivity doesn't say much and in general one cannot expect equality (although various hyperbolicity results can help). For instance, any elliptic K3 surface is fibred over $\mathbb P^1$ and gives an example when the inequality is strict.

One should probably add that there is a more sophisticated version of subadditivity (due to Viehweg) which predicts that if $\kappa(Y)\geq 0$, then $$ \kappa(X)\geqslant\kappa(X_y) + \mathrm{max}\{\kappa(Y), \mathrm{Var}(f)\}, $$ where $\mathrm{Var}(f)$ is the variation of $f$, i.e., the measure of how much the family varies in moduli (if there is a corresponding moduli space then this is the dimension of the image of the corresponding moduli map). So, if $\kappa(Y)\lneq \mathrm{Var}(f)$, then the inequality in the original format is expected to be strict.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy