Consider the Hodge numbers of smooth projective varieties over $\mathbb C$. I am aware of the fact that only the outer Hodge numbers (essentially $h^{p,0}$) are invariant under birational equivalence. I would like to know whether there will be more invariances if we have some additional conditions. More precisely, say $X$ and $Y$ are birational equivalence via some rational map $f: X\dashrightarrow Y$, and we moreover assume that
the birational map $f$ can be defined on a larger open set such that the complement has some higher codimension $d$,
or
both $X$ and $Y$ are fiber bundles over some curve $T$ and the rational map $f: X \dashrightarrow Y$ is an isomorphism of fiber bundles over some open subset of $T$,
then, will there be more birational invariances?
The only example I know of non-birationally-invariant Hodge numbers comes from blow-up, and I guess basically this is the only reason. From this, I believe the answer would be quite positive. But I do not know how to prove it.
Any references would be helpful. Thanks in advance.
Best Answer
One can use the Mayer-Vietoris sequence to show if one blows up a smooth subvariety of dimension $d$, then only the $H^{p,q}$ with $|p-q|\leq d$ will be changed. Together with the weak factorization theorem, it proves the first statement we want.