Higher cohomology of line bundles and small modifications

Question

I am a PhD student in algebraic geometry and I am blocked on a question that I asked to myself for my research. I am not yet very comfortable with the traditional tools. Also, I apologize in advance if the question has already been asked (I checked and I do not believe so, thought there are similarities to what it is asked in the question "Higher cohomology of line bundle and flops", which is however unanswered).

Suppose that we have a small modification of smooth algebraic varieties $f : X \dashrightarrow X'$ (for me, this is a birational map such that there exists two open subsets $U \subset X$ and $U' \subset X'$ such that $ \operatorname{codim} X \backslash U, \operatorname{codim} X' \backslash U' \geq 2 $, $f\vert_U : U \to U'$ is an isomorphism) and a line bundle $\mathcal O_X(D)$ associated to a divisor on $X$. Consider the induced pushforward $$f_* : \text{Div}(X) \to \text{Div}(X'), ~A \mapsto \overline{f(A\vert_U)}$$ and write $D' := f_*(D)$. It is known by Hartog's theorem that $$H^0(X, \mathcal O_X(D)) \simeq H^0(X', \mathcal O_{X'}(D'))$$ and this has already been repeatedly demonstrated here (see for example the Q&A "Zero-cohomology of birational varieties").

Is there a way to compare the higher cohomology groups $H^i(X,\mathcal O_{X}(D))$ and $H^i(\mathcal O_{X'}(D'))$ for $i > 0$ in general ? Or do you have special cases where a comparison is possible (for example, projective varieties with trivial canonical divisor) ? I have not found anything on the net and I have no idea personally. Thank you.

Answer 1

A straightforward way to relate these bundles is by taking a smooth variety $\hat{X}$ dominating both $X$ and $X′$, and relating the pullbacks of your line bundles to $\hat{X}$. They will differ by a linear combination of exceptional divisors, so there will be exact sequences relating the bundles which can be used to compare their higher cohomology.