TITLE: Is the Euler characteristic a birational invariant
QUESTION [13 upvotes]: Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have  that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the Hirzebruch-Riemann-Roch theorem, this boils down to showing that $$ \chi(X,\mathcal{O}_X) = \chi(Y,\mathcal{O}_Y).$$ 
This is probably a basic fact. A stronger statement is apparently shown in Birationale Transformation von linearen Scharen auf algebraischen Mannigfaltigkeiten by van der Waerden. The only problem is that I can't seem to find it in that article (probably because I don't read that well German).
For $\dim X =2$ one can prove this as Hartshorne does as follows.
Any birational transformation of nonsingular projective curves can be factored into a sequence of monoidal transformations and their inverses. For such a monoidal transformation, the result follows from Proposition 3.4 in Chapter V of Hartshorne. 
Does this work in the general case?

REPLY [17 votes]: The dimensions $h^i(\mathcal O_X)$ of the cohomology groups of $\mathcal O_X$, and thus the Euler characteristic, are birational invariants of smooth proper varieties in positive characteristic as well, by a recent work of Andre Chatzistamatiou and Kay Rülling. It is not published yet but a preprint is available.