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?