Lemma 19.3.2224E66EB949E4

Proof. We first prove independence of the choice of $h$. Let $h_ i\colon \widehat Z_ i\rightarrow Z$, $i=1,2$, be two choices, and let $(W^ i_{\bullet },g_ i)$ be the corresponding liftings, with $g_ i=\operatorname{cor}(Y_{\bullet },\pi \circ h_ i)$. Choose a common modification over $Z$, $q_ i\colon U\rightarrow \widehat Z_ i$ such that $Y_{\bullet }$ admits a lifting $W_{\bullet }$ to $U$. Then $W_{\bullet }$ is also the lifting of $W^ i_{\bullet }$ with respect to $q_ i$. Put

By proposition 19.3.14, proposition 19.3.17,

Since $h_1\circ q_1=h_2\circ q_2$, the left-hand side is independent of $i$. This proves that (19.3.21) is well-defined.

For the birational invariance, choose a proper bimeromorphic morphism $q\colon \widehat Z\rightarrow Z'$ from a compact Kähler manifold such that $Y_{\bullet }$ admits a lifting to $\widehat Z$ with respect to $\pi \circ p\circ q$. Computing $\nu _{Y_{\bullet }}(p^*S)$ using $q$, and computing $\nu _{Y_{\bullet }}(S)$ using $p\circ q$, gives the same expression. $\square$