PomeFE6CD72EA03AProposition 10.16.9.FE6CD72EA03A Let $W$ be a connected compact Kähler manifold of dimension $k>0$, let $g:W\to X$ be holomorphic, and suppose $\dim g(W)<k$. Then for every class $a\in H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$,
Proof. If the image is a point, then $g^*a=0$ and (10.16.6) gives the assertion. Otherwise, choose a Kähler resolution $r:Q\to g(W)$ and resolve the graph of the meromorphic lift of $g$ to $Q$. This gives a Kähler modification $\rho :\widehat W\to W$ and a holomorphic map $F:\widehat W\to Q$ with $g\circ \rho =r\circ F$, where $r$ is also viewed as a map into $X$.
Choose a smooth representative $\eta \in a$, a Kähler form $\kappa _ Q$ on $Q$, and $C>0$ such that $C\kappa _ Q-r^*\eta \geq 0$. Since $\rho ^*g^*\eta =F^*r^*\eta $, the forms
are closed and semipositive. Since $\dim Q<k$, we have $P^ k=0$. For any positive current $S\in g^*a$, the current $\rho ^*S+A$ is positive and lies in the nef class $\{ P\} $. Hence
Modification invariance (Step 3 in the proof of proposition 10.16.3) now gives the result. If the class has no positive representative, its invariant is zero by definition. $\square$
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