Lemma 18.22.24A14B90776C1

Proof. The positive $(0,0)$-current $S=p_*(\gamma ^{n-1})$ on $C$ satisfies $\mathrm{dd}^{\mathrm{c}}S=p_*\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$. A harmonic distribution is smooth, so $S$ is a constant $\kappa _\gamma $. On the complement of the critical values, smooth fiber integration identifies it with the positive integral of $\gamma ^{n-1}$ on a regular fiber. Thus $\kappa _\gamma >0$.

First let $\mu $ be positive with mass $M>0$. Choose smooth positive measures $\mu _ j$ with the same mass and $\mu _ j\rightharpoonup \mu $. Their mean-zero solutions $q_ j$ have a common lower curvature bound $-M\eta $. Normalized compactness bounds $\sup _ Cq_ j$: their mean-zero condition makes this supremum the negative mean of their supremum-zero normalization. Hence $(q_ j)$ is precompact in $L^1(C)$. Every limit has mean zero and Hessian $\mu -M\eta $. Two such limits differ by a harmonic function, hence by a constant, which their means force to be zero. Therefore $q_ j\to q_\mu $ in $L^1(C)$.

By 5.43.10, $q_ j\circ p\to q_\mu \circ p$ in $L^1(X)$. Thus $p^*\mu _ j\to p^*\mu $ as currents and their smooth Hermitian traces converge weakly. For smooth $\mu _ j$, the definition of $S$ gives $p_*L_\gamma (\mu _ j)=\kappa _\gamma \mu _ j$. Passing to the limit proves the identity for $\mu $. The case $M=0$ is zero, and linearity proves the signed identity.

For a positive $\mu $, split it into $\mathbf1_ B\mu $ and $\mathbf1_{C\setminus B}\mu $. Their traces are positive; the pushforward identity shows that they are respectively carried by $p^{-1}(B)$ and its complement. Additivity then gives the restriction identity. Apply the same argument to the positive and negative Jordan parts of a signed measure. Their disjoint Borel carriers pull back to disjoint carriers, so their traces are the Jordan parts upstairs. The pushforward identity gives the asserted total-variation norm.

Finally the Poincaré–Lelong formula in the book normalization gives $p^*\delta _ s=[F_ s]$, including every zero multiplicity of a local equation of the fiber. Its trace has total mass $\kappa _\gamma $, by the pushforward identity with $\mu =\delta _ s$. $\square$