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Proposition 15.22.10D1ADF0EFDD30

Proposition 15.22.10.D1ADF0EFDD30 Under (15.22.1), assume $\mathrm{dd}^{\mathrm{c}}(h^ k)=0$ for $1\leq k\leq n-1$, and put $H=\pi ^*h$. Then

00D02AC76853 15.22.11
\begin{equation} \int _ Xh^ n-\int _ Y\beta ^ n =\sum _ jc_ j\sum _{i=0}^{n-1} \int _{D_ j}H^ i\wedge \beta ^{n-1-i}>0. \end{equation}

Thus the constructed solution of theorem 15.22.2 has $\kappa =\int _ Y\beta ^ n/\int _ Xfh^ n$. If $\int _ Xfh^ n=\int _ Xh^ n$, its scalar is strictly less than one.

Proof. Let $C=\sum _ jc_ j\Theta _ j-\mathrm{dd}^{\mathrm{c}}b$, so $\beta =H-C$ with $C$ closed. The mixed forms $Q_ i=H^ i\wedge \beta ^{n-1-i}$ satisfy $\mathrm{dd}^{\mathrm{c}}Q_ i=0$: expand $\beta =H-C$ and apply the power condition to each $H^ k$. With $\Psi =\psi \circ \pi $, $\mathrm{dd}^{\mathrm{c}}\Psi =\sum _ jc_ j[D_ j]-C$. Distributional integration by parts against the smooth $\mathrm{dd}^{\mathrm{c}}$-closed $Q_ i$ gives

\[ \int _ Y(H^ n-\beta ^ n)=\sum _ i\int _ Y C\wedge Q_ i =\sum _ jc_ j\sum _ i\int _{D_ j}Q_ i. \]

Change of variables gives $\int _ YH^ n=\int _ Xh^ n$. Every integrand on the right is nonnegative, since $H$ is semipositive and $\beta $ positive; the $i=0$ term on each nonempty $D_ j$ is strictly positive. This includes exceptional divisors. By theorem 11.1.7, the residual solution has mass $\int _ Y\beta ^ n$, giving the scalar. $\square$

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