Pome12F28DB77548When $h$ is not Kähler, the scalar in theorem 15.22.2 is still determined by masses if $h$ satisfies the power condition $\mathrm{dd}^{\mathrm{c}}(h^ k)=0$, $1\leq k\leq n-1$, of theorem 11.1.7, which fixes the mass of every bounded solution. The condition persists under pullback and under addition of a smooth closed form: expand each power and apply the original identities to its terms. For $n=2$ the condition is simply $\mathrm{dd}^{\mathrm{c}}h=0$, so it includes the nonclosed Hopf metric of proposition 19.8.13. The residual form then has strictly smaller mass than $h$, the difference being carried by the divisor.
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
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
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. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.