PomeProposition 10.21.9.86DDC28E0989 Let $u\in \mathrm{PSH}(X,\theta )$ be of minimal type, and suppose that its residual current $S=(\theta +\mathrm{dd}^{\mathrm{c}}u)-c[E]$ extends smoothly and semipositively across $E$. Then, on smaller adapted charts, $0\leq S^ n/\mathrm{d} V\leq C|z|^{2(r-1)}$. Consequently, if the Monge–Ampère measure of $u$ has density $|z|^{2\ell }g$ with $g$ smooth and strictly positive, then $\ell \geq r-1$. This order condition is necessary but not sufficient for a smooth residual. In particular, in this background a smooth strictly positive density is incompatible with a smooth residual for a potential of minimal type.
Proposition 10.21.986DDC28E0989
Proof. Write $u=cL+v$. Since $S=\pi ^*h+\mathrm{dd}^{\mathrm{c}}v$ is smooth, the distributional Laplacian of $v$ is smooth, and local elliptic regularity makes $v$ smooth. On each fiber of $E\to Z$, a compact projective space, $\pi ^*h$ restricts to zero and $\mathrm{dd}^{\mathrm{c}}v$ restricts to the semipositive form $S$; the maximum principle therefore makes $v$ constant on the fiber. Hence every vertical diagonal entry of $S$ is a smooth nonnegative function vanishing at $z=0$, and is therefore $O(|z|^2)$. The other diagonal entries are bounded, so Hadamard's determinant inequality gives the stated bound. The pullback $\pi ^*h$ attains this order, while the examples below show that the order condition is not sufficient. The argument does not apply to prescribed nonminimal singularities, whose residuals can be positive on the exceptional fibers. $\square$
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