Theorem 9.11.343DAEEE3C82F5

Proof. We first establish the divisor formula

Indeed, $\log x=\log |t|^2+\log S-\log (1+|t|^2S)$ and $\mathrm{dd}^{\mathrm{c}}\log |t|^2=[t=0]$. Along $E$ the kernels of $\beta $ and $\sigma $ are complementary, and $\beta >0$ off $E$. Thus $\beta _ a$ is a smooth Kähler form. Fiber integration also gives

By (9.11.31), $P_ a(0)=a$, $P_ a(1)=1$ and

The numerator defining $Q_ a'$ vanishes at both ends, so $Q_ a$ is smooth. On $0<x<1$, a local potential for the full positive current is $F_ a(\rho _\psi )$, where

Direct differentiation gives $F_ a'=P_ a(s)$ and $F_ a''=c_ a k(s)s^ n(1-s)/P_ a(s)^{n-1}$. Consequently

The coefficients are smooth since $P_ a\geq a>0$. On $E$ the restriction is $a\sigma _\psi $; at infinity the horizontal coefficient is one and the transverse coefficient relative to $B_\psi -\sigma _\psi $ is $c_ a k(1)>0$. This proves all the asserted positivity properties, including along both sections.

The functions $\varphi \circ \pi $ and $\varphi \circ \pi +q_ a$ are quasi-plurisubharmonic on $Y$. By (9.11.36), their curvature currents in the background $\beta $ are $a[E]+\beta _ a$ and $a[E]+A_ a$, respectively, so both are positive. The descent theorem theorem 9.11.5 gives $\varphi \in \mathrm{PSH}(X,\theta _0)$ and a potential $u_0$ with $u_0\circ \pi =\varphi \circ \pi +q_ a$. The bounded residual $q_ a$ puts $u_0$ in the ordinary class of $\varphi $ and gives the uniform logarithmic asymptotics at $p$. Its finite supremum is the constant $K$ of the theorem.

Let $H$ be admissible. Self-adjointness makes $\exp H$ positive, and $\det \exp H=\exp (\operatorname {tr}H)=1$. Hence

The function $u=u_0-K$ does not depend on $H$. On the regular locus its $\theta _ H$-curvature form is the positive form in (9.11.39); near $p$ the background and the potential are unchanged. Thus $u$ is $\theta _ H$-plurisubharmonic. Formula (9.11.36), together with the deformation, gives $\pi ^*\theta _ H+\mathrm{dd}^{\mathrm{c}}(u\circ \pi )=a[E]+A_ a+\pi ^*\eta _ H$; removing the divisor proves the equation. The mass of the solution is $c_ aM=V$. This calculation uses the preservation of the determinant, rather than a mass identity for the class of a nonclosed form.

Bounded residuals and uniqueness. Let $w\in \mathcal C_{a,H}$. On $Y\setminus E$, its residual $g=w\circ \pi -\varphi \circ \pi $ is bounded and $\beta _{a,H}$-plurisubharmonic. By lemma 5.24.7, it extends across $E$ with the same background. Conversely, for any bounded $\beta _{a,H}$-plurisubharmonic $g$, the function $\varphi \circ \pi +g$ has the positive curvature current $a[E]+\beta _{a,H}+\mathrm{dd}^{\mathrm{c}}g$ in the pulled-back background. It descends by theorem 9.11.5, and its uniform bounds relative to $\varphi \circ \pi $ put the descended potential in $\mathcal C_{a,H}$. Thus bounded residuals correspond exactly to members of the prescribed ordinary class.

The bounded product is finite on the compact manifold $Y$ and gives $E$ zero mass, by theorem 5.27.1, proposition 5.28.2. Thus

For $H=0$, closedness of $\beta _ a$ makes every term of positive degree in $\mathrm{dd}^{\mathrm{c}}g$ in the bounded expansion integrate to zero, so (9.11.37) gives mass $V$ for every member of the class. In the nonclosed case only finiteness is used here.

If $w$ solves the equation with a scalar $c'>0$, its lifted measure is $c'\nu $, since both sides give $E$ zero mass. Both residuals, $g$ and $q_ a-K$, are continuous by corollary 14.19.29, and the uniqueness of the scalar in proposition 14.19.34 gives $c'=c_ a$. The density $\nu /\beta _{a,H}^ n$ is smooth and nonnegative, and its positive locus is exactly $Y\setminus E$. This locus is connected and dense: it is a bundle over the connected base $B$ whose fibers, $\mathbb P^1$ minus a point, are connected. Theorem 14.24.1 shows that $g-(q_ a-K)$ is constant. The normalization fixes this constant, and both full functions equal $-\infty $ at $p$. This proves uniqueness in the ordinary class.

The uniform logarithmic asymptotics give the continuity of the full solution with values in $[-\infty ,+\infty )$. Its residual restricts to $E$ as $a\psi +Q_ a(0)-K$. If $\psi $ is nonconstant, this restriction is nonconstant, so the residual is not a pullback from $X$, since a pullback is constant on $E$. This concerns only the residual; the full potential is $-\infty $ on the whole fiber $E$ and does descend.

Nonclosed deformations. Fix $z_0\in O$ and a smooth $A_ a$-unitary frame on a smaller ball. Let $K_0=\operatorname {diag}(1,-1,0,\ldots ,0)$ in this frame. Choose a real compactly supported function $\chi $ with $\chi (z_0)=0$ such that $\mathrm{d}\chi (z_0)$ is a nonzero covector in the first two complex directions, and, for $\epsilon \in \mathbb R$, put $H=\epsilon \chi K_0$. At $z_0$,

Terms containing derivatives of the frame and of $A_ a$ vanish at $z_0$ because $\chi (z_0)=0$. The restriction of $\gamma $ to these four real directions is nondegenerate, so wedge multiplication by $\gamma $ sends a nonzero one-form to a nonzero three-form. Thus $\mathrm{d}\theta _ H(z_0)\ne 0$ for $\epsilon \ne 0$. On the fixed compact support, both strict inequalities (9.11.33) persist for sufficiently small $|\epsilon |$, and $H\to 0$ smoothly as $\epsilon \to 0$. This proves the existence of arbitrarily small nonclosed admissible choices; admissibility itself does not force nonclosedness.

For any resolution $\tau :Z\to X$, theorem 9.11.26 identifies the pushforward of its non-pluripolar product with (9.11.40), which proves (9.11.35). Its total mass equals the finite mass on $X$. Since the ordinary class, the equation and the normalization are intrinsic to $X$, uniqueness gives the same solution and scalar on every resolution and common refinement. $\square$