20.3.72 Pullback and strict representativesD36C5D62DF35

Proof. By proposition 5.43.18, every $\pi ^*\theta $-plurisubharmonic potential descends. Since $\pi $ is surjective, ordinary inequalities are equivalent before and after pullback. Thus the competitors are in bijection. On the locus where $\pi $ is an isomorphism, locality identifies their products. Both extended products give the excluded analytic subsets zero mass, so

Taking the infimum and supremum proves the identities. Smooth gauge preserves the actual currents and measures; it preserves multiplier ideals because $\mathrm e^{sf}$ is locally bounded above and below. $\square$

Proof. Put $\beta =(c-b)w_1+aw_2$. The representatives of lemma 20.3.65 show that $U$ satisfies (18.27.1); by lemma 18.27.2 so does $\pi ^*U$. The logarithmic divisor is $b\pi ^*D$, whose smooth curvature is $b\pi ^*w_1$, so its residual background is $\pi ^*\beta $. If a strict current existed in the ordinary type of $\pi ^*r$, theorem 18.27.4 would give a smooth $f$ with $\pi ^*\beta +\mathrm{dd}^{\mathrm{c}}f>0$. The exceptional curve $E$ of the last blowup is contracted by the composite map. Restriction to this smooth curve would give $\mathrm{dd}^{\mathrm{c}}(f|_ E)>0$, contrary to its zero integral by Stokes. Only the smooth function $f$ is restricted, so this argument applies also when the original potential has a pole along $E$.

For the one-sided assertion, start with the strict representative $v_0=bG_ p+U_ L$ from theorem 20.3.66. Suppose $v_ j$ is strict with lower bound $\delta _ jh_ j>0$ on the current surface, and let $\pi _{j+1}$ be the next point blowup. For its exceptional divisor $E_{j+1}$, choose a smooth metric on $\mathcal O(E_{j+1})$ with curvature $\eta _{j+1}$ negative along the vertical tangent, and write $\mathrm{dd}^{\mathrm{c}}\ell _{j+1}=[E_{j+1}]-\eta _{j+1}$. For all sufficiently small $t_ j>0$, $\pi _{j+1}^*h_ j-t_ j\eta _{j+1}>0$. To see this, in a blowup chart $(x,z)\mapsto (x,xz)$ the pullback has a uniformly positive horizontal coefficient, mixed coefficient $O(|x|)$ and Schur complement at least $c_1|x|^2$. The perturbation adds a positive vertical term at least $t_ jc_2$, while its error in the Schur complement is bounded by $C(t_ j|x|+t_ j|x|^2+t_ j^2)$. Shrink the exceptional neighborhood and then choose $t_ j$ small; on its compact complement positivity persists under small perturbations. Thus

is strict. These genuine quasi-plurisubharmonic pullbacks are defined even over pole centers. The logarithmic terms are bounded above, so induction gives $v'\leq \pi ^*r+C$. Finally proposition 20.3.73 preserves both volumes in the pulled-back ordinary type. The additional exceptional poles of the one-sided witness are not competitors in that fixed type. $\square$