12.22 Comparison under an arbitrary modification9BED3D691F82

Proof. First let $q:M'\to M$ be one blowup along a smooth center, with exceptional divisor $E$, and let $h_ M$ be a Hermitian form on $M$. The restriction of $\mathcal O(-E)$ to each projective fiber of $E$ is $\mathcal O(1)$. A Hermitian metric on the normal bundle of the center therefore induces a metric on $\mathcal O(-E)|_ E$ whose curvature is positive on the tangent spaces of the fibers. Extend this metric to $\mathcal O(-E)$ on $M'$ by extending the smooth logarithm of its ratio to a fixed reference metric from the closed submanifold $E$, and let $\beta $ be its curvature. The semipositive form $q^*h_ M$ is positive off $E$, and at points of $E$ its kernel is exactly the tangent space of the fiber, on which $\beta $ is positive. Hence $Cq^*h_ M+\beta >0$ for every sufficiently large $C$: otherwise, unit tangent vectors violating the inequality for $C\to \infty $ would converge, by compactness, to a vector in the kernel of $q^*h_ M$ on which $\beta $ is nonpositive. With $\eta _ E=-\beta $, the curvature of the induced metric on $\mathcal O(E)$, this reads $Cq^*h_ M-\eta _ E>0$.

For a composition of blowups, apply this step successively, with $h_ M$ the positive form already obtained on the previous blowup. Each step multiplies the previous divisor by a positive constant, pulls it back and adds the new exceptional divisor, so the divisor stays effective and exceptional. This proves (12.22.2). Each exceptional divisor maps into the image of its center, which has dimension at most $n-2$. $\square$

Proof. Use the factorization $p=\pi \circ F$ above and lemma 12.22.1. By compactness there is $a>0$ with $h_ Z\geq aF^*h_ Y$. Pulling back the semipositive form $\Psi $ and integrating, we obtain

Here the changes of variables use that $F$ and $p$ have degree one, and the Poincaré–Lelong formula, together with $\mathrm{dd}^{\mathrm{c}}p^*\Psi =0$, replaces the pairing with $\eta _ D$ by the integral over $D$. This last integral vanishes: on the regular locus of each component of $D$, the restriction of $p$ has rank at most $n-2$, so the restriction of the $(n-1,n-1)$-form $p^*\Psi $ is zero. Taking $K=C_0/a$ proves the assertion. $\square$

Proof. Step 1: nefness. Choose $C'$ with $\pi ^*h_ X\leq C'h_ Y$. For $\epsilon >0$, pull back a smooth representative $\theta +\mathrm{dd}^{\mathrm{c}}u_\tau \geq -\tau h_ X$ with $\tau <\epsilon /C'$; this gives a positive representative of $\pi ^*\theta +\epsilon h_ Y$. Thus $\pi ^*\theta $ is nef, and both volumes in (12.22.6) are positive.

Step 2: regularized tests. The ratio $\pi ^*\mu _ X/\mu _ Y$ is smooth, nonnegative and bounded on the compact manifold $Y$, so there is $c>0$ with $\mu _ Y\geq c\pi ^*\mu _ X$. Fix a smooth Gauduchon metric $\gamma _0$ on $Y$, given by theorem 11.4.2. For an arbitrary Gauduchon metric $\gamma $ on $X$, write $\Psi =\gamma ^{n-1}$, and for $\delta >0$ define $\Gamma _\delta $ by

The right side is a strictly positive $(n-1,n-1)$-form, and such a form has a unique smooth positive $(n-1)$st root, obtained by solving for the Hermitian matrix of the root from the positive cofactor matrix. Since the right side is $\mathrm{dd}^{\mathrm{c}}$-closed, $\Gamma _\delta $ is a smooth Gauduchon metric on $Y$.

Step 3: the numerator. For $\epsilon >0$, lemma 12.22.3 gives

The first term in the middle is nonnegative because $\theta $ is nef, and the second is positive because $\pi ^*\Psi $ is positive on the set where $\pi $ is biholomorphic, so the limit is positive.

Step 4: the denominator. We claim that

To justify the limit on the exceptional locus, represent $\Gamma _\delta ^{n-1}$ by its positive cofactor matrix. The density of $(\Gamma _\delta ^ n)^{(n-1)/n}$ is a fixed dimensional constant times the $1/n$th power of the determinant of that matrix, and this expression extends continuously to positive semidefinite matrices. Since $Y$ is compact, dominated convergence applies. The inequality follows from $\mu _ Y\geq c\pi ^*\mu _ X$ and the change of variables for the degree-one map $\pi $.

Step 5: conclusion. Test the infimum defining $\lambda _{\mu _ Y}(\pi ^*\theta +\epsilon h_ Y)$ with $\Gamma _\delta $, use (12.22.7) and (12.22.8), and let $\delta \downarrow 0$. For every Gauduchon metric $\gamma $ on $X$ this gives

Taking the infimum over $\gamma $ and then the $n$th power proves (12.22.6) with $C=c^{-1}$.

Dividing the logarithm of (12.22.6) by $\log \epsilon <0$ reverses the inequality. The bounded error $\log C$ and the rescaling $\epsilon \mapsto K\epsilon $ do not affect lower and upper limits, as in the proof of proposition 12.20.1. This gives $\underline a(\pi ^*\theta )\geq \underline a(\theta )$ and $\overline a(\pi ^*\theta )\geq \overline a(\theta )$. Finally, a proper holomorphic map of degree one between compact Riemann surfaces is an isomorphism, which settles the case $n=1$. $\square$