PomeB9786385D8CDB9786385D8CD
On the elliptic threefold of section 12.19, closed classes can be described completely. The product of the two horizontal forms is $\mathrm{dd}^{\mathrm{c}}$-exact, so every positive $\mathrm{dd}^{\mathrm{c}}$-closed $(1,1)$-current has zero vertical component, and the fiberwise descent of lemma 11.22.4 identifies it with a current on the base. Closedness descends as well. The closed positive cones, the singularity envelopes and the spaces of sections selected by multiplier ideals are therefore all computed on the base surface.
Let $X$ be the compact threefold of section 12.19, and let
be its elliptic fibration. Write $\sigma _1,\sigma _2$ for the Fubini–Study forms of the two factors, each of integral one, and put $\alpha =\pi ^*\sigma _1$ and $\beta =\pi ^*\sigma _2$. The product coordinates of that construction show that $\pi $ is locally a holomorphic product with fiber the elliptic curve $E$. The logarithmic ratio $t$ of that construction is defined only on the covering space, but its differentials descend to $X$. We take for $\omega $ the balanced metric $\omega =\alpha +\beta +D$, where $D=\mathrm{d}t\wedge \mathrm d^{\mathrm c}t$. By proposition 12.19.18, it satisfies
The arguments below use this identity, not the vanishing of the lower Monge–Ampère mass.
Lemma 10.19.1.464DA68EB4DA Every positive $\mathrm{dd}^{\mathrm{c}}$-closed real $(1,1)$-current $T$ on $X$ has the form
for a unique positive $\mathrm{dd}^{\mathrm{c}}$-closed $(1,1)$-current $S$ on $B$. Moreover, $T$ is closed if and only if $S$ is closed. Here $\pi ^*S$ is the pullback of currents under the holomorphic submersion $\pi $.
Proof. The measure $T\wedge \alpha \wedge \beta $ is positive. Since $\mathrm{dd}^{\mathrm{c}}T=0$, Stokes' theorem for currents gives
so this measure vanishes. All products in this identity pair a current with smooth forms. Now apply lemma 11.22.4 to $\pi $, with the base Hermitian form $\eta =\sigma _1+\sigma _2$. Since $\eta ^2=2\sigma _1\wedge \sigma _2$, we have $T\wedge \pi ^*\eta ^2=0$, and since the fiber is a compact connected elliptic curve, the lemma gives the unique positive $\mathrm{dd}^{\mathrm{c}}$-closed current $S$. The same lemma shows that $S$ is closed when $T$ is; conversely, pullback commutes with $\mathrm{d}$. $\square$
This description concerns $(1,1)$-currents. It does not conflict with the Gauduchon forms of lemma 12.19.13: those have bidegree $(2,2)$ and satisfy $\mathrm{dd}^{\mathrm{c}}\Gamma _ j=0$, with bounded degree and divergent volume of their positive roots, and they belong to a different cone of currents.
Proposition 10.19.2.24BCA82D5034 Pullback
is injective. The pseudoeffective and nef cones on $X$ coincide and are
For the trace functional of the Gauduchon metric $\omega $, the slice of this cone at trace one is the segment $a+b=(8\pi )^{-1}$. Every class in the cone has a smooth semipositive representative, and none is big. The real form $\beta -\alpha $ is de Rham exact but represents a nonzero Bott–Chern class.
Proof. Suppose a smooth closed real form $\tau $ on $B$ satisfies $\pi ^*\tau =\mathrm{dd}^{\mathrm{c}}u$ with $u$ smooth on $X$. The restriction of $u$ to each compact connected elliptic fiber is harmonic, hence constant, and local holomorphic sections show that $u=v\circ \pi $ for a smooth function $v$ on $B$. Thus $\pi ^*(\tau -\mathrm{dd}^{\mathrm{c}}v)=0$, which implies $\tau =\mathrm{dd}^{\mathrm{c}}v$ because $\pi $ is a submersion. This proves injectivity.
The usual cohomology computation for $\mathbb P^1\times \mathbb P^1$, together with the $\partial \bar\partial $-lemma on this Kähler surface, gives
If $S$ is a positive closed current on $B$, the two coefficients of its class are
By lemma 20.4.10 and lemma 10.19.1, every pseudoeffective class on $X$ is the pullback of the class of such a current, and it therefore lies in the cone of the statement. Conversely, every class in that cone has the smooth semipositive representative $a\alpha +b\beta $ and is therefore nef. A nef class is pseudoeffective by the compactness argument following proposition 10.11.2. This proves the equality of the cones; the whole Bott–Chern group of $X$ is not needed.
Since $\omega ^2=2\alpha \wedge \beta +2(\alpha +\beta )\wedge D$ and $\int _ E D=4\pi $ on each fiber, integration along the fibers gives
This proves the assertion about the slice. By proposition 12.19.22, $X$ carries no closed Kähler current, so none of these classes is big. Finally, the one-form $\mathrm d^{\mathrm c}t$ is globally defined on $X$ and satisfies
The class of $\sigma _2-\sigma _1$ on $B$ is nonzero, and injectivity makes its pullback nonzero in Bott–Chern cohomology. Thus de Rham exactness does not imply $\mathrm{dd}^{\mathrm{c}}$-exactness in this example. $\square$
Consequently, every pseudoeffective class on this threefold is nef and has empty non-nef locus, whereas its non-Kähler locus is all of $X$, since no class is big. All its closed positive representatives come from the two-dimensional base; in particular $(a\alpha +b\beta )^3=0$.
7CFF33409C17Descent also applies to potentials and preserves both ordinary singularity types and multiplier ideals, so the singularity envelopes in these backgrounds are computed on the base. The three-dimensional Monge–Ampère measures of these potentials nevertheless vanish, because their currents have only two horizontal directions.
We use the ordinary singularity order $\preceq $ and equivalence $\sim $ of definition 16.7.1, the ideal order $\preceq _{\mathcal I}$ and equivalence $\sim _{\mathcal I}$, which compare the multiplier ideals $\mathcal I(ku)$ at all weights $k>0$, and the ordinary and multiplier-ideal envelopes $P_\theta [u]$ and $P_\theta [u]_{\mathcal I}$ of definition 5.40.4, definition 18.4.1.
Proposition 10.19.4.2555699F5657 Let $a,b\geq 0$, put $\theta _ B=a\sigma _1+b\sigma _2$ and $\theta =\pi ^*\theta _ B$. Pullback is a bijection
It preserves the ordinary singularity order and equivalence, as well as the ideal order and equivalence. For every $v\in \mathrm{PSH}(B,\theta _ B)$,
Nevertheless, every $u\in \mathrm{PSH}(X,\theta )$ satisfies
Proof. The form $\theta _ B$ is closed, and $\pi $ is a proper holomorphic submersion with connected fibers, so proposition 5.43.5 gives the asserted bijection. Since $\pi $ is surjective, an ordinary singularity inequality holds upstairs exactly when it holds downstairs, with the same constant.
By lemma 5.37.7, multiplier ideals of potentials on $B$ pull back to those of the composed potentials on $X$, at every positive weight. Restriction to local holomorphic sections shows conversely that an inclusion of multiplier ideals on $X$ implies the corresponding inclusion on $B$. Hence the ideal orders and equivalences correspond as well.
The competitors defining both envelopes correspond under the bijection, including the bound by zero and the required ordinary or ideal condition, and their pointwise suprema commute with pullback. Upper semicontinuous regularization also commutes with pullback: in a product chart, the supremum of a pulled-back function over a sufficiently small product neighborhood equals its supremum over the corresponding base neighborhood. This proves the two envelope identities.
Finally, write $u=v\circ \pi $ and, for positive integers $j$, set $u_ j=\max \{ u,-j\} $, which descends to $v_ j=\max \{ v,-j\} $. Since $\theta _ B$ is semipositive, these are bounded admissible potentials. By lemma 5.28.4, $(\theta +\mathrm{dd}^{\mathrm{c}}u_ j)^3=0$ because the base has dimension two. The definition of the extended non-pluripolar measure by increasing truncations now proves the last assertion. $\square$
Thus the ordinary and ideal envelopes coincide in these backgrounds exactly when they coincide on the base, while every three-dimensional Monge–Ampère measure vanishes. Criteria from the closed theory that require positive volume do not apply to these classes, which are not big. The Hermitian background $\omega $ behaves differently: its Monge–Ampère equations with smooth positive densities are solvable, even though the infimum of its masses is zero.
E99505EBA4B4Descent also identifies the spaces of holomorphic sections selected by multiplier ideals, together with their weighted $L^2$ norms. In particular, sections of pulled-back line bundles grow quadratically, as on the two-dimensional base. For integers $a,b$, write $\mathcal O_ B(a,b)$ for the tensor product of the pullbacks of $\mathcal O_{\mathbb P^1}(a)$ and $\mathcal O_{\mathbb P^1}(b)$ from the two factors of $B$.
Proposition 10.19.6.CDBA8166D33B Let $a,b$ be positive integers, let $L_ B=\mathcal O_ B(a,b)$ and $L=\pi ^*L_ B$. For every quasi-plurisubharmonic $v$ on $B$ and positive integer $m$, pullback gives an isomorphism
For $v=0$ these spaces have dimension $(am+1)(bm+1)$.
Equip $L_ B$ with a smooth Hermitian metric and $L$ with its pullback, choose a continuous real function $\varphi $ on $B$, let $\mathrm{d}V_ X$ be a smooth positive volume form on $X$, and put $\nu =\pi _*\mathrm{d}V_ X$. Use the norms
Then the isomorphism is isometric. Let $K_{m,B}$ and $K_{m,X}$ be the corresponding partial Bergman kernels, that is, the sums of the weighted pointwise squared norms $|s|^2\mathrm{e}^{-m\varphi }$ and $|\pi ^*s|^2\mathrm{e}^{-m\varphi \circ \pi }$ over orthonormal bases of the two spaces. Then
Proof. The identification of the section spaces is proposition 18.36.1, applied to $L_ B$ and $v$ on the base. For $v=0$, the usual basis of bihomogeneous monomials on $\mathbb P^1\times \mathbb P^1$ has $(am+1)(bm+1)$ elements.
Integration along the fibers of the proper submersion shows that $\nu $ is a smooth positive volume form, and the defining identity $\int _ Bg\, \mathrm d\nu =\int _ X(g\circ \pi )\, \mathrm dV_ X$ proves equality of the two norms. Pulling back an orthonormal basis and comparing pointwise norms gives the first kernel identity, and the same integration identity gives the second. The assertions include the case of a zero selected space, for which both kernels vanish. $\square$
Even without a singular selecting weight, the growth is therefore quadratic:
The next calculation shows that the variational volume of a small Hermitian perturbation detects this missing dimension of growth.
714E787B0503Although no class $a\{ \alpha \} +b\{ \beta \} $ is big, its small Hermitian perturbations have positive Gauduchon variational volume $\mathcal V_\mu $ (definition 12.2.1). For the volume form of the balanced metric, the zero potential solves the normalized Monge–Ampère equation for every positive combination of $\alpha $, $\beta $ and $D$, so the volume is an explicit cubic polynomial in the perturbation parameter. Its order of vanishing at the nef boundary records how many of the two horizontal coefficients vanish.
Proposition 10.19.8.141C0ACF5576 On the elliptic threefold, let $a,b\geq 0$, put $\theta =a\alpha +b\beta $, and let $\mu _0=\omega ^3/(24\pi )$. Then for every $\epsilon >0$,
E7BC529546A7
10.19.9
In particular, $\mathcal V_\mu (\theta )=0$ for every smooth positive probability measure $\mu $. Let $q$ be the number of positive coefficients among $a,b$. Then the degeneration orders of proposition 12.20.1 are
for every smooth positive probability measure and every Hermitian perturbation direction.
Proof. The forms $\alpha ,\beta ,D$ are semipositive of rank one, so their squares vanish, and their sum $\omega $ is strictly positive. Thus
Since $\omega ^3=6\alpha \wedge \beta \wedge D$ and $\int _ X\omega ^3=24\pi $, the wedge expansion gives
Hence the zero potential solves the smooth normalized equation for $\Omega _\epsilon $ and $\mu _0$. By theorem 12.2.3, the scalar of this equation is $\mathcal V_{\mu _0}(\Omega _\epsilon )$, which proves (10.19.9). The continuity of the volume at nef classes in theorem 12.11.1 and comparison of smooth densities then give $\mathcal V_\mu (\theta )=0$.
As $\epsilon \downarrow 0$, the leading term of (10.19.9) is $24\pi ab\epsilon $ if $a$ and $b$ are both positive, $24\pi (a+b)\epsilon ^2$ if exactly one of them is positive, and $24\pi \epsilon ^3$ if both vanish. The logarithmic quotients defining the orders therefore have limits $1,2,3$, respectively. Their independence of the density and of the perturbing metric is part of proposition 12.20.1. $\square$
The same number $q$ governs the growth of sections. For nonnegative integers $a,b$, the line bundle $L=\pi ^*\mathcal O_ B(a,b)$ satisfies
since the descent along compact fibers and the monomial count in the proof of proposition 10.19.6 also apply when one or both integers vanish. The growth degree of these spaces is exactly $q$, whereas the boundary order of the smooth curvature form $a\alpha +b\beta $ of $L$ is $3-q$. In this explicit family, the boundary order thus records the loss of algebraic growth dimension, even though the three-dimensional Monge–Ampère measures in the unperturbed background vanish.
Formula (10.19.9) concerns one fixed smooth density; it does not assert that a total mass is conserved over all potentials. Indeed, by theorem 12.19.1 every Hermitian form on $X$, in particular each $\Omega _\epsilon $, has zero lower mass, while its upper mass is finite because $X$ has the bounded mass property (definition 13.14.14, proposition 12.19.18). Whether comparable identifications hold beyond such explicit families is part of 12.33.4.
7128CD1C74CBThe Gauduchon metrics that minimize the quotient defining the volume in (10.19.9) can be identified explicitly: they are the equation metrics themselves. Their normalized squares converge smoothly at the nef boundary, although the normalized metrics may diverge. Recall from definition 12.2.1 that the root volume $\lambda _{\mu _0}$, with $\lambda _{\mu _0}^3=\mathcal V_{\mu _0}$, is the infimum over Gauduchon metrics $\gamma $ of $\int _ X\theta \wedge \gamma ^2$ divided by the denominator $D_{\mu _0}(\gamma )=\int _ X(\mu _0/\gamma ^3)^{1/3}\gamma ^3$. For positive functions of $\epsilon $, we write $f\asymp g$ if $f/g$ is bounded above and below by positive constants as $\epsilon \downarrow 0$.
Proposition 10.19.11.983D2EEDBE68 Let $a,b\geq 0$, $\theta =a\alpha +b\beta $ and $\Omega _\epsilon =\theta +\epsilon \omega $. For every $\epsilon >0$, the metric $\Omega _\epsilon $ is Gauduchon, and it minimizes the quotient defining $\lambda _{\mu _0}(\Omega _\epsilon )$. Put
Then $\int _ X\omega \wedge T_\epsilon =1$, $\mathrm{dd}^{\mathrm{c}}T_\epsilon =0$ and $T_\epsilon =\gamma _\epsilon ^2$. As $\epsilon \downarrow 0$, the forms $T_\epsilon $ converge smoothly to
In every case $\theta \wedge T_0=0$. If $q$ is the number of positive coefficients among $a,b$, the denominator of the normalized minimizing metric satisfies
where the two comparison constants may depend on the fixed pair $(a,b)$.
Proof. Write $x=a+\epsilon $, $y=b+\epsilon $ and $z=\epsilon $. Since $\alpha ^2=\beta ^2=D^2=0$,
The forms $\alpha ,\beta $ are closed and $\mathrm{dd}^{\mathrm{c}}D=2\alpha \wedge \beta $. Hence the first term is closed, and $\mathrm{dd}^{\mathrm{c}}$ of each of the last two terms contains $\alpha ^2$ or $\beta ^2$ and therefore vanishes. Thus $\mathrm{dd}^{\mathrm{c}}(\Omega _\epsilon ^2)=0$. Since the zero potential solves the normalized equation for $\Omega _\epsilon $ and $\mu _0$, the characterization of minimizing metrics in theorem 12.2.3 shows that $\Omega _\epsilon $ is a minimizer.
Integration along the fibers gives $\int _ X\alpha \wedge \beta \wedge D=4\pi $. Hence
which is the trace normalization. Dividing the three coefficients in the wedge expansion of $\Omega _\epsilon ^2$ by $I_\epsilon $ gives each of the stated limits; the convergence holds in every smooth norm, because the forms $\alpha \wedge \beta $, $\alpha \wedge D$ and $\beta \wedge D$ are fixed. The identities $\alpha ^2=\beta ^2=0$ then give $\theta \wedge T_0=0$ in all four cases.
Set $c_\epsilon =24\pi \epsilon (a+\epsilon )(b+\epsilon )$. The equation for $\Omega _\epsilon $ gives
Substituting this in the definition of $D_{\mu _0}$ and using $\int _ X\mu _0=1$ yields $D_{\mu _0}(\gamma _\epsilon )=c_\epsilon ^{2/3}/I_\epsilon $. For fixed $(a,b)$ one has $c_\epsilon \asymp \epsilon ^{3-q}$ and $I_\epsilon \asymp \epsilon ^{2-q}$, so the quotient is of order $\epsilon ^{q/3}$. $\square$
In a coframe adapted to the three rank-one forms $\alpha $, $\beta $ and $D$, the three complementary products in the proof show that the limiting positive $(2,2)$-forms have coefficient matrices of rank $1$, $2$ and $3$ when $q=2$, $1$ and $0$, respectively. If $a,b>0$, the normalized metrics themselves converge smoothly to
If $a>0$ and $b=0$, however, the coefficient of $\alpha $ in $\gamma _\epsilon $ grows like a positive constant times $\epsilon ^{-1/2}$, while its other two coefficients tend to zero. Thus the metrics $\gamma _\epsilon $ are unbounded relative to the fixed metric $\omega $, although their squares converge smoothly; the same holds with $\alpha $ and $\beta $ interchanged. This divergence is unavoidable. In dimension three, a positive $(2,2)$-form of rank two is not the square of a semipositive $(1,1)$-form: the coefficients of such a square are the pairwise products of the eigenvalues of the $(1,1)$-form, and the number of nonzero products is three, one or zero. At the vertex $a=b=0$, the normalized metric instead stays equal to $\omega /\sqrt{24\pi }$.
For a nonzero boundary class, the smooth limit $T_0$ annihilates $\theta $, and the denominators $D_{\mu _0}(\gamma _\epsilon )$ tend to zero; no concentration of the coefficient measures occurs in this family. This degeneration of the minimizing tests for a fixed density differs from the sequence of Gauduchon forms used in lemma 12.19.13 to prove that the lower mass vanishes. At the vertex, the denominator stays positive, and the order three comes entirely from the scaling $\Omega _\epsilon =\epsilon \omega $ of the background.
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