Pome269CE892EA5B269CE892EA5B
Suppose we require a $\theta $-plurisubharmonic function to have generic Lelong numbers at least $a_ j$ along prescribed prime divisors $D_ j$. Removing the divisor $D=\sum _ ja_ jD_ j$ leaves the residual background $\theta -\eta _ D$, where $\eta _ D$ is a curvature form of $D$, and in each Gauduchon quotient this removal subtracts the Gauduchon degree of $D$ from the numerator. Lamari's duality therefore decides when such functions exist, and the volume of the residual background measures what remains. By the correspondence of theorem 11.41.1, equations in the ordinary singularity type of the logarithmic potential of $D$ reduce to equations in the residual background. When the residual background has a smooth positive representative, the smooth maximizer of the dual problem forces every subsolution with the prescribed Lelong numbers and the residual volume as scalar to be the logarithmic potential of $D$ plus a smooth function, up to a constant.
Fix a smooth real $(1,1)$-form $\theta $ on $X$. Let $D=\sum _{j=1}^ N a_ jD_ j$ be an effective real divisor, with distinct prime divisors $D_ j$ and coefficients $a_ j>0$. Choose a smooth Hermitian metric on each line bundle $\mathcal O_ X(D_ j)$, write $s_ j$ for its canonical section, and let $\eta _ j$ be its Chern curvature form. Set
The Poincaré–Lelong formula reads
For a Gauduchon metric $\gamma $ on $X$, define the Gauduchon degree
where integration over a singular divisor means integration over its regular locus, or equivalently against its current of integration. This degree is finite, positive for $D\ne 0$, and equal to $\int _ X\eta _ D\wedge \gamma ^{n-1}$ by distributional Stokes. We write $\nu (u,D_ j)$ for the generic Lelong number of $u$ along $D_ j$, as in section 5.34.
Proposition 12.35.1.C43B33AA4A18 The following are equivalent:
there is $u\in \mathrm{PSH}(X,\theta )$ with $\nu (u,D_ j)\geq a_ j$ for every $j$;
the residual background $\beta _ D$ is pseudoeffective;
for every Gauduchon metric $\gamma $,
The first condition does not require $u-\sigma _ D$ to be bounded, nor the Lelong numbers of $u$ to equal the prescribed ones.
Proof. Suppose first that $u$ satisfies the first condition. By lemma 5.38.1, these Lelong numbers can be subtracted: $\theta +\mathrm{dd}^{\mathrm{c}}u-[D]\geq 0$. Consequently
where the difference $u-\sigma _ D$ is first taken in $L^1$ and then replaced by its quasi-plurisubharmonic representative. This proves the second condition. Conversely, if $v\in \mathrm{PSH}(X,\beta _ D)$, then $u=\sigma _ D+v$ satisfies
so its generic Lelong numbers along the $D_ j$ are at least $a_ j$. For $n\geq 2$, Lamari's duality, theorem 11.6.1, says that a smooth background is pseudoeffective exactly when all its Gauduchon integrals are nonnegative; applied to $\beta _ D$, together with the degree identity, it gives the equivalence of the second and third conditions. For $n=1$ the same criterion reads $\int _ X\beta _ D\geq 0$: necessity follows from Stokes' theorem, and for sufficiency one solves $\mathrm{dd}^{\mathrm{c}}f=(\int _ X\beta _ D)\mu -\beta _ D$ for a smooth real function $f$ with the scalar Green operator. $\square$
For a nonzero effective real divisor $D$ and a pseudoeffective $\theta $, the exact threshold is therefore
D8ACD0FADFBD
12.35.2
It is finite and nonnegative, and a function $u\in \mathrm{PSH}(X,\theta )$ with $\nu (u,D_ j)\geq ta_ j$ for all $j$ exists if and only if $0\leq t\leq \tau _\theta (D)$. The endpoint is included, because the non-strict Gauduchon inequalities still hold there and duality gives a positive current. This is a threshold for the existence of a positive current, not for the existence of a bounded residual potential.
Whenever $\beta _ D$ is pseudoeffective, define the residual volume
The definition and the degree identity give the exact formula
44DAB02F7C7C
12.35.3
and the residual volume is finite and nonnegative. Changing the auxiliary metrics on the line bundles changes $\eta _ D$ by a smooth $\mathrm{dd}^{\mathrm{c}}$-exact term, so the residual volume does not depend on these metrics, nor on a smooth $\mathrm{dd}^{\mathrm{c}}$ change of $\theta $. It depends on the prescribed divisor and on the smooth probability measure $\mu $.
Here $\mathrm{MA}^{\mathrm{ext}}_\theta $ denotes the extended non-pluripolar Monge–Ampère measure of theorem 13.14.3.
Theorem 12.35.4.7B32818E6010 Suppose that $\beta _ D$ has a smooth positive representative, and put $c=\mathcal V_\mu (\theta ;D)$. There is a $\theta $-plurisubharmonic function $u_0=\sigma _ D+v$, with $v$ smooth, satisfying
Every $u\in \mathrm{PSH}(X,\theta )$ such that
differs from $u_0$ by a constant. In particular, a solution of a normalized equation $\mathrm{MA}^{\mathrm{ext}}_\theta (u)=c'\mu $ in the ordinary singularity type of $\sigma _ D$ has scalar $c'=c$, has a smooth remainder $u-\sigma _ D$, and is unique up to constants.
Proof. Existence. The smooth equation and theorem 12.2.3 give a smooth $v$ with
Then $\theta +\mathrm{dd}^{\mathrm{c}}(\sigma _ D+v)=[D]+\Omega \geq 0$. Off $D$, the extended measure of $u_0=\sigma _ D+v$ is $\Omega ^ n$, and both measures give zero mass to $D$. This proves existence.
Reduction to the residual background. Let $u$ satisfy the two inequalities. Removing the divisor as in the proof of proposition 12.35.1, we obtain
where the difference is first taken in $L^1$ and then replaced by its canonical representative. On $X\setminus D$, the two currents differ only by smooth changes in the decomposition of their local potentials, so their extended measures agree there, and both measures vanish on $D$. Hence
Rigidity. Take $\mathrm{d}V=\mu $. The zero function solves the smooth equation $\Omega ^ n=c\mu $ in the background $\Omega $, so its subsolution threshold is $b_\Omega (1)=c$, by proposition 15.28.7. For $n\geq 2$, the normalized $(n-1)$st power of the Gauduchon metric conformal to $\Omega $ is a smooth strictly positive maximizer of the dual problem. By corollary 12.10.12, any two threshold subsolutions differ by a constant; comparing $w$ with zero shows that $w$ is constant. For $n=1$, the positive current $\Omega +\mathrm{dd}^{\mathrm{c}}w$ dominates its extended measure, hence dominates $c\mu =\Omega $, while both have total mass $\int _ X\Omega =c$ by Stokes' theorem. Thus $\mathrm{dd}^{\mathrm{c}}w=0$, and $w$ is again constant.
Solutions in the ordinary type. Finally, let $u$ be in the ordinary singularity type of $\sigma _ D$ with $\mathrm{MA}^{\mathrm{ext}}_\theta (u)=c'\mu $ for some $c'\geq 0$. Then $\nu (u,D_ j)=a_ j$, and the same reduction gives a bounded $w\in \mathrm{PSH}(X,\Omega )$ with $\mathrm{MA}_\Omega (w)=c'\mu $. Suppose $c'<c$. For every constant $C$, the bounded functions $w$ and $C$ satisfy $\mathrm{MA}_\Omega (w)=(c'/c)\mathrm{MA}_\Omega (C)$, so theorem 14.19.10 gives $w\geq C$, which is impossible for a bounded $w$. The case $c'>c$ is symmetric, with the roles of $w$ and the constants exchanged. Therefore $c'=c$, and the rigidity just proved gives $u=u_0+\mathrm{constant}$. $\square$
When $\theta $ is Hermitian, write $\mu =f\, \mathrm{d}V$ relative to a smooth strictly positive probability volume $\mathrm{d}V$. For every $u\in \mathrm{PSH}(X,\theta )$ with $\theta +\mathrm{dd}^{\mathrm{c}}u\geq [D]$ and $\mathrm{MA}^{\mathrm{ext}}_\theta (u)\geq cf\, \mathrm{d}V$, in particular for the potential $u_0$ of theorem 12.35.4 with $c=\mathcal V_\mu (\theta ;D)$, proposition 10.18.1 bounds the coefficient $c$ in terms of the subsolution threshold $b_\theta (f)$ of theorem 15.28.4 and the linear threshold of the class of the divisor. The residual problem prescribes the divisor, whereas the threshold $b_\theta (f)$ allows all $\theta $-plurisubharmonic functions.
Along a family $D_ t=tD$, $t\geq 0$, in the feasible interval, the root of (12.35.3) is concave and nonincreasing in $t$: every test quotient is affine in $t$ with nonpositive slope, and both properties pass to the infimum. On a subinterval where $\theta -t\eta _ D$ has a smooth positive representative, choose a minimizing Gauduchon metric $\gamma _ t$ for that background. Proposition 12.3.2 then gives the sharper formula
3EE856593807
12.35.5
In dimension one the derivative is $-\deg D$, since the denominator is one. If the residual background at an endpoint of the interval is nef, theorem 12.11.1 also expresses its volume as the limit of the scalars of the smooth normalized equations for the perturbations by $\epsilon h$, with $h$ a fixed Hermitian form and $\epsilon \downarrow 0$.
2AD000453E7CFor a fixed support, the residual equation depends smoothly on the coefficients as long as the residual background stays in $\mathscr P$. The smooth remainders then give convergence in capacity of the solutions, and they determine all generic Lelong numbers of the limit on every modification. Keep $D_ j$, $s_ j$ and $\eta _ j$ as above, and put $\sigma _ j=\log |s_ j|^2$.
For $a=(a_1,\ldots ,a_ N)\in \mathbb R_{\geq 0}^ N$ put
where zero coefficients are omitted from the logarithmic sum. Let $\mathcal A$ be the set of such $a$ for which $\beta _ a$ has a smooth positive representative.
Proposition 12.35.7.280181A9DE22 There are unique smooth functions $v_ a$, $a\in \mathcal A$, such that
The maps $a\mapsto v_ a$ and $a\mapsto c(a)$ are smooth, and every smooth norm of $v_ a$ is bounded on compact subsets of $\mathcal A$. Here smoothness at a parameter with a zero coefficient means that the family is the restriction of a smooth family defined on an open subset of $\mathbb R^ N$.
Set $u_ a=\sigma _ a+v_ a$. If $a^{(k)}\in \mathcal A$ and $a^{(k)}\to a\in \mathcal A$, then $u_{a^{(k)}}\to u_ a$ in $L^1$ and in capacity, and smoothly on compact subsets of $X\setminus \bigcup _ jD_ j$. Moreover,
where $\| \cdot \| _{\mathrm{TV}}$ is the total variation norm. For every modification $\pi :Y\to X$ with $Y$ smooth and every prime divisor $E\subset Y$,
where $\operatorname {ord}_ E(\pi ^*D_ j)$ is the vanishing order along $E$ of the pullback of a local equation of $D_ j$. In particular, the limit $u_ a$ has exactly the ordinary singularity type of $\sigma _ a$, and no further singularity appears in this finite family.
Proof. The set of parameters in $\mathbb R^ N$ for which $\beta _ a$ has a smooth positive representative is open, and the smooth equation gives $v_ a$ and $c(a)$ on it. The argument in the proof of proposition 12.3.2 gives smooth dependence of both quantities on the parameter, with the fixed mean normalization; uniqueness makes the local families agree. Covering a compact set of parameters by finitely many such neighborhoods gives the asserted bounds.
The Poincaré–Lelong formula gives
so $u_ a\in \mathrm{PSH}(X,\theta )$. Its extended measure equals $\Omega _ a^ n=c(a)\mu $ off the union of the divisors, and both measures vanish on that union; this also covers the case $D_ a=0$. Each fixed $\sigma _ j$ is locally integrable, so the convergence of the coefficients and the smooth convergence of $v_ a$ give convergence in $L^1$. Off the union of the divisors every $\sigma _ j$ is smooth, which gives smooth convergence there. For convergence in capacity, remove a neighborhood of this analytic set of arbitrarily small Hermitian capacity; on its compact complement the convergence is uniform. Since the union of the divisors has capacity zero, this proves convergence in capacity. The identity for the total variation distances follows from the equations and $\mu (X)=1$.
Finally, after pullback the logarithmic potential $\sigma _ j\circ \pi $ has generic Lelong number $\operatorname {ord}_ E(\pi ^*D_ j)$ along $E$, whereas $v_ a\circ \pi $ is smooth. Additivity of Lelong numbers for the finite sum proves the formula on every modification. Since $v_ a$ is smooth also at the limiting parameter, $u_ a$ has the ordinary singularity type of $\sigma _ a$. $\square$
The smooth bounds above use that the limiting parameter lies in $\mathcal A$. When the sections and weights move, theorem 11.41.3 gives $L^1$ convergence of the potentials normalized by $\sup _ X=0$; when the residual background is fixed, it also allows uniformly bounded measurable right-hand sides converging in $L^1$. On a modification, the pullback of a positive probability density can vanish along the exceptional divisor, so the smooth equation with positive density no longer applies directly. For a minimal exceptional pole, proposition 10.21.9 gives a vanishing order of the density that is necessary for a smooth residual, but this order alone does not ensure smoothness.
Multiplier ideals need not stabilize in this family. At a smooth point of $D_ j$ outside the other divisors, take a local coordinate $z_1$ defining $D_ j$. There $u_ a=a_ j\log |z_1|^2+O(1)$, so
by the one-variable integral in example 5.33.3 and Fubini's theorem. Thus, if the parameter stays in $\mathcal A$ while $a_ j$ increases to an integer from below, this local ideal jumps at the limit. Convergence of the Lelong numbers to their limiting values does not imply that the multiplier ideals are eventually equal to those of the limit.
E45BF47B8568On the ruled surface of subsection 20.3.63, the Gauduchon variational volume has a different dependence on the data from the two volumes of an ordinary type. Keep $Y=\mathbb P^1\times \mathbb P^1$, $\theta =cw_1+aw_2$, $0\leq b<c$ and $\beta =(c-b)w_1+aw_2$. Let $m>0$ be smooth on the first curve, with $\int mw=1$, and put
In dimension two the Gauduchon tests are all smooth positive $(1,1)$-forms $T$ with $\mathrm{dd}^{\mathrm{c}}T=0$, and $D_\mu (T)=\int _ Y\sqrt{T^2/\mu }\, \mu $.
Proposition 12.35.9.E45E9799C813 The all-test infimum is attained and
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12.35.10
Writing $V=2k/H$ and $g=Vm/(2a)$, a minimizing test is the closed positive form
Proof. Since $\int gw=k$, solve $\mathrm{dd}^{\mathrm{c}}f=(g-k)w$ on the first sphere. Then $\Omega =\beta +\mathrm{dd}^{\mathrm{c}}f=gw_1+aw_2>0$ and $\Omega ^2=V\mu $. For every Gauduchon test, Stokes and the matrix arithmetic–geometric mean inequality give
The displayed $T$ is closed, since $g/a$ depends only on the first factor, and $T=a^{-1}\Omega $, so equality holds pointwise. Explicitly,
This proves the infimum over all tests. The curvature of $D$ is $w_1$, so the divisorial residual is indeed $\beta $. Replacing $k$ by $c$ gives the formula for $\theta $. $\square$
Fix a real constant $d$ and a smooth real first-factor function $e$. On an interval where $c_ s=c-sd>b$ and $a_ s=a-se>0$, keep $b,m$ fixed and put
Corollary 12.35.11.F402CC8C0A7E For the residual background $\beta _ s=k_ sw_1+a_ sw_2$,
10918993D19D
12.35.12
Consequently, for every compact segment $[s_0,s_1]$ in that interval,
Proof. We have $H_ s'=\int em/a_ s^2\, w$, $D_\mu (T_ s)=\sqrt{V_ s}H_ s$ and
Differentiating $V_ s=2k_ s/H_ s$ gives (12.35.12); integration proves the second identity. This is the explicit first variation of proposition 12.3.2 along the residual family. $\square$
For $c=1$, $b=1/4$, $a=2+\tau $ and $m=1$, the height $\tau $ has distribution $\mathrm d\tau /2$ under $w$. Hence $H=(\log 3)/2$, and the residual and original variational volumes are $3/\log 3$ and $4/\log 3$. The ordinary-type volumes from (20.3.68) are instead $3/2$ and $9/2$, independent of $m$. The subtraction identity differentiates the density-dependent variational volume. Pure-pullback invariance of the ordinary-type masses does not identify an arbitrary smooth positive probability density upstairs with the degenerate pullback of $\mu $.
The distinction persists along a segment ending at a nef background. The following surface example combines the subtraction cost with an imposed nonanalytic ordinary type, including the range in which its closed first moment is infinite.
Example 12.35.13.76B0694EBA2F Let $\mu =w_1\wedge w_2$, let $\tau (z)=(|z|^2-1)/(|z|^2+1)$ be the height on the first sphere, and set
Fix the ordinary type of $U=1-(1-G_ p)^\alpha $ with $1/2\leq \alpha <1$, and write $\sigma _-(s),\sigma _+(s)$ for its lower and upper volumes in $\theta _ s$. Then
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12.35.14
The improper subtraction cost is finite and equals $2/\sqrt{\log 3}$, while $\sigma _+(s)\to 4$. The endpoint $\theta _1=w_1+(1+\tau )w_2$ is nef, has zero variational volume, and is not big.
Proof. Take $c=1$, $b=0$, $a_ s=2+\tau -s$ and $m=1$ in proposition 12.35.9. Radial integration on the first sphere gives the law $\mathrm d\tau /2$ on $[-1,1]$, so
Thus $V_ s=2/H_ s$. Since $b=0<c$, theorem 20.3.66 gives all rough representatives of the fixed type finite positive mass, with the extrema in (12.35.14). The reference has zero Lelong numbers and trivial multiplier ideals but no analytic representative, by proposition 20.3.71. Its infinite closed first moment in this range of $\alpha $ was computed after (20.3.69); no such moment is needed for either extremum.
For the minimizing tests $T_ s=V_ s(2+\tau -s)^{-2}w_1/2+w_2$, corollary 12.35.11 gives
Integrating to $r<1$ gives $\sqrt{V_0}-\sqrt{V_ r}$. Since $V_ r\to 0$, the improper integral is $\sqrt{V_0}=2/\sqrt{\log 3}$. By theorem 12.2.3, every minimizing test is a constant multiple of $T_ s$, so the cost is independent of this choice.
The endpoint is smooth semipositive, hence nef. Monotonicity gives
so its variational volume vanishes. To exclude bigness, suppose that $\theta _1+\mathrm{dd}^{\mathrm{c}}q\geq \epsilon (w_1+w_2)$ for a quasi-plurisubharmonic $q$ and $\epsilon >0$. Choose a nonzero smooth nonnegative $(1,1)$-form $\zeta $ on the first sphere supported where $1+\tau <\epsilon $. Its pullback is closed, so distributional Stokes gives
a contradiction. Thus the positive segment ends outside the big cone, although its upper ordinary-type volume has a positive limit. $\square$
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