20.1 The theory of b-divisors1EAF89CE07E1

Proof. First observe that $\mathbb {D}$ is pseudo-effective, since $\mathbb {D}_ i\geq 0$ for all $i$.

Let $\pi \colon Y\rightarrow X$ be a modification. Then

The inequality (20.1.8) follows. As for the decreasing case, it suffices to observe that both sides of (20.1.8) can be written as

This completes the proof. $\square$

Proof. Let $\omega $ be a Kähler form on $X$.

Step 1. Reduce to the case where $T$ is a Kähler current.

Note that $\mathbb {D}(\omega )$ is the Cartier b-divisor realized by $(X,\{ \omega \} )$. The b-divisors $\mathbb {D}(T+\epsilon \omega )=\mathbb {D}(T)+\epsilon \mathbb {D}(\omega )$ decrease to $\mathbb {D}(T)$ as $\epsilon \searrow 0$. Hence, by 20.1.7 and proposition 18.1.27, it suffices to prove our assertion with $T+\epsilon \omega $ in place of $T$.

Step 2. We prove the assertion under the additional assumption that $T$ has analytic singularities.

Let $\pi \colon Y\rightarrow X$ be a modification so that

where $D$ is an effective $\mathbb {Q}$-divisor on $Y$ and $R$ is a closed positive $(1,1)$-current with locally bounded potentials. See theorem 5.18.8 for the existence of $\pi $. Then $\mathbb {D}(T)$ is the nef Cartier b-divisor realized by $(\pi ,\{ R\} )$. Note that (20.1.16) is immediate in this case.

Step 3. We prove the assertion for a general Kähler current $T$. Next, take a closed smooth real $(1,1)$-form $\theta $ in the cohomology class of $T$ and write $T=\theta _{\varphi }$ for some $\varphi \in \mathrm{PSH}(X,\theta )$. Since $T$ is a Kähler current, we choose the quasi-equisingular approximation $(\varphi _ j)_ j$ of $\varphi $ so that, after discarding finitely many terms, each $\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ j$ is a Kähler current. This is possible by Demailly regularization, in the form used in the proof of theorem 18.1.2. We claim that

By definition of this convergence, we need to establish the following: for every modification $\pi \colon Y\rightarrow X$, we have

This follows immediately from theorem 16.9.55 if $\operatorname{Sing}(\pi ^*T)$ has only finitely many components.

Fix a Kähler form $\omega _ Y$ on $Y$. We want to show that for any $\epsilon >0$, we can find $j_0>0$ so that when $j\geq j_0$,

Write the divisorial part of $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}\pi ^*\varphi _ j$ and $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}\pi ^*\varphi $ as

Then $a_ i^ j\leq a_ i$.

We can find an integer $N>0$ large enough so that, in the pseudo-effective order of cohomology classes,

By theorem 16.9.55, we can take $j_0$ large enough so that for $j>j_0$,

Then (20.1.18) follows, and (20.1.17) is established.

As a consequence, $\mathbb {D}(T)$ is nef and the volume can be computed using 20.1.7:

Hence, (20.1.16) follows. $\square$

Proof. Fix a modification $\pi \colon Y\rightarrow X$. Fix a Kähler form $\omega $ on $X$. It suffices to establish the following:

As a map from the pseudometric space $\mathrm{PSH}(X,\theta )$ to the finite-dimensional Euclidean space $\mathrm{H}^{1,1}(Y,\mathbb {R})$, the continuity of $\mathbb {D}(\bullet )_ Y$ can be tested on sequences. Thus, without loss of generality, we may assume that $(\varphi _ i)_ i$ is a sequence and $I=\mathbb {Z}_{>0}$. It is enough to prove that every subsequence has a further subsequence for which (20.1.22) holds.

By corollary 16.9.11, $\pi ^*\varphi _ i\xrightarrow {d_ S}\pi ^*\varphi $. Hence it suffices to prove the claim on $Y$, so we may assume that $\pi $ is the identity. Therefore, we are reduced to the following assertion:

After adding a Kähler form to $\theta $, we may also assume that $\theta _{\varphi }$ is a Kähler current. By theorem 16.9.49, replacing all potentials by their $\mathcal{I}$-envelopes does not change the divisorial parts and preserves $d_ S$-convergence. The masses are then bounded from below along a tail, so after passing to a subsequence proposition 16.9.17 supplies increasing and decreasing envelopes $\eta _ i\leq \varphi _ i\leq \psi _ i$ which both converge to $\varphi $ in $d_ S$. Since

it is enough to treat the case where $(\varphi _ i)_ i$ is monotone.

Write the divisorial parts of $\theta _{\varphi _ i}$ and $\theta _{\varphi }$ as

By theorem 16.9.55, $a_ m^ i\to a_ m$ for each fixed $m$. If $(\varphi _ i)_ i$ is decreasing, then $a_ m^ i\leq a_ m$ for all $i,m$. The finite-tail argument in the proof of (20.1.18) then gives

for all sufficiently large $i$.

If $(\varphi _ i)_ i$ is increasing, then $a_ m\leq a_ m^ i\leq a_ m^1$. We first choose $N$ so that the tails of both $\sum _ m a_ m[D_ m]$ and $\sum _ m a_ m^1[D_ m]$ are at most $(\epsilon /3)\{ \omega \} $ in the pseudo-effective order. The convergence of the finitely many coefficients $a_ m^ i$, $m\leq N$, then gives

for all sufficiently large $i$. This proves (20.1.22). $\square$

Proof. By Hironaka's Chow lemma theorem 9.10.8 and 16.8.26, we may assume that $\pi $ is a modification.

By 18.1.38,

where $F_ T$ and $F_ S$ are effective $\pi $-exceptional divisors. Since $\{ T\} =\{ S\} $, the exceptional divisors $F_ T$ and $F_ S$ have the same cohomology class. Factoring $\pi $ into blow-ups and using proposition 20.1.2 at each step, the classes of $\pi $-exceptional prime divisors are linearly independent in the relative cohomology; hence $F_ T=F_ S$. It follows that

Our assertion then follows from 16.8.26. $\square$

Proof. Fix a big and nef b-divisor $\mathbb {D}$ over $X$.

For each $\pi \colon Y\rightarrow X$, we take a current with minimal singularities $T_ Y$ in $\mathbb {D}_ Y$. We claim that $\mathbb {D}(\pi _*T_ Y)$ coincides with $\mathbb {D}$ up to the level of $Y$: For any modification $\pi '\colon Z\rightarrow X$ dominated by $\pi $ through a morphism $g\colon Y\rightarrow Z$, we have

The notations are summarized in the following commutative diagram:

36BC6A8ED233 20.1.25

After unfolding the definitions, this means

Note that

Due to proposition 5.19.17 and proposition 16.6.17, we know that $\mathbb {D}_ Y$ is modified nef and big. In particular, $T_ Y$ is non-divisorial, hence so is $g_*T_ Y$ by 5.19.18. It follows from 18.1.38 that

Note that

Next we claim that the $P$-singularity types of the net $(\pi _*T_ Y)_ Y$ form a decreasing net.

To see this, let us fix a diagram as (20.1.25). We need to show that

Since $T_ Z$ has minimal singularities, we have $g_*T_ Y\preceq _{\mathcal{I}} T_ Z$. In particular, 20.1.23 guarantees that $\pi _*T_ Y\preceq _{\mathcal{I}} \pi '_*T_ Z$. But by corollary 18.1.41 and example 18.1.11, both $\pi _*T_ Y$ and $\pi '_*T_ Z$ are $\mathcal{I}$-good, so there is no difference between the $P$-preorder and the $\mathcal{I}$-preorder in this case. Our assertion follows.

Next observe that the net $(\pi _*T_ Y)_ Y$ is decreasing with respect to the $P$-preorder and has a $d_ S$-limit as a consequence of (20.1.26) and corollary 16.9.24. Take a closed positive $(1,1)$-current $T\in \mathbb {D}_ X$ such that

It follows from proposition 20.1.19 that

Therefore, we conclude that

Thanks to theorem 20.1.15, $\operatorname{vol}T>0$. Write $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ for some $\varphi \in \mathrm{PSH}(X,\theta )$, then

is $\mathcal{I}$-good, non-divisorial and $\mathbb {D}(T')=\mathbb {D}(T)$. $\square$

Proof. 1 $\implies $ 2. Suppose that 1 holds. Fix a modification $\pi \colon Y\rightarrow X$. Then we need to show that $\mathbb {D}_ Y$ is modified nef. Since modified nefness is a closed condition, after approximating $\mathbb {D}$ by nef Cartier b-divisors, we may assume that $\mathbb {D}$ itself is a nef Cartier b-divisor. We can then find a modification $\pi '\colon Z\rightarrow X$ dominating $\pi $ so that $\mathbb {D}$ is realized by a nef class $\alpha $ on $Z$. Then $\mathbb {D}_ Y$ is nothing but the pushforward of $\alpha $, and hence modified nef thanks to proposition 5.19.17.

2 $\implies $ 1. Suppose that 2 holds. We need to show that $\mathbb {D}$ is nef. Fix a Kähler form $\omega $ on $X$. It suffices to show that $\mathbb {D}+\epsilon \mathbb {D}(\omega )$ is nef for each $\epsilon >0$. After replacing $\mathbb {D}$ by the latter, we may further assume that $\mathbb {D}_ Y$ is big for each modification $\pi \colon Y\rightarrow X$, and $\operatorname{vol}\mathbb {D}_ Y$ has a uniform positive lower bound. In this case, the argument of theorem 20.1.24 shows that $\mathbb {D}=\mathbb {D}(T)$ for some closed positive $(1,1)$-current in $\mathbb {D}_ X$. Therefore, $\mathbb {D}$ is nef, thanks to theorem 20.1.15. $\square$

Proof. Given $T\in \mathcal{G}(\alpha )$, we associate the b-divisor $\mathbb {D}(T)$. It is big and nef due to theorem 20.1.15. This map clearly descends to $\mathcal{G}(\alpha )/\sim _{\mathcal{I}}$.

This map is surjective by theorem 20.1.24. Now we show that it is injective. Let $T,T'\in \mathcal{G}(\alpha )$. Assume that $\mathbb {D}(T)=\mathbb {D}(T')$, we want to show that $T\sim _{\mathcal{I}} T'$.

Let $E$ be a prime divisor over $X$, it suffices to show that

We may assume that $E$ is not a prime divisor on $X$, as otherwise both sides vanish.

Choose a sequence of blow-ups with smooth connected centers

so that $E$ is a prime divisor on $Y$, exceptional with respect to $X_ k\rightarrow X_{k-1}$. Denote the composition by $\pi \colon Y\rightarrow X$. Thanks to proposition 20.1.2,

where $E_ k=E$ is the exceptional divisor of $X_ k\rightarrow X_{k-1}$.

By induction,

where $E_ i$ is the exceptional divisor of $X_ i\rightarrow X_{i-1}$. Now by 18.1.38,

In particular, the cohomology class of $\operatorname{Reg}\pi ^*T$ determines $\nu (T,E)$. Hence, (20.1.29) follows. $\square$

Proof. For each Kähler form $\omega $, theorem 20.1.28 identifies $\mathcal G(\alpha +\omega )/\sim _{\mathcal I}$ with the set of big and nef b-divisors whose root on $X$ is $\alpha +\{ \omega \} $. The transition map is exactly addition of $\mathbb D(\omega '-\omega )$. Passing to the inverse limit gives precisely nef b-divisors with root $\alpha $. $\square$

Proof. This is a direct consequence of theorem 20.1.28 and Siu's decomposition 5.19.12. $\square$

Proof. Apply corollary 20.1.31 to the non-divisorial parts and use Siu's decomposition on $X$ to record the divisorial part. The convergence condition in definition 20.1.32 is exactly the condition that the roots of the non-divisorial parts converge to the prescribed cohomology class $\alpha $. $\square$

Proof. We only handle the nontrivial implication. Assume that $\mathbb {D}$ is nef in the sense of definition 20.1.12. We want to show that $\mathbb {D}$ is nef in the sense of definition 20.1.11. After replacing $X$ by a realization of $\mathbb {D}$, we may assume that $\mathbb {D}$ is realized by $(X,\alpha )$ for some cohomology class $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$.

We first assume that $\mathbb {D}$ is big. Choose a non-divisorial closed positive $(1,1)$-current $T$ on $X$ such that $\mathbb {D}=\mathbb {D}(T)$. Now $\mathbb {D}=\mathbb {D}(T)$ means that for each modification $\pi \colon Y\rightarrow X$, the current $\pi ^*T$ is non-divisorial. In particular, $T$ has vanishing generic Lelong number along each prime divisor over $X$, see (20.1.30). Equivalently, $T$ has vanishing Lelong number everywhere. By Demailly's regularization theorem, a pseudo-effective class carrying a positive current with vanishing Lelong number at every point is nef. Hence $\alpha =\{ T\} $ is nef.

In the general case, fix a Kähler form $\omega $ on the realization of $\mathbb {D}$. For every $\epsilon >0$, the Cartier b-divisor $\mathbb {D}+\epsilon \mathbb {D}(\omega )$ is big and nef in the sense of definition 20.1.12, hence is nef in the sense of definition 20.1.11 by the big case. Thus $\alpha +\epsilon \{ \omega \} $ is nef for all $\epsilon >0$, and the closedness of the nef cone shows that $\alpha $ is nef. $\square$

Proof. Let $E$ be a prime divisor over $X$. Choose a sequence of blow-ups with smooth connected centers as in the proof of theorem 20.1.28, so that $E$ appears as one of the exceptional prime divisors on the resulting model $\pi \colon Y\rightarrow X$. By (20.1.30), the difference

is represented in the relative cohomology by

Using proposition 20.1.2 inductively, or equivalently the negativity lemma for exceptional classes, pseudo-effectivity of this exceptional class is equivalent to the non-negativity of all coefficients. Hence $\mathbb {D}(T)\leq \mathbb {D}(T')$ is equivalent to

for every prime divisor $E$ over $X$, which is exactly $T\preceq _{\mathcal{I}}T'$ by proposition 16.8.9. $\square$

Proof. Take a Kähler form $\omega $ on $X$. By theorem 20.1.24, for each $i>0$, we can find a non-divisorial Kähler current $T_ i\in \mathbb {D}_ X+i^{-1}\{ \omega \} $ such that

We observe that $T_ i$ and $T_{i+1}$ have the same $\mathcal{I}$-singularity type. Indeed, applying corollary 20.1.36 to $T_ i$ and $T_{i+1}+(i^{-1}-(i+1)^{-1})\omega $ shows that these two currents are $\mathcal{I}$-equivalent; adding the smooth form does not change the singularity type. Let $(T_ i^ j)_ j$ be quasi-equisingular approximations of $T_ i$ such that

1. 7B39195D3D69

   $T_ i^ j$ is a Kähler current in $\mathbb {D}_ X+i^{-1}\{ \omega \} $ for $j\geq j_0(i)$, and

2. CD8709C3E97A

   for each fixed $j$, the singularity type of $T_ i^ j$ is independent of $i$.

Note that 2 is possible by using the Bergman kernel construction of the quasi-equisingular approximations, since the $T_ i$'s have the same multiplier ideals.

For fixed $j$, the b-divisor

is independent of $i$. Since the quasi-equisingular approximations are decreasing in $j$, the sequence $(\mathbb {B}_ j)_ j$ is decreasing, and proposition 20.1.19 gives $\mathbb {B}_ j\to \mathbb {D}$. It suffices to take $\mathbb {D}_ i=\mathbb {B}_{j_ i}+i^{-1}\mathbb {D}(\omega )=\mathbb {D}(T_ i^{j_ i})$, where $j_ i$ is chosen to be a sufficiently fast increasing sequence of positive integers with $j_ i\geq j_0(i)$. $\square$

Proof. 1, 2 and 3 follow immediately from the corresponding results for mixed volumes, as proved in proposition 18.1.27.

4 Take $\mathcal{I}$-good non-divisorial closed positive $(1,1)$-currents $T_1,\ldots ,T_ n$ and $T'$ so that $\mathbb {D}(T_ i)=\mathbb {D}_ i$ for all $i=1,\ldots ,n$ and $\mathbb {D}(T')=\mathbb {D}'$. Furthermore, we may assume that the $T_ i$'s and $T'$ are Kähler currents by the perturbation argument.

Let $(T_{i}^ j)_ j$ be a quasi-equisingular approximation of $T_ i$ for $i=2,\ldots ,n$. It follows from theorem 16.9.30 that

It suffices to show that for all $j\geq 1$,

Therefore, we have reduced to the case where $T_2,\ldots ,T_ n$ have analytic singularities. After a resolution, we may assume that they have log singularities along $\mathbb {Q}$-divisors. By proposition 18.1.27 5, we can further reduce to the case where $T_2,\ldots ,T_ n$ have bounded local potentials. Perturbing $T_2,\ldots ,T_ n$ by a Kähler form, we may further assume that $\{ T_2\} ,\ldots , \{ T_ n\} $ are Kähler classes. By proposition 18.1.27 4, we finally reduce to the case where $T_2,\ldots ,T_ n$ are Kähler forms. In this case, the assertion follows from the pseudo-effectivity of $\{ T'\} -\{ T_1\} $. $\square$

Proof. 1, 2, 3 and 4 These are immediate consequences of the corresponding results in the big and nef case established in 20.1.40.

5 By (18.1.26) and (20.1.42), we may assume that $T_1,\ldots ,T_ n$ are Kähler currents. In this case, the assertion follows from the definitions. $\square$

Proof. By 20.1.40 4, the left-hand side is non-negative. For the upper bound, by the multi-linearity of the product proved in proposition 20.1.43, we can rewrite the difference as a sum of $2^ n-1$ terms like

where $J\subseteq \{ 1,\ldots ,n\} $ is a non-empty subset of cardinality $m$, and $\{ i_1,\ldots ,i_{n-m}\} $ is the complement of $J$. We need to show that these coefficients are uniformly bounded by a constant depending only on the given data. For this purpose, we add $\mathbb {D}(\omega )$ to each $\mathbb {D}_ i$ and then replace $K$ by $K+\{ \omega \} $. Therefore, we reduce to the case where each $\mathbb {D}_ i$ is nef and big, and hence $\mathbb {D}_ i=\mathbb {D}(T_ i)$ for some $\mathcal{I}$-good closed positive $(1,1)$-currents on $X$ with positive volumes. Then we need to bound

For this purpose, let $D\geq 0$ be the minimal non-negative number so that $D\{ \omega \} -\alpha $ is nef for any $\alpha \in K$. Then the integral is bounded by the cohomological intersection $\{ \omega \} ^ m\cap (D\{ \omega \} )^{n-m}$. Our assertion follows. $\square$

Proof. Using 20.1.44 and 20.1.7, we may easily reduce to the case where $\mathbb {D}$ is nef and big. In this case, take a non-divisorial closed positive $(1,1)$-current $T$ in $\mathbb {D}_ X$ such that $\mathbb {D}(T)=\mathbb {D}$. Then we need to show that

which is proved in theorem 20.1.15. $\square$

Proof. We may assume that $\operatorname{vol}\mathbb {D}_ i>0$ for each $i=1,\ldots ,n$ since there is nothing to prove otherwise. In this case, our assertion follows from proposition 18.1.36. $\square$

Proof. Step 1. We first assume that the $\mathbb D_ i^ j$'s and the $\mathbb D_ i$'s are all big.

Choose $\mathcal I$-good non-divisorial currents $T_ i^ j$ and $T_ i$ such that $\mathbb D(T_ i^ j)=\mathbb D_ i^ j$ and $\mathbb D(T_ i)=\mathbb D_ i$. Fix a Kähler form $\omega $ on $X$ and fix $\epsilon >0$. Since $\mathbb D_{i,X}^ j\to \mathbb D_{i,X}$, for all sufficiently large $j$ the class

is Kähler. Choose a Kähler form $\chi _ i^ j$ in this class and set $S_ i^ j=T_ i^ j+\chi _ i^ j$. Then $S_ i^ j$ lies in the fixed class $\mathbb D_{i,X}+\epsilon \{ \omega \} $ and has the same divisorial Lelong numbers as $T_ i^ j$. Hence, by theorem 18.1.42 and proposition 18.1.27 2,

Therefore,

Letting $\epsilon \to 0+$ gives (20.1.48).

Step 2. Next we handle the general case.

Take a Kähler form $\omega $ on $X$. Then by 20.1.44, for any $\epsilon \in (0,1]$, we have

Since $\epsilon $ is arbitrary, our assertion follows. $\square$

Proof. This is a straightforward consequence of 20.1.40 4 and proposition 20.1.47. $\square$