2.7. Intersection numbers and Monge-Ampère measures [019I]

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2.7. Intersection numbers and Monge-Ampère measures

The Monge-Ampère operator that we will use arises from intersection theory on models.

Let $\mathcal{X}$ be a model of $X$ , and pick numerical classes $\theta_{1,\mathcal{X}},\dots,\theta_{n,\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ . For any vertical divisor $D\in\Div_{0}(\mathcal{X})$ we define

D\cdot\theta_{1}\cdot\ldots\cdot\theta_{n}:=\sum_{E}\ord_{E}(D)\,(\theta_{1,\mathcal{X}}|_{E}\cdot\ldots\cdot\theta_{n,\mathcal{X}}|_{E}),

where $E$ ranges over all irreducible components of the special fiber $\mathcal{X}_{0}$ . We obtain a pairing that is linear in each entry and symmetric in the $\theta_{i}$ ’s.

Proposition-Definition 2.18.

To any $n$ -tuple $(\theta_{1},\dots,\theta_{n})$ of closed $(1,1)$ -forms we can associated a signed atomic measure $\theta_{1}\wedge\dots\wedge\theta_{n}$ supported on $X^{\mathrm{div}}$ such that

(2.2)

\int_{X}f\,\theta_{1}\wedge\dots\wedge\theta_{n}=\sum_{i\in I}b_{i}f(x_{i})\,(\theta_{1,\mathcal{X}}|_{E_{i}}\cdot\ldots\cdot\theta_{n,\mathcal{X}}|_{E_{i}})

for any common determination $\mathcal{X}$ of the forms $\theta_{i}$ , and for any model function $f$ . Here we have written the special fiber as $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$ and $x_{i}=x_{E_{i}}$ is the divisorial point associated to $E_{i}$ .

Further, $(\theta_{1},\dots,\theta_{n})\mapsto\theta_{1}\wedge\dots\wedge\theta_{n}$ is multilinear and symmetric.

Proof.

Choose a common determination of the forms $\theta_{i}$ , and define $\int_{X}f\,(\theta_{1}\wedge\dots\wedge\theta_{n})$ using (2.2). The fact that $\int_{X}f\,(\theta_{1}\wedge\dots\wedge\theta_{n})$ does not depend on the choice of a determination $\mathcal{X}$ is a consequence of the projection formula

\pi_{*}D\cdot\theta_{1,\mathcal{X}}\cdot\ldots\cdot\theta_{n,\mathcal{X}}=D\cdot\pi^{*}\theta_{1,\mathcal{X}}\cdot\ldots\cdot\pi^{*}\theta_{n,\mathcal{X}}

if $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ , and $D$ is any vertical divisor in $\mathcal{X}^{\prime}$ .

Then by construction $\theta_{1}\wedge\dots\wedge\theta_{n}$ can be identified with the atomic measure $\sum_{i}w_{i}\delta_{x_{i}}$ with $w_{i}=(\theta_{1,\mathcal{X}}|_{E_{i}}\cdot\ldots\cdot\theta_{n,\mathcal{X}}|_{E_{i}})$ . This measure is supported on the divisorial points associated to the irreducible components of $\mathcal{X}_{0}$ . The last statement is clear. ∎

Proposition 2.19.

If the forms $\theta_{1},\dots,\theta_{n}$ are semipositive, then $\theta_{1}\wedge\dots\wedge\theta_{n}$ is a positive measure, of mass

(2.3)

\int_{X}\theta_{1}\wedge\dots\wedge\theta_{n}=\{\theta_{1}\}\cdot\ldots\cdot\{\theta_{n}\}.

Proof.

Pick a model $\mathcal{X}$ such that each $\theta_{i}$ is determined by a nef class $\theta_{i,\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ . The restriction of $\theta_{i,\mathcal{X}}$ to each component $E_{\omega}$ of $\mathcal{X}_{0}$ is then also nef, and it follows that the intersection number $(\theta_{1,\mathcal{X}}|_{E}\cdot...\cdot\theta_{n,\mathcal{X}}|_{E})$ is non-negative, hence the first assertion. Since the constant function $1$ corresponds to the vertical divisor $\mathcal{X}_{0}$ we have by definition

\int_{X}\theta_{1}\wedge...\wedge\theta_{n}=\mathcal{X}_{0}\cdot\theta_{1}\cdot\ldots\cdot\theta_{n}.

By [Ful98, Example 20.3.3] this is the same as the intersection number against the generic fiber of $\mathcal{X}$ , and this is equal to $\{\theta_{1}\}\cdot\ldots\cdot\{\theta_{n}\}$ by definition. ∎

As a special case, fix $\theta\in\mathcal{Z}^{1,1}(X)$ . To any $\theta$ -psh model functions $\varphi_{1},\dots,\varphi_{n}$ we then associate a mixed Monge-Ampère measure

(\theta+dd^{c}\varphi_{1})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n}).

This is an atomic positive measure on $X$ of mass $\{\theta\}^{n}$ .

Analogously to the complex case we have the following integration by parts formula:

Proposition 2.20.

If $f,g\in\mathcal{D}(X)$ are model functions and $\theta_{1},\dots,\theta_{n-1}$ are closed $(1,1)$ -forms then we have

\int f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}=\int g\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}.

Proof.

Pick a common determination $\mathcal{X}$ of $f,g$ and the $\theta_{i}$ ’s, and divisors $D,D^{\prime},D_{i}$ such that $f=\varphi_{D}$ , $g=\varphi_{D^{\prime}}$ and $\theta_{i}$ is the class in $N^{1}(\mathcal{X}/S)$ induced by $D_{i}$ . Then by definition we have

\int f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}=\sum_{E}\ord_{E}(D)(D^{\prime}|_{E}\cdot D_{1}|_{E}\cdot...\cdot D_{n-1}|_{E})\\ =\sum_{E,E^{\prime}}\ord_{E}(D)\ord_{E^{\prime}}(D^{\prime})\,(D_{1}|_{E})|_{E^{\prime}\cap E}\cdot...\cdot(D_{n-1}|_{E})|_{E^{\prime}\cap E}\\ =\sum_{E,E^{\prime}}\ord_{E}(D)\ord_{E^{\prime}}(D^{\prime})\,(D_{1}|_{E^{\prime}})|_{E\cap E^{\prime}}\cdot...\cdot(D_{n-1}|_{E^{\prime}})|_{E\cap E^{\prime}}\\ =\int g\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}

where the third equality follows from [Ful98, Theorem 2.4]. ∎

The next result follows from the Hodge index theorem, compare [YZ10, Theorem 2.1.1].

Proposition 2.21.

Suppose $\theta_{1},\dots,\theta_{n-1}$ are semipositive closed $(1,1)$ -forms. Then the symmetric bilinear form

(f,g)\mapsto\int_{X}f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}

on $\mathcal{D}(X)$ is negative semidefinite. In particular, for any two model functions $f$ , $g$ , the following Cauchy-Schwarz inequality holds:

(2.4)

\left|\int_{X}f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\right|\leq\\ \left(-\int_{X}f\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\right)^{1/2}\,\left(-\int_{X}g\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\right)^{1/2}.

Proof of Proposition 2.21.

Fix a model function $f$ . We need to prove

I:=\int_{X}f\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\leq 0

Choose a common determination $\mathcal{X}$ of $\varphi$ and all the $\theta_{i}$ . By continuity, we may assume $\varphi=\varphi_{D}$ for some $D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ , and each form $\theta_{i}$ is determined by a $\mathbf{Q}$ -line bundle $\mathcal{L}_{i}$ on $\mathcal{X}$ . Then $I=D^{2}\cdot\mathcal{L}_{1}\cdot\ldots\cdot\mathcal{L}_{n-1}$ and the result follows from [YZ10, Theorem 2.1.1 (a)]. ∎

Remark 2.22.

In the complex case we have by Stokes’ theorem

\int f\,dd^{c}g\wedge\theta_{1}\wedge...\wedge\theta_{n-1}=-\int df\wedge d^{c}g\wedge\theta_{1}\wedge...\wedge\theta_{n-1},

and negativity comes from that of the $(1,1)$ -form $df\wedge d^{c}f$ . Recall also that $df\wedge d^{c}f\wedge\omega^{n-1}=|df|_{\omega}^{2}\,\omega^{n}$ when $\omega$ is a Kähler form, so that $\left(-\int f\,dd^{c}f\wedge\omega^{n-1}\right)^{1/2}$ is the $L^{2}$ -norm of the gradient of $f$ .