B.3. Multiplier ideals [01I1]

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B.3. Multiplier ideals

Let us first give the definition of multiplier ideals in our setting:

Definition B.4.

Let $\mathcal{X}$ be a regular model and let $\mathfrak{a}$ be a vertical ideal sheaf on $\mathcal{X}$ . For each rational number $c>0$ the multiplier ideal of $\mathfrak{a}^{c}$ is the vertical ideal sheaf of $\mathcal{X}$ defined as

\mathcal{J}(\mathfrak{a}^{c}):=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}\left(K_{\mathcal{X}^{\prime}/\mathcal{X}}-\lfloor c\,D\rfloor\right)

where $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ is a vertical blow-up with $\mathcal{X}^{\prime}$ SNC such that $\pi^{-1}\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}$ is locally principal and $D\in\Div_{0}(\mathcal{X}^{\prime})$ is the corresponding effective Cartier divisor.

This definition only depends on the model function $c\log|\mathfrak{a}|$ (cf. [JM11]), and would in fact make sense for an arbitrary non-positive model function $\varphi\in\mathcal{D}(X)$ .

If $\mathfrak{a}_{\bullet}$ is a graded sequence of ideals as above then $\mathcal{J}(\mathfrak{a}_{\bullet}^{c})$ is defined as the largest element of the family of coherent ideals $\mathcal{J}(\mathfrak{a}_{m}^{c/m})$ , $m\geq 1$ .

As a matter of terminology, if $\mathcal{L}$ is a line bundle on a model $\mathcal{X}$ , $\mathfrak{a}$ is a vertical coherent ideal sheaf and $c>0$ then we shall say that $\mathcal{L}\otimes\mathfrak{a}^{c}$ is nef if $\pi^{*}\mathcal{L}-cD$ is nef, where $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ is the normalization of the blow-up of $\mathcal{X}$ along $\mathfrak{a}$ and $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(-D)$ . In other words, the model function $c\log|\mathfrak{a}|$ is required to be $\theta$ -psh, where $\theta$ is the curvature form of the model metric on $L$ induced by $\mathcal{L}$ .

Using Theorem B.3 we may follow the usual line of arguments to prove the following basic vanishing property of multiplier ideals:

Theorem B.5 (Nadel Vanishing).

Let $\mathcal{X}$ be a regular model of $X$ and $\mathcal{L}\in\Pic(\mathcal{X})$ a line bundle whose restriction to $X$ is ample. If $\mathfrak{a}$ is a vertical coherent ideal sheaf on $\mathcal{X}$ and $c>0$ is a rational number such that $\mathcal{L}\otimes\mathfrak{a}^{c}$ is nef, then we have

H^{q}\left(\mathcal{X},\omega_{\mathcal{X}}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}^{c})\right)=0\,\,\text{ for all }q\geq 1.

In particular, if $\mathfrak{a}_{\bullet}$ is a graded sequence of vertical coherent ideal sheaves on $\mathcal{X}$ such that $\mathcal{L}^{m}\otimes\mathfrak{a}_{m}$ is globally generated for all sufficiently divisible $m$ , then

H^{q}\left(\mathcal{X},\omega_{\mathcal{X}}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}_{\bullet})\right)=0\,\,\text{ for all }q\geq 1.

Proof.

Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be an SNC model dominating the blow-up of $\mathcal{X}$ along $\mathfrak{a}_{m}$ , so that we have $\mathfrak{a}_{m}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(-D)$ for some effective divisor $D\in\Div_{0}(\mathcal{X}^{\prime})$ . By the projection formula we have

\omega_{\mathcal{X}}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}^{c})=\pi_{*}\left(\omega_{\mathcal{X}^{\prime}}\otimes\pi^{*}\mathcal{L}(-\lfloor c\,D\rfloor)\right).

Now $\pi^{*}\mathcal{L}-\tfrac{1}{m}D$ is nef and $c\,D-\lfloor c\,D\rfloor$ has coefficients in $[0,1[$ . Lemma B.6 below together with the projection formula yields

R^{q}\pi_{*}\left(\omega_{\mathcal{X}^{\prime}}\otimes\pi^{*}\mathcal{L}\left(-\lfloor c\,D\rfloor\right)\right)=0\,\,\text{ for all }q\geq 1.

The Leray spectral sequence is thus degenerate and we conclude using Theorem B.3. ∎

Lemma B.6 (Local vanishing).

Let $\mathcal{X}$ be a regular model, let $\mathfrak{a}$ be a vertical ideal sheaf on $\mathcal{X}$ and let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be an SNC model such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(-D)$ with $D\in\Div_{0}(\mathcal{X}^{\prime})$ . Then we have

R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)=0\,\,\text{ for all }q\geq 1.

Proof.

We argue as in the last part of the proof of Theorem B.3. Let $\mathcal{A}\in\Pic(\mathcal{X})$ be sufficiently ample to guarantee:

(i)

$\pi^{*}\mathcal{A}-cD$ is nef.
(ii)

$\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)$ is globally generated on $\mathcal{X}$ .
(iii)

$H^{p}\left(\mathcal{X},\mathcal{A}\otimes R^{m}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\right)=0\,\,\text{ for all }p\geq 1\text{ and }m\geq 0$ .

Note that the first condition can be achieved since $-D$ is $\pi$ -globally generated. The degeneration of the Leray spectral sequence shows that

H^{0}\left(\mathcal{X},\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\right)=H^{q}\left(\mathcal{X}^{\prime},\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\otimes\pi^{*}\mathcal{A}\right),

which vanishes by Theorem B.3. It follows that $\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)=0$ by global generation, whence the result. ∎

We may now deduce from the above results the following two consequences that we need in the proof of Theorem B.

Theorem B.7 (Subadditivity).

Let $\mathcal{X}$ be a regular model, $\mathfrak{a},\mathfrak{b}$ vertical coherent ideal sheaves on $\mathcal{X}$ and $c,d>0$ . Then we have

\mathcal{J}(\mathfrak{a}^{c}\cdot\mathfrak{b}^{d})\subset\mathcal{J}(\mathfrak{a}^{c})\cdot\mathcal{J}(\mathfrak{b}^{d}).

Proof.

This is proved exactly as in [Laz, Theorem 9.5.20] using local vanishing. (See also [JM11, Theorem A.2] for a different proof.) ∎

Theorem B.8 (Uniform generation property).

Let $\mathcal{X}$ be a regular model. Then there exists an ample line bundle $\mathcal{A}$ on $\mathcal{X}$ such that the following holds. Given $\mathcal{L}\in\Pic(\mathcal{X})$ , a vertical ideal sheaf $\mathfrak{a}$ and a rational number $c>0$ such that $\mathcal{L}\otimes\mathfrak{a}^{c}$ is nef, the sheaf

\mathcal{A}\otimes\mathcal{L}\otimes\mathcal{J}(\mathfrak{a}^{c})

is globally generated. In particular, if $\mathfrak{a}_{\bullet}$ is a graded sequence of vertical coherent ideal sheaves on $\mathcal{X}$ such that $\mathcal{L}^{m}\otimes\mathfrak{a}_{m}$ is globally generated for all sufficiently divisible $m$ , then

\mathcal{A}\otimes\mathcal{L}^{m}\otimes\mathcal{J}(\mathfrak{a}_{\bullet}^{m})

is globally generated for all $m$ .

Proof.

Let $\mathcal{B}$ be a given very ample line bundle such that $\mathcal{A}:=\omega_{\mathcal{X}}\otimes\mathcal{B}^{n+1}$ is ample. By the Castelnuovo-Mumford criterion it is enough to check that

H^{q}\left(\mathcal{X},\mathcal{A}\otimes\mathcal{L}\otimes\mathcal{B}^{-q}\otimes\mathcal{J}(\mathfrak{a}^{c})\right)=0

for $q=1,\dots,n$ , and this is a consequence of Theorem B.5. ∎