motivation for multiplier ideal sheaves

Question

What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly elaborate on the motivation behind defining multiplier ideal sheaves.I think there are lots of experts here in mathoverflow who are experts in these things like diverio and many others.http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/trieste.pdf this is I think one of the most standard places to learn about it.

Answer 1

Here is a sketch of Nadel's original motivation. Classical results of Aubin and Yau imply the existence of Kahler-Einstein metrics on manifolds with ample canonical bundle and and for all polarisations of Calabi-Yau manifolds. The method involved is a continuity method for the complex Monge-Ampère equation (see for example Tian's Canonical metrics in Kaehler geometry for an introduction to this stuff), together with certain a priori $C^0$ estimates.

When one searches for Kaehler-Einstein metrics on Fano manifolds ($-K_X$ ample), things are harder. In Nadel's time, certain obstructions were known (for example Matsushima showed the lie algebra of the automorphism group must be reductive), but few sufficient conditions were known. However, on Fano manifolds without a Kaehler-Einstein metric, the continuity method must fail. Nadel's idea was to study consequences of the failure of the continuity method. Specifically, Nadel showed that if the continuity method fails, then there must exist a singular hermitian metric written locally $h=h_0e^{-\phi}$ on $-K_X$, where $h_0$ is a genuine smooth hermitian metric, and $\phi$ satisfies some mild regularity assumptions, such that $h$ has semipositive curvature current and $\phi$ has non-trivial multiplier ideal sheaf $\mathcal{I}(\gamma \phi)$ for all $\gamma \in (\frac{n}{n+1},1)$. Here, one views the multiple ideal sheaf as the functions where certain integrals don't converge (equivalently, if certain integrals converge, the continuity method doesn't fail and there is a Kaehler-Einstein metric). Moreover, one can assume that for any compact $G\subset Aut(X)$, $h_0$ and $\phi$ are $G$-invariant.

Nadel combined this with his vanishing result: $H^q(X,\mathcal{I}(\gamma \phi))=0$ for all $q>0$. Here we're using that $h$ is a singular hermitian metric on $-K_X$. This form of Nadel vanishing has strong geometric consequences: associating a $G$-invariant subscheme $Z_{\gamma}$ to $\mathcal{I}(\gamma \phi)$, this implies that $H^q(Z_{\gamma}, \mathcal(O_{Z_{\gamma}}))=0$ for all $q>0$ and equals $\mathbb{C}$ for $q=0$. A simple corollary is that $Z_{\gamma}$ is connected, so if $G$ acts without fixed points, cannot be of dimension $0$. Then, if $X$ is of dimension $3$, $Z_{\gamma}$ must be $1$ dimensional, and Nadel showed $Z_{\gamma}$ must be a tree of rational curves, the existence of which can sometimes be ruled out. Nadel's construction therefore gave new examples of Fano manifolds with Kaehler-Einstein metrics.

One can also think about multiplier ideal sheaves as follows. This probably isn't how Nadel thought about them at the time, however it is slightly more appealing algebro-geometrically. Given an anti-canonical divisor $D$, one can naturally associate a singular hermitian metric on $-K_X$. One property of the pair $(X,D)$ is whether or not it is log canonical - algebraically this means it is not too singular, analytically this tells you a certain integral converges. The multiplier ideal sheaf associated to $D$ refines this, essentially giving a scheme structure to the set at which the pair $(X,\gamma D)$ is not log canonical, for all $\gamma$. Nadel vanishing then tells you, for example, that the set at which $\gamma D$ is not log canonical (i.e. is highly singular) is connected. In this case then one can view Nadel's result on Kaehler-Einstein metrics as saying that the non-existence of such a metric implies the existence of a highly singular anti-canonical divisor, and moreover the "highly-singular" locus of this divisor satisfies certain geometric conditions which can be ruled out in certain cases (at least in the case that the singular hermitian metric in Nadel's theorem arises from an anti-canonical divisor - I suspect this is the case due to certain approximation results).

I think a good reference for this is section $6$ of the Demailly-Kollár paper "Semi-continuity of complex singularity exponents and Kaehler-Einstein metrics on Fano orbifolds". It explains what I have described above fully and precisely (gives definitions etc.), and proves Nadel's result on Kaehler-Einstein metrics in a simpler way than Nadel originally did.

Answer 2

There's a parallel history of multiplier ideals (especially of the non-dynamic multiplier ideal sheaves on algebraic varieties, say as described in Lazarsfeld's book).

From this perspective, for $\mathfrak{a}$ an ideal sheaf on $X$, the multiplier ideal of $(X, \mathfrak{a}^t)$ is defined as follows (assuming $K_X$ is $\mathbb{Q}$-Cartier, which always holds if $X$ is smooth). Choose $\pi : Y \to X$ a log resolution of $(X, \mathfrak{a})$ with $\mathfrak{a} \cdot O_Y = O_Y(-G)$. Then $$ \mathcal{J}(X, \mathfrak{a}^t) = \pi_* O_Y( \lceil K_Y - \pi^* K_X - t G\rceil) $$ These ideal sheaves are older than Nadel's work. For instance, they were extremely common in the work of Esnault and Viehweg in the early 1980s (see for instance their notes which survey some of this work Lectures on vanishing theorems), also see the works of Kawamata and Kollar. Indeed, these sheaves and slight variants appeared frequently whenever Kawamata-Viehweg vanishing theorems were applied throughout the 1980s. Essentially, the reason why they show up in this context is as follows. You want to prove some Kodaira-type vanishing theorem on a variety that is either non-smooth or with respect to a not-necessarily-ample line bundle. The multiplier ideal lets you correct for this.

If you assume that $\mathfrak{a} = O_X$ and if you remove the $\pi^* K_X$ from the definition, then you get a subsheaf of $\omega_X$. This subsheaf appeared in the work Grauert and Riemenschneider (1970) and was used frequently by Lipman in his work in the 1970s especially in his work on resolution of singularities of excellent two-dimensional rings (the fact that the multiplier submodule of $\omega_X$ is not equal to $\omega_X$ is a measure of singularities).

In the case that $t = 1$ and $X$ is regular, this appeared in the work of Lipman in the 1980s and 1990s (especially in relation to questions of integral closure of powers of ideals).