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://wwwfourier.ujfgrenoble.fr/~demailly/manuscripts/trieste.pdf this is I think one of the most standard places to learn about it.

4$\begingroup$ Since you wrote my name explicitly, I'll try to write something later :) $\endgroup$– diveriettiSep 23, 2013 at 9:57

9$\begingroup$ Multiplier ideal sheaves are important in the study of KahlerEinstein metric:Let $M$ be a Fano manifold and $g$ an initial Kähler metric on $M$ whose Kähler form represents the first Chern class $c_1(M)$ of $M$.If the closedness does not hold for the continuity method,then there is a multiplier ideal subvariety $V⊂M$ and for any global holomorphic vector field on $M$ with vanishing Futaki invariant we have $V⊄Z^+(X)$, where $Z^+(X)$ is the set of all points in $M$ at which $X$ vanishes and has divergence of positive real part.This is due to Nadel.Lelong also studied multiplier ideal sheaves $\endgroup$– user21574Jul 23, 2017 at 16:47

8$\begingroup$ On a Kähler manifold that does not admit KählerEinstein metrics there is a nontrivial coherent ideal sheaf, which he called by Nadel a "multiplier ideal sheaf'' But before Nadel , the first person who introduced Multiplier Ideal sheaves was J. Kohn $\endgroup$– user21574Jul 23, 2017 at 17:03

9$\begingroup$ Mori's used a nice method of constructing rational curves in a Fano manifold and later Siu by using study of dynamics of Multiplier ideal sheaves gave a new proof of Mori's theorem, See Siu, YumTong Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds. Complex analysis and digital geometry, 323–360, $\endgroup$– user21574Jul 23, 2017 at 17:10

9$\begingroup$ To be more precise to my second previous comment: If $X$ does not admit a KählerEinstein metric, then there exists for each $p>1$ a function $ψ$ which is the limit of $ϕ_t−\frac{1}{V}\int_Xϕ_tω^n_0$ in the $L^1$ topology and such that the multiplier ideal sheaf $\mathcal I(ψ)$ is a proper coherent analytic sheaf on $X$ with acyclic cohomology, i.e., $$H^q(X,K^{−[p]}_X⊗\mathcal I(pψ))=0$$ Here, one defines $\mathcal I(ψ)⊂\mathcal O_X $by $$Γ(U,\mathcal I(ψ))=\left\{f∈\mathcal O_X(U): f^2e^{−pψ}∈L^1_{loc}(U)\right\}$$. Here $ ϕ_t $ is the potential evolving the MongeAmpere equation $\endgroup$– user21574Jul 23, 2017 at 17:35
2 Answers
Here is a sketch of Nadel's original motivation. Classical results of Aubin and Yau imply the existence of KahlerEinstein metrics on manifolds with ample canonical bundle and and for all polarisations of CalabiYau manifolds. The method involved is a continuity method for the complex MongeAmpère 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 KaehlerEinstein 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 KaehlerEinstein 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 nontrivial 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 KaehlerEinstein 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 KaehlerEinstein 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 algebrogeometrically. Given an anticanonical 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 KaehlerEinstein metrics as saying that the nonexistence of such a metric implies the existence of a highly singular anticanonical divisor, and moreover the "highlysingular" 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 anticanonical divisor  I suspect this is the case due to certain approximation results).
I think a good reference for this is section $6$ of the DemaillyKollár paper "Semicontinuity of complex singularity exponents and KaehlerEinstein metrics on Fano orbifolds". It explains what I have described above fully and precisely (gives definitions etc.), and proves Nadel's result on KaehlerEinstein metrics in a simpler way than Nadel originally did.

5$\begingroup$ I don't think I have much to add to this very detailed answer. It is anyhow amazing how many others applications there are, completely independent of the original motivation! For an excellent reference, see Lazarsfeld's book "Positivity in Algebraic Geometry". $\endgroup$ Sep 23, 2013 at 12:57

$\begingroup$ thanks for an outstanding and highly informative answer. $\endgroup$– KoushikSep 24, 2013 at 4:43

$\begingroup$ “Here, one views the multiple ideal sheaf as the functions where certain integrals don't converge”, is there a concrete example(of complex manifold,or paper) that clarifies this point? $\endgroup$– Henry.LApr 8, 2017 at 23:35
There's a parallel history of multiplier ideals (especially of the nondynamic 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 KawamataViehweg 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 Kodairatype vanishing theorem on a variety that is either nonsmooth or with respect to a notnecessarilyample 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 twodimensional 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).