AA SHORT COURSE ON MULTIPLIER IDEALS
ROBERT LAZARSFELD
Contents
Introduction 11. Construction and Examples of Multiplier Ideals 22. Vanishing Theorems for Multiplier Ideals 103. Local Properties of Multiplier Ideals 164. Asymptotic Constructions 245. Extension Theorems and Deformation Invariance of Plurigenera 31References 36
Introduction
These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aimto give an introduction to the algebro-geometric side of the theory, with an emphasis on itsglobal aspects. Besides serving as warm-up for the lectures of Hacon, my hope was to conveyto the audience a feeling for the sorts of problems for which multiplier ideals have proveduseful. Thus I have focused on concrete examples and applications at the expense of generaltheory. While referring to [21] and other sources for some technical points, I have tried toinclude sufficient detail here so that the conscientious reader can arrive at a reasonable graspof the machinery by working through these lectures.The revolutionary work of Hacon–McKernan, Takayama and Birkar–Cascini–Hacon–McKernan ([14], [15], [28], [3]) appeared shortly after the publication of [21], and thesepapers have led to some changes of perspectives on multiplier ideals. In particular, the firstthree made clear the importance of adjoint ideals as a tool in proving extension theorems;these were not so clearly in focus at the time [21] was written. I have taken this new viewpointinto account in discussing the restriction theorem in Lecture 3. Adjoint ideals also open thedoor to an extremely transparent presentation of Siu’s theorem on deformation-invarianceof plurigenera of varieties of general type, which appears in Lecture 5.Besides Part III of [21], I have co-authored an overview of multiplier ideals once before,in [2]. Those notes focused more on the local and algebraic aspects of the story. The analytictheory is surveyed in [26], as well as in other lecture series in this volume.
Research partially supported by NSF grant DMS 0652845. a r X i v : . [ m a t h . AG ] J a n ROBERT LAZARSFELD
I wish to thank Eugene Eisenstein, Christopher Hacon, J´anos Koll´ar and Mircea Mustat¸˘afor valuable suggestions. I am particularly grateful to Sam Grushevsky, who read throughin its entirety a draft of these lectures, and made copious suggestions.1.
Construction and Examples of Multiplier Ideals
This preliminary lecture is devoted to the construction and first properties of multiplierideals. We start by discussing the algebraic and analytic incarnations of these ideals. Aftergiving the example of monomial ideals, we survey briefly some of the invariants of singularitiesthat can be defined via multiplier ideals.
Definition of Multiplier Ideals.
In this section, we will give the definition of multiplierideals.We work throughout with a smooth algebraic variety X of dimension d defined over C . For the moment, we will deal with two sorts of geometric objects on X : an ideal sheaf a ⊆ O X together with a weighting coefficient c >
0, and an effective Q -divisor D on X .Recall that the latter consists of a formal linear combination D = (cid:88) a i D i , where the D i are distinct prime divisors and each a i ∈ Q is a non-negative rational number.We will attach to these data multiplier ideal sheaves J ( a c ) , J ( D ) ⊆ O X . The intuition is that these ideals will measure the singularities of D or of functions f ∈ a ,with “nastier” singularities being reflected in “deeper” multiplier ideals.Although we will mainly focus on algebraic constructions, it is perhaps most intuitiveto start with the analytic avatars of multiplier ideals. Definition 1.1 (Analytic multiplier ideals) . Given D = (cid:80) a i D i as above, choose localequations f i ∈ O X for each D i . Then the (analytic) multiplier ideal of D is given locally by J an ( X, D ) = locally (cid:110) h ∈ O X (cid:12)(cid:12)(cid:12) | h | (cid:81) | f i | a i is locally integrable (cid:111) . Similarly, if f , . . . , f r ∈ a are local generators, then J an ( X, a c ) = locally (cid:110) h ∈ O X (cid:12)(cid:12)(cid:12) | h | (cid:0) (cid:80) | f i | (cid:1) c is locally integrable (cid:111) . (One checks that these do not depend on the choice of the f i .) (cid:3) Equivalently, J an ( D ) and J an ( a c ) arise as the multiplier ideal J ( φ ), where φ is theappropriate one of the two plurisubharmonic functions φ = (cid:88) log | f i | a i or φ = c · log (cid:0) (cid:88) | f i | (cid:1) . SHORT COURSE ON MULTIPLIER IDEALS 3
Note that this construction exhibits quite clearly the yoga that “more” singularities give riseto “deeper” multiplier ideals: the singularities of f ∈ a or of D are reflected in the rate atwhich the real-vallued functions 1 (cid:81) | f i | a i or 1 (cid:0) (cid:80) | f i | (cid:1) c blow-up along the support of D or the zeroes of a , and this in turn is measured by thevanishing of the multipliers h required to ensure integrability. Exercise 1.2.
Suppose that D = (cid:80) a i D i has simple normal crossing support. Then J an ( X, D ) = O X ( − [ D ]) , where [ D ] = (cid:80) [ a i ] D i is the round-down (or integer part) of D . ( Hint : This boils downto the assertion that if z , . . . , z d are the standard complex coordinates in C d , and if h ∈ C { z , . . . , z d } is a convergent power series, then | h | | z | a · . . . · | z d | a d is locally integrable near the origin if and only if z [ a ]1 · . . . · z [ a d ] d | h in C { z , . . . , z d } . By separating variables, this in turn follows from the elementary compu-tation that the function 1 / | z | c of one variable is locally integrable if and only if c < (cid:3) Multiplier ideals can also be constructed algebro-geometrically. Let µ : X (cid:48) −→ X be a log resolution of D or of a . Recall that this means to begin with that µ is a propermorphism, with X (cid:48) smooth. In the first instance we require that µ ∗ D + Exc( µ ) have simplenormal crossing (SNC) support, while in the second one asks that a · O X (cid:48) = O X (cid:48) ( − F )where F is an effective divisor and F + Exc( µ ) has SNC support. We consider also therelative canonical bundle K X (cid:48) /X = det( dµ ) , so that K X (cid:48) /X ≡ lin K X (cid:48) − µ ∗ K X . Note that this is well-defined as an actual divisor supportedon the exceptional locus of µ (and not merely as a linear equivalence class). Definition 1.3 (Algebraic multiplier ideal) . The multiplier ideals associated to D and to a are defined to be: J ( D ) = µ ∗ O X (cid:48) ( K X (cid:48) /X − [ µ ∗ D ]) J ( a c ) = µ ∗ O X (cid:48) ( K X (cid:48) /X − [ cF ]) . (As in the previous Exercise, the integer part of a Q -divisor is defined by taking the integerpart of each of its corefficients.) (cid:3) ROBERT LAZARSFELD
Observe that these are subsheaves of µ ∗ O X (cid:48) ( K X (cid:48) /X ) = O X , i.e. they are indeed ideal sheaves.One can rephrase the definition more concretely in terms of discrepancies. Write(1.1) µ ∗ D = (cid:88) r i E i , K X (cid:48) /X = (cid:88) b i E i , where the E i are distinct prime divisors on X (cid:48) : thus the r i are non-negative rational numbersand the b i are non-negative integers. We view each of the E i as defining a valuation ord E i on rational or regular functions on X . Then it follows from Definition 1.3 that J ( D ) = (cid:8) f ∈ C ( X ) (cid:12)(cid:12) ord E i ( f ) ≥ [ r i ] − b i , with f otherwise regular (cid:9) , with a similar interpretation of J ( a c ). (Note that we are abusing notation a bit here: J ( D )is actually the sheaf determined in the evident manner by the recipe on the right.) Observethat b i > E i is µ -exceptional, so the conditionord E i ( f ) ≥ [ r i ] − b i does not allow f to have any poles on X . Thus we see again that J ( D ) is a sheaf of ideals. Remark 1.4.
The definitions of J an ( D ) and J ( D ) may seem somewhat arbitrary or un-motivated, but they are actually dictated by the vanishing theorems that multiplier idealssatisfy. In the algebraic case, this will become clear for example in the proof of Theorem2.4. (cid:3) Example 1.5.
We work out explicitly one (artificially) simple example. Let X = C , let A , A , A ⊆ X be three distinct lines through the origin, and set D = 23 ( A + A + A ) . Then D is resolved by simply blowing up the origin: µ : X (cid:48) = Bl ( X ) −→ X. Writing E for the exceptional divisor of µ , and A (cid:48) i for the proper transform of A i , one has µ ∗ ( A + A + A ) = ( A (cid:48) + A (cid:48) + A (cid:48) ) + 3 Eµ ∗ D = 23 ( A (cid:48) + A (cid:48) + A (cid:48) ) + 2 E [ µ ∗ D ] = 2 E. Moreover K X (cid:48) /X = E , and hence J ( D ) = µ ∗ O X (cid:48) ( − E )is the maximal ideal of functions vanishing at the origin. Observe that this computation alsoshows that rounding does not in general commute with pull-back of Q -divisors. (cid:3) The algebraic construction of multiplier ideals started by choosing a resolution of singu-larities. Therefore it is important to establish:
SHORT COURSE ON MULTIPLIER IDEALS 5
Proposition 1.6.
The multiplier ideals J ( D ) and J ( a c ) do not depend on the resolutionused to construct them. In brief, using the fact that any two resolutions can be dominated by a third, one reduces tochecking that if X is already a log resolution of the data at hand, then nothing is changedby passing to a further blow-up: Lemma 1.7.
Assume that D has SNC support, and let µ : X (cid:48) −→ X be a further logresolution of ( X, D ) . Then µ ∗ O X (cid:48) (cid:0) K X (cid:48) /X − [ µ ∗ D ] (cid:1) = O X (cid:0) − [ D ] (cid:1) . This in turn can be checked by an elementary direct calculation. We refer to [21, 9.2.19] fordetails.The next point is to reconcile the analytic and algebraic constructions of multiplierideals.
Proposition 1.8.
Let D be an effective Q -divisor on X . Then J an ( X, D ) = J ( X, D ) , and similarly J an ( X, a c ) = J ( X, a c ) for any ideal sheaf a . (Strictly speaking, the analytic multiplier ideals are the analytic sheaves associated to theiralgebraic counterparts, but we do not dwell on this distinction.)For the Proposition, the key point is that both species of multiplier ideals transform thesame way under birational morphisms: Lemma 1.9.
Let µ : X (cid:48) −→ X be a proper birational map, and let D be an effective Q -divisor on X . Then: J an ( X, D ) = µ ∗ (cid:0) O X (cid:48) ( K X (cid:48) /X ) ⊗ J an ( X (cid:48) , µ ∗ D ) (cid:1) J ( X, D ) = µ ∗ (cid:0) O X (cid:48) ( K X (cid:48) /X ) ⊗ J ( X (cid:48) , µ ∗ D ) (cid:1) . In the analytic setting this is a consequence of the change of variables formula for integrals,while the algebraic statement is established with a little computation via the projectionformula. The Proposition follows at once from the Lemma. In fact, one is reduced toproving Proposition 1.8 when D or a are already in normal crossing form, and this case ishandled by Exercise 1.2. Remark 1.10 (Multiplier ideals on singular varieties) . Under favorable circumstances, Def-inition 1.3 makes sense even when X is singular. The main point at which non-singularityis used in the discussion above is to be able to define the relative canonical bundle K X (cid:48) /X = K X (cid:48) − µ ∗ K X of a log resolution µ : X (cid:48) −→ X of ( X, D ). For this it is enough that X is normal and that K X is Cartier or even Q -Cartier,so that µ ∗ K X is defined. Thus Definition 1.3 goes through without change provided that ROBERT LAZARSFELD X is Gorenstein or Q -Gorenstein. For an arbitrary normal variety X , one can introduce a“boundary” Q -divisor ∆ such that K X + ∆ is Q -Cartier, and define multiplier ideals J (( X, ∆); D ) ⊆ O X . These generalizations are discussed briefly in [21, § X . (cid:3) We conclude this section with two further exercises for the reader.
Exercise 1.11.
Assume that X is affine, and let a ⊆ C [ X ] be an ideal. Given c >
0, choose k > c general elements f , . . . , f k ∈ a , let A i = div( f i ), and put D = ck ( A + . . . + A k ). Then J ( D ) = J ( a c ) . (By a “general element” of an ideal, one means a general C -linear combination of a collectionof generators of the ideal.) (cid:3) Exercise 1.12.
Let D = (cid:80) a i D i be an effective Q -divisor on X . Assume thatmult x ( D ) = def (cid:88) a i · mult x ( D i ) ≥ dim X for some point x ∈ X . Then J ( X, D ) is non-trivial at x , i.e. J ( D ) x ⊆ m x ⊆ O x X, where m x ⊆ O X is the maximal ideal of x . (Compute the multiplier ideal in question usinga resolution µ : X (cid:48) −→ X that dominates the blow-up of X at x , and observe thatord E (cid:0) K X (cid:48) /X − [ µ ∗ D ] (cid:1) ≤ − , where E is the proper transform of the exceptional divisor over x .) (cid:3) Monomial Ideals.
It is typically very hard to compute the multiplier ideal of an explicitlygiven divisor or ideal. One important class of examples that has been worked out is that ofmonomial ideals on affine space. These are handled by a theorem of Howald [16].Let X = C d , and let a ⊆ C [ x , . . . , x d ]be an ideal generated by monomials in the x i . Observe that such a monomial is specified byan exponent vector w = ( w , . . . , w d ) ∈ N d : we write x w = x w · . . . · x w d d . The
Newton polyhedron P ( a ) ⊆ R d of a is the closed convex set spanned by the exponent vectors of all monomials in a . This isillustrated in Figure 1, which shows the Newton polyhedron for the monomial ideal(1.2) a = (cid:10) x , x y, xy , y (cid:11) . SHORT COURSE ON MULTIPLIER IDEALS 7
Figure 1.
Newton Polyhedron of (cid:104) x , x y, xy , y (cid:105) Finally, put = (1 , . . . , ∈ N d .Howald’s statement is the following: Theorem 1.13.
For any c > , J ( a c ) ⊆ C [ x , . . . , x d ] is the monomial ideal spanned by all monomials x w where w + ∈ int (cid:0) c · P ( a ) (cid:1) . Once one knows the statement for which one is aiming, the proof is relatively straight-forward: see [16] or [21, § Example 1.14.
For a = (cid:104) x , x y, xy , y (cid:105) , one has J ( a ) = ( x , xy, y ), while if 0 < c < x, y ) ⊆ J ( a c ). (cid:3) Example 1.15.
Let a = (cid:0) x e , . . . , x e d d (cid:1) ⊆ C [ x , . . . , x d ] . Writing ξ , . . . , ξ d for the natural coordinates on R d adapted to N d ⊆ R d , the Newtonpolyhedron P ( a ) ⊆ R d of a is the region in the first orthant given by the equation ξ e + . . . + ξ d e d ≥ . Hence J ( a c ) is the monomial ideal spanned by all monomials x w whose exponent vectorssatisfy the equation w + 1 e + . . . + w d + 1 e d > c. (cid:3) Invariants Defined by Multiplier Ideals.
Multiplier ideals lead to invariants of thesingularities of a divisor or the functions in an ideal. The most important and well-knownis the following:
ROBERT LAZARSFELD
Definition 1.16 (Log-canonical threshold) . Let D be an effective Q -divisor on a smoothvariety X , and let x ∈ X be a fixed point. The log-canonical threshold of D at x islct x ( D ) = def min (cid:8) c > (cid:12)(cid:12) J ( cD ) is non-trivial at x (cid:9) . The log-canonical threshold of an ideal sheaf a ⊆ O X is defined analogously. (cid:3) Thus small values of lct x ( D ) reflect more dramatic singularities. Definition 1.25 below ex-plains the etymology of the term. Exercise 1.17.
Let µ : X (cid:48) −→ X be a log resolution of D , and as in equation (1.1) write µ ∗ D = (cid:88) r i E i , K X (cid:48) /X = (cid:88) b i E i . Then lct x ( D ) = min µ ( E i ) (cid:51) x (cid:110) b i + 1 r i (cid:111) , the minimum being taken over all i such that x lies in the image of the correspondingexceptional divisor. In particular, lct x ( D ) is rational. Example 1.18 (Complex singularity exponent) . One of the early appearances of the log-canonical threshold was in the work [29] of Varchenko, who studied the complex singularityexponent c ( f ) of a polynomial or holomorphic function f in a neighborhood of 0 ∈ C d .Specifically, set(*) c ( f ) = sup (cid:110) c > (cid:12)(cid:12)(cid:12) | f | c is locally integrable near 0 (cid:111) . Writing lct ( f ) for the log-canonical threshold of the divisor determined by f (or equivalentlyof the principal ideal generated by f ), it follows from Proposition 1.8 that c ( f ) = lct ( f ) . In view of the previous exercise, this establishes the fact – which is certainly not obviousfrom (*) – that c ( f ) is rational. (The rationality of c ( f ) was proven in this manner byVarchenko, although his work pre-dates the language of multiplier ideals.) (cid:3) Exercise 1.19. If D = { x − y = 0 } ⊆ C , then lct ( D ) = . (cid:3) Exercise 1.20.
Consider as in Example 1.15 the monomial ideal a = ( x e , . . . , x e d d ) ⊆ C [ x , . . . , x d ] . Then lct ( a ) = (cid:80) e i . (cid:3) The log-canonical threshold is the first of a sequence of invariants defined by the “jump-ing” of multiplier ideals. Specifically, observe that the ideals J ( cD ) become deeper as thecoefficient c grows. So one is led to: Proposition/Definition 1.21.
In the situation of Definition 1.16, there exists a discretesequence of rational numbers ξ i = ξ i ( D ; x ) with ξ < ξ < ξ < . . . SHORT COURSE ON MULTIPLIER IDEALS 9 characterized by the property that ( the stalks at x of ) the multiplier ideals J ( cD ) x are con-stant exactly for c ∈ [ ξ i , ξ i +1 ) . The ξ i are called the jumping numbers of D at x . Jumping numbers of an ideal sheaf a ⊆ O X are defined similarly.It follows from the definition that lct x ( D ) = ξ ( D ; x ). In the notation of (1.1), the ξ i occur among the rational numbers ( b i + m ) /r i for various m ∈ N . First appearing implicitlyin work of Libgober and Loeser-Vaqui´e, these quantities were studied systematically in [9].In particular, this last paper establishes some connections between jumping coefficients andother invariants. Exercise 1.22.
Compute the jumping numbers of the ideal a = ( x e , . . . , x e d d ). (cid:3) Exercise 1.23.
Let ξ i < ξ i +1 be consecutive jumping coefficients of an ideal a ⊆ O X at apoint x ∈ X . Then √ a · J ( a ξ i ) x ⊆ J ( a ξ i +1 ) x in O x X . In particular, ( √ a ) m ⊆ J ( a ξ m ) x for every m >
0. (This was pointed out to us by M. Mustat¸˘a. See Example 3.19 for anapplication.) (cid:3)
Remark 1.24.
In an analogous fashion, one can define the log-canonical threshold lct( D )and lct( a ), as well as jumping numbers ξ i ( D ) and ξ i ( a ), globally on X , without localizing ata particular point. We leave the relevant definitions – as well as the natural extension of theprevious Exercise – to the reader. (cid:3) Finally, we note that multiplier ideals lead to some natural classes of singularities for apair (
X, D ) consisting of a smooth variety X and an effective Q -divisor D on X . Definition 1.25.
One says that (
X, D ) is
Kawamata log-terminal (KLT) if J ( X, D ) = O X . The pair (
X, D ) is log-canonical if J ( X, (1 − ε ) D ) = O X for 0 < ε (cid:28) (cid:3) These concepts (and variants thereof) play an important role in the minimal model program,although in that setting one does not want to limit oneself to smooth ambient varieties.
Remark 1.26 (Characteristic p analogues) . Work of Smith, Hara, Yoshida, Watanabe,Takagi, Mustat¸˘a and others has led to the development of theory in characteristic p > (cid:3) Vanishing Theorems for Multiplier Ideals
In this Lecture we discuss the basic vanishing theorems for multiplier ideals, and givesome first applications. As always, we work with varieties over C . The Kawamata–Viehweg–Nadel Vanishing Theorem.
We start by recalling some def-initions surrounding positivity for divisors. Let X be an irreducible projective variety ofdimension d , and let B be a (Cartier) divisor on X . One says that B is nef (or numericallyeffective ) if ( B · C ) ≥ C ⊆ X. Nefness means in effect that B is a limit of ample divisors: see [21, Chapter 1.4] for a preciseaccount. A divisor B is big if the spaces of sections of mB grow maximally with m , i.e. if h (cid:0) X, O X ( mB ) (cid:1) ∼ m d for m (cid:29) Q -divisors. We will first deal with divisorsthat are both nef and big: a typical example arises by pulling back an ample divisor undera birational morphism.A basic fact is that for a nef divisor, bigness is tested numerically: Lemma 2.1.
Assume that B is nef. Then B is big if and only if its top self-intersectionnumber is strictly positive: ( B d ) > . See [21, § Q -divisors.The fundamental result for our purposes was proved independently by Kawamata andViehweg in the early 1980’s. Theorem 2.2 (Kawamata–Viehweg Vanishing Theorem) . Let X be a smooth projectivevariety. Consider an integral divisor L and an effective Q -divisor D on X . Assume that (i). L − D is nef and big; and (ii). D has simple normal crossing support.Then H i (cid:0) X, O X ( K X + L − [ D ]) (cid:1) = 0 for i > . As in the previous Lecture, the integer part (or round-down) [ D ] of a Q -divisor D is obtainedby taking the integer part of each of its coefficients.When D = 0 and L is ample, this is the classical Kodaira vanishing theorem. Stilltaking D = 0, the Theorem asserts in general that the statement of Kodaira vanishingremains true for divisors that are merely big and nef: this very useful fact – also due toKawamata and Viehweg – completes some earlier results of Ramanujam, Mumford, andGrauert–Riemenschneider. However the real power (and subtlety) of Theorem 2.2 lies in thefact that while the positivity hypothesis is tested for a Q -divisor, the actual vanishing holdsfor a round of this divisor. As we shall see, this apparently technical improvement vastly SHORT COURSE ON MULTIPLIER IDEALS 11 increases the power of the result: taking integer parts can significantly change the shape ofa divisor, so in favorable circumstances one gets a vanishing for divisors that are far frompositive.The original proofs of the theorem proceeded by using covering constructions to reduceto the case of integral divisors. An account of this approach, taking into account simplifica-tions introduced by Koll´ar and Mori in [20], appears in [21, § § L ∂ -machinery gives yetanother proof. In any event, it is nowadays not substantially harder to establish Theorem2.2 than to prove the classical Kodaira vanishing.The main difficulty in applying Theorem 2.2 is that in practice the normal crossinghypothesis is rarely satisfied directly. Given an arbitrary effective Q -divisor D on a variety X , a natural idea is to apply vanishing on a resolution of singularities and then “push down”to get a statement on X . Multiplier ideals appear inevitably in so doing, and this leads tothe basic vanishing theorems for these ideals.There are two essential results. Theorem 2.3 (Local vanishing theorem) . Let X be a smooth variety, D an effective Q -divisor on X , and µ : X (cid:48) −→ X a log resolution of D . Then R j µ ∗ O X (cid:48) (cid:0) K X (cid:48) /X − [ µ ∗ D ] (cid:1) = 0 for j > . The analogous statement holds for higher direct images of the sheaves computing the mul-tiplier ideals J ( a c ). Theorem 2.4 (Nadel Vanishing Theorem) . Let X be a smooth projective variety, and let L and D be respectively an integer divisor and an effective Q -divisor on X . Assume that L − D is nef and big. Then H i (cid:0) X, O X ( K X + L ) ⊗ J ( D ) (cid:1) = 0 for i > .Proof of Theorem 2.4 (granting Theorems 2.2 and 2.3). Let µ : X (cid:48) −→ X be a log resolu-tion of D , and set L (cid:48) = µ ∗ L , D (cid:48) = µ ∗ D. Thus L (cid:48) − D (cid:48) is a nef and big Q -divisor on X (cid:48) , and by construction D (cid:48) has SNC support.Therefore Kawamata–Viehweg applies on X (cid:48) to give(2.1) H i (cid:0) X (cid:48) , O X (cid:48) ( K X (cid:48) + L (cid:48) − [ D (cid:48) ]) (cid:1) = 0for i >
0. Now note that K X (cid:48) + L (cid:48) − [ D (cid:48) ] = K X (cid:48) /X − [ µ ∗ D ] + µ ∗ ( K X + L ) . On the other hand, one finds using the projection formula and the definition of J ( D ): µ ∗ O X (cid:48) (cid:0) K X (cid:48) /X − [ µ ∗ D ] + µ ∗ ( K X + L ) (cid:1) = µ ∗ O X (cid:48) (cid:0) K X (cid:48) /X − [ µ ∗ D ] (cid:1) ⊗ O X ( K X + L )= O X ( K X + L ) ⊗ J ( D ) . But thanks to Theorem 2.3, the vanishing (2.1) is equivalent to the vanishing H i (cid:0) X, O X ( K X + L ) ⊗ J ( D ) (cid:1) = 0of the direct image of the sheaf in question, as required. (cid:3) The proof of Theorem 2.3 is similar: one reduces to the case when X is projective, andapplies the result of Kawamata and Viehweg in the global setting. See [21, § Singularities of Plane Curves and Projective Hypersurfaces.
As a first illustration,we will apply these theorems to prove some classical results about singularities of planecurves and their extensions to hypersurfaces of higher dimension. Starting with a singularhypersurface, the strategy is to build a Q -divisor having a non-trivial multiplier ideal. Thenthe vanishing theorems give information about the postulation of the singularities of theoriginal hypersurface.Consider to begin with a (reduced) plane curve C ⊆ P of degree m , and letΣ = Sing( C ) , considered as a reduced finite subset of the plane. Our starting point is the classical Proposition 2.5.
The set Σ imposes independent conditions on curves of degree k ≥ m − ,i.e. (2.2) H (cid:0) P , I Σ ( k ) (cid:1) = 0 for k ≥ m − . Here I Σ denotes the ideal sheaf of Σ. We give a proof using Nadel vanishing momentarily,but first we discuss a less familiar extension due to Zariski.Specifically, suppose that C ⊆ P has a certain number of cusps, defined in local analyticcoordinates by an equation x − y = 0. ( C may have other singularities as well.) LetΞ = Cusps( C ) , again regarded as a reduced finite subset of P . Zariski [30] proved that one gets a strongerresult for the postulation of Ξ. In fact: Proposition 2.6.
One has H (cid:0) P , I Ξ ( k ) (cid:1) = 0 for k > m − . Interestingly enough, Zariksi proved this by considering the irregularity of the cyclic coverof P branched along C . One can see the Kawamata–Viehweg–Nadel theorem as a vastgeneralization of this approach. SHORT COURSE ON MULTIPLIER IDEALS 13
Proof of Proposition 2.6.
This is a direct consequence of Nadel vanishing. In fact, considerthe Q -divisor D = C . Since the log-canonical threshold of a cusp is = , one has J ( D ) ⊆I Ξ . But as C is reduced the multiplier ideal J ( D ) is non-trivial only at finitely many points.Thus I Ξ / J ( D ) is supported on a finite set, and therefore the map H (cid:0) P , O P ( k ) ⊗ J ( D ) (cid:1) −→ H (cid:0) P , O P ( k ) ⊗ I Ξ (cid:1) is surjective for all k . So it suffices to prove that the group on the left vanishes for k > m −
3. But this follows immediately from Nadel vanishing upon recalling that O P ( K P ) = O P ( − (cid:3) Proof of Proposition 2.5.
Here an additional trick is required in order to produce a Q -divisorwhose multiplier ideal vanishes on finite set including Σ = Sing( C ). Specifically, fix 0 < ε (cid:28)
1, and let Γ be a reduced curve of degree (cid:96) , not containing any components of C , passingthrough Σ. Consider the Q -divisor D = (1 − ε ) C + 2 ε Γ . This has multiplicity ≥ C , and hence J ( D ) vanishes on Σ thanksto Exercise 1.12. Moreover J ( D ) is again cosupported on a finite set since no component of D has coefficient ≥
1. Therefore, as in the previous proof, it suffices to show that(*) H (cid:0) P , O P ( k ) ⊗ J ( D ) (cid:1) = 0for k ≥ m −
2. But deg D = (1 − ε ) m + 2 ε(cid:96) < m + 1for ε (cid:28)
1, and so (*) again follows from Nadel vanishing. (cid:3)
Example 2.7.
When C is the union of m general lines, the bound k ≥ m − k ≥ m − C isirreducible. (cid:3) Finally, we present a generalization of Proposition 2.5 to higher dimensional hypersur-faces.
Proposition 2.8.
Let S ⊆ P r be a ( reduced ) hypersurface of degree m ≥ having onlyisolated singularities, and set Σ = Sing( S ) . Then Σ imposes independent conditions on hypersurfaces of degree ≥ m ( r − − (2 r − ,i.e. H (cid:0) P r , I Σ ( k ) (cid:1) = 0 for k ≥ m ( r − − (2 r − . When r = 3 the statement was given by Severi. The general case, as well as the proof thatfollows, is due to Park and Woo [24]. Proof of Proposition 2.8.
We may suppose that r ≥
3, in which case the hypotheses implythat S is irreducible. Write Σ = (cid:8) P , . . . , P t (cid:9) , and denote by Λ ⊆ |O P r ( m − | the linear series spanned by the partial derivatives of adefining equation of S ; observe that every divisor in Λ passes through the points of Σ. Foreach P i ∈ Σ, there exists a divisor Γ i ∈ Λ with mult P i (Γ i ) ≥ Then for 0 < ε (cid:28) (cid:96) (cid:29)
0, set D = (1 − ε ) S + ε · t (cid:88) i =1 Γ i + (cid:0) r − − ε ( t − (cid:1) (cid:96) · (cid:96) (cid:88) j =1 A j , where A , . . . , A (cid:96) ∈ Λ are general divisors. As S is irreducible, none of the Γ i or A j occuras components of S , and therefore J ( D ) is cosupported on a finite set provided that ε (cid:28) t (cid:29)
0. One hasmult P i ( D ) ≥ (2 − ε ) + ε ( t + 1) + (cid:0) ( r − − ε ( t − (cid:1) ≥ r, which guarantees that J ( D ) ⊆ I Σ . Moreover:deg( D ) = m (1 − ε ) + εt ( m −
1) + (cid:0) ( r − − ε ( t − (cid:1) ( m − < m ( r − − ( r − , and so the required vanishing follows from Theorem 2.4. (cid:3) Singularities of Theta Divisors.
We next discuss a theorem of Koll´ar concerning thesingularities of theta divisors.Let ( A, Θ) be a principally polarized abelian variety (PPAV) of dimension g . Recall thatby definition this means that A = C g / Λ is a g -dimensional complex torus, and Θ ⊆ A is anample divisor with the property that h (cid:0) A, O A (Θ) (cid:1) = 1 . The motivating example historically is the polarized Jacobian (
J C, Θ C ) of a smooth projec-tive curve of genus g .In their classical work [1], Andreotti and Meyer showed that Jacobians are genericallycharacterized among all PPAV’s by the condition that dim Sing(Θ) ≥ g − In view of this,it is interesting to ask what singularities can occur on theta divisors. Koll´ar used vanishingfor Q -divisors to prove a very clean statement along these lines.Koll´ar’s result is the following: Theorem 2.9.
The pair ( A, Θ) is log-canonical. In particular, mult x (Θ) ≤ g for every x ∈ A .Proof. Suppose to the contrary that J ((1 − ε )Θ) (cid:54) = O A for some ε >
0. We will derive acontradiction from Nadel vanishing. To this end, let Z ⊆ A denote the subscheme defined This uses that m ≥
3: see [24, Lemma 3.2]. The precise statement is that the Jacobians form an irreducible component of the locus of all ( A, Θ)defined by the stated condition.
SHORT COURSE ON MULTIPLIER IDEALS 15 by J ((1 − ε )Θ). Then Z ⊆ Θ: this is clear set-theoretically, but in fact it holds on the levelof schemes thanks to the inclusion O A ( − Θ) = J (Θ) ⊆ J ((1 − ε )Θ) . Now consider the short exact sequence0 −→ O A (Θ) ⊗ J ((1 − ε )Θ) −→ O A (Θ) −→ O Z (Θ) −→ H of the term on the left vanishes thanks to Theorem 2.4 and the fact that K A = 0.Therefore the map H (cid:0) A, O A (Θ) (cid:1) −→ H (cid:0) Z, O Z (Θ) (cid:1) is surjective. On the other hand, the unique section of O A (Θ) vanishes on Z , and so weconclude that(*) H (cid:0) Z, O Z (Θ) (cid:1) = 0 . To complete the proof, it remains only to show that (*) cannot hold. To this end, let a ∈ A be a general point. Then Θ + a meets Z properly, and hence H (cid:0) Z, O Z (Θ + a ) (cid:1) (cid:54) = 0. Letting a →
0, if follows by semicontinuity that H (cid:0) Z, O Z (Θ) (cid:1) (cid:54) = 0 , as required. (cid:3) Remark 2.10.
In the situation of the theorem, the fact that ( A, Θ) is log-canonical impliesmore generally that the locusΣ k (Θ) = def (cid:8) x ∈ A (cid:12)(cid:12) mult x Θ ≥ k (cid:9) of k -fold points of Θ has codimension ≥ k in A . Equality is achieved when( A, Θ) = ( A , Θ ) × . . . × ( A k , Θ k )is the product of k smaller PPAV’s. It was established in [6] that this is the only situationin which codim A Σ k (Θ) = k . It was also shown in that paper that if Θ is irreducible, then Θis normal with rational singularities. (cid:3) Uniform Global Generation.
As a final application, we prove a useful result to the effectthat sheaves of the form O ( L ) ⊗ J ( D ), where D is a Q -divisor numerically equivalent to L , become globally generated after twisting by a fixed divisor. This was first observed byEsnault and Viewheg, and later rediscovered independently by Siu. The statement plays animportant role in the extension theorems of Siu discussed in Lecture 5.The theorem for which we are aiming is the following: Theorem 2.11.
Let X be a smooth projective variety of dimension d . There exists a divisor B on X with the following property: • For any divisor L on X ; and • For any effective Q -divisor D ≡ num L ,the sheaf O X ( L + B ) ⊗ J ( D ) is globally generated. Note that the hypothesis implies that L is Q -effective, i.e. that H (cid:0) X, O X ( mL ) (cid:1) (cid:54) = 0 forsome m (cid:29)
0. The crucial point is that B is independent of the choice of L and D . Corollary 2.12.
There is a fixed divisor B on any smooth variety X with the property that H (cid:0) X, O X ( L + B ) (cid:1) (cid:54) = 0 for any big ( or even Q -effective ) divisor L on X . (cid:3) The Theorem is actually an immediate consequence of Nadel vanishing and the elemen-tary lemma of Castelnuovo–Mumford:
Lemma 2.13 (Castelnouvo–Mumford) . Let F be a coherent sheaf on a projective variety X , and let H be a basepoint-free ample divisor on X . Assume that H i (cid:0) X, F ⊗ O X ( − iH ) (cid:1) = 0 for i > . Then F is globally generated. We refer to [21, § Proof of Theorem 2.11.
As above, let d = dim X . It sufficies to take B = K X + ( d + 1) H for a very ample divisor H , in which case Theorem 2.4 gives the vanishings required for theCastelnuovo–Mumford lemma. (cid:3) Local Properties of Multiplier Ideals
In this Lecture we will discuss some local properties of multiplier ideals. First we takeup the restriction theorem: here we emphasize the use of adjoint ideals, whose importancehas lately come into focus. The remaining sections deal with the subadditivity and Skodatheorems. As an application of the latter, we give a down-to-earth discussion of the recentresults of [22] concerning syzygetic properties of multiplier ideals.
Adjoint Ideals and the Restriction Theorem.
Let X be a smooth complex variety,let D be an effective Q -divisor on X , and let S ⊆ X be a smooth irreducible divisor, notcontained in any component of D . Thus the restriction D S of D to S is a well-defined Q -divisor on S .There are now two multiplier-type ideals one can form on S . First, one can take themultiplier ideal J ( X, D ) of D on X , and then restrict this ideal to S . On the other hand,one can form the multiplier ideal J ( S, D S ) on S of the restricted divisor D S . In general,these two ideals are different: Example 3.1.
Let X = C , let S be the x -axis, and let A = { y − x = 0 } be a parabolatangent to S . If D = A , then J ( X, D ) = O X , J ( S, D S ) = O S ( − P ) , where P ∈ S denotes the origin. (cid:3) SHORT COURSE ON MULTIPLIER IDEALS 17
However a very basic fact is that there is a containment between these two ideal sheaveson S . Theorem 3.2 (Restriction Theorem) . One has an inclusion J ( S, D S ) ⊆ J ( X, D ) · O S . This result is perhaps the most important local property of multiplier ideals. In the ana-lytic perspective, it comes from the Osahawa–Takegoshi extension theorem: an element inthe ideal on the left is a function on S satisfying an integrability condition, and Osahawa–Takegoshi guarantees that it is the restriction of a function satisfying the analogous integra-bility condition on X .We will prove Theorem 3.2 by constructing and studying the adjoint ideal Adj S ( X, D )of D along S . This is an ideal sheaf on X that governs the multiplier ideal J ( S, D S ) of therestriction D S of D to S . Theorem 3.3.
With hypotheses as above, there exists an ideal sheaf
Adj S ( X, D ) ⊆ O X sitting in an exact sequence: (3.1) 0 −→ J ( X, D ) ⊗ O X ( − S ) · S −→ Adj S ( X, D ) −→ J ( S, D S ) −→ . Moreover, for any < ε ≤ S ( X, D ) ⊆ J ( X, D + (1 − ε ) S ) . The sequence (3.1) shows thatAdj S ( X, D ) · O S = J ( S, D S ) . Therefore (3.2) not only yields the Restriction Theorem, it implies that in fact J ( S, D S ) ⊆ J ( X, D + (1 − ε ) S ) · O S for any 0 < ε ≤ Corollary 3.4.
Let Y ⊆ X be a smooth subvariety not contained in the support of D , sothat the restriction D Y of D to Y is defined. Then J ( Y, D Y ) ⊆ J ( X, D ) · O Y . (This follows inductively from the restriction theorem since Y is locally a complete intersec-tion in X .) (cid:3) Corollary 3.5.
In the situation of the Theorem, assume that J ( S, D S ) is trivial at a point x ∈ S . Then J ( X, D + (1 − ε ) S ) ( and hence also J ( X, D )) are trivial at x . (cid:3) Exercise 3.6. If D is an effective Q -divisor on X such that mult x ( D ) <
1, then J ( X, D )is trivial at x . (Using the previous corollary, take hyperplane sections to reduce to the casedim X = 1, where the result is clear.) (cid:3) Proof of Theorem 3.3.
Let µ : X (cid:48) −→ X be a log resolution of ( X, D + S ), and denote by S (cid:48) ⊆ X (cid:48) the proper transform of S , so that in particular µ S : S (cid:48) −→ S is a log resolution of( S, D S ). Write µ ∗ S = S (cid:48) + R, and put B = K X (cid:48) /X − [ µ ∗ D ] − R . We define:Adj S ( X, D ) = µ ∗ O X (cid:48) (cid:0) B ) . To establish the exact sequence (3.1), note first that K S ≡ lin (cid:0) K X + S (cid:1) | S , K S (cid:48) ≡ lin (cid:0) K X (cid:48) + S (cid:48) (cid:1) | S (cid:48) , and hence K S (cid:48) /S = (cid:0) K X (cid:48) /X − R (cid:1) | S (cid:48) . (One can check that this holds on the level of divisors, and not only for linear equivalenceclasses.) On X (cid:48) , where the relevant divisors have SNC support, rounding commutes withrestriction. Therefore J ( S, D S ) = µ S, ∗ O S (cid:48) ( K S (cid:48) /S − [ µ ∗ S D S ])= µ ∗ O S (cid:48) ( B S (cid:48) ) . Observing that B − S (cid:48) = K X (cid:48) /X − [ µ ∗ D ] − µ ∗ S, the adjoint exact sequence (3.1) follows by pushing forward0 −→ O X (cid:48) ( B − S (cid:48) ) −→ O X (cid:48) ( B ) −→ O S (cid:48) ( B S (cid:48) ) −→ B = K X (cid:48) /X − [ µ ∗ D ] − R (cid:52) K X (cid:48) /X − (cid:2) µ ∗ D + (1 − ε ) R (cid:3) = K X (cid:48) /X − (cid:2) µ ∗ D + (1 − ε ) R + (1 − ε ) S (cid:48) (cid:3) , which yields (3.2). (cid:3) Remark 3.7.
We leave it to the reader to show that Adj S ( X, D ) is independent of the choiceof log resolution. (cid:3) If a ⊆ O X is an ideal that does not vanish identically on S , then for c > S ( X, a c ) ⊆ O X sitting in exact sequence0 −→ J ( X, a c ) ⊗ O X ( − S ) −→ Adj S ( X, a c ) −→ J ( S, ( a S ) c ) −→ , where a S = def a · O S is the restriction of a to S . In particular, the analogue of the restrictiontheorem holds for the multiplier ideals associated to a .Finally, Theorem 3.3 works perfectly well if S is allowed to be singular, as in Remark1.10. Since in any event S is Gorenstein, when in addition it is normal there is no questionabout the meaning of the multiplier ideals on S appearing in the Theorem. In this case the SHORT COURSE ON MULTIPLIER IDEALS 19 statement and proof of 3.3 remain valid without change. For arbitrary S one can twist by O X ( K X + S ) and rewrite (3.1) as(3.3) 0 −→ O X ( K X ) ⊗ J ( X, D ) −→ O X ( K X + S ) ⊗ Adj S ( X, D ) −→ µ ∗ (cid:0) O S (cid:48) ( K S (cid:48) ) ⊗ J ( S (cid:48) , D (cid:48) S ) (cid:1) −→ , where D S (cid:48) = µ ∗ D S denotes the pullback of D S to S (cid:48) . When D = 0 this is the adjoint exactsequence appearing in [6] and [21, Section 9.3.E].We conclude with some exercises for the reader. Exercise 3.8 (Irreducible plane curves) . Let C ⊆ P be an irreducible (reduced) planecurve of degree m , and as in Proposition 2.5 put Σ = Sing( C ). Then the points of Σimpose independent conditions on curves of degree ≥ m −
3. (Let f : C (cid:48) −→ C be thedesingularization of C , and use the adjoint sequence0 −→ O P ( − −→ O P ( m − ⊗ Adj C −→ f ∗ O C (cid:48) ( K C (cid:48) ) −→ D = 0 to show that H (cid:0) P , O P ( k ) ⊗ Adj C (cid:1) = 0when k ≥ m − (cid:3) Example 3.9 (Condition for an embedded point) . It is sometimes interesting to know thata multiplier ideal has an embedded prime ideal. As usual, let X be a smooth variety ofdimension d , and fix a point x ∈ X with maximal ideal m . Consider an effective Q -divisor D with integral multiplicity s = mult x ( D ) ≥ d , and denote by D the effective Q -divisor ofdegree s on P ( T x X ) = P d − arising in the natural way as the “projectivized tangent cone”of D at x . The following proposition asserts that if there is a hypersurface of degree s − d on P d − vanishing along the multiplier ideal J ( P d − , D ), then J ( X, D ) has an embeddedpoint at x : Proposition. If H (cid:16) P d − , J ( P d − , D ) ⊗ O P d − ( s − d ) (cid:17) (cid:54) = 0 , then m is an associated prime of J ( X, D ) . Observe that the statement is interesting only when J ( X, D ) is not itself cosupported at x .For the proof, let π : X (cid:48) −→ X be the blowing up of x , with exceptional divisor E = P d − , so that K X (cid:48) /X = ( d − E . Write D (cid:48) for the proper transform of D , so that D (cid:48) ≡ num π ∗ D − sE , and note that D E = D . Now consider the adjoint sequence for D (cid:48) :0 −→ J ( X (cid:48) , D (cid:48) ) ⊗ O X (cid:48) ( − E ) −→ Adj E ( X (cid:48) , D (cid:48) ) −→ J ( P d − , D ) −→ , and twist through by O X (cid:48) (cid:0) ( d − s ) E (cid:1) . The resulting term on the left pushes down withvanishing higher direct images to J ( X, D ) thanks to Lemma 1.9. One finds an exact sequencehaving the shape0 −→ J ( X, D ) −→ A −→ H (cid:16) E, J ( E, D ) ⊗ O E (( d − s ) E ) (cid:17) −→ , where A is an ideal on X , and the vector space on the right is viewed as a sky-scraper sheafsupported at x . But the hypothesis of the Proposition is exactly that this vector space isnon-zero, and the assertion follows. (cid:3) The Subadditivity Theorem.
The Restriction Theorem was applied in [5] to prove aresult asserting that the multiplier ideal of a product of two ideals must be at least as deepas the product of the corresponding multiplier ideals. This will be useful at a couple ofpoints in the sequel.We start by defining “mixed” multiplier ideals:
Definition 3.10.
Let a , b ⊂ O X be two ideal sheaves. Given c, e ≥
0, the multiplier ideal J ( a c · b e ) ⊆ O X is defined by taking a common log resolution µ : X (cid:48) −→ X of a and b , with a · O X (cid:48) = O X (cid:48) ( − A ) , b · O X (cid:48) = O X (cid:48) ( − B )for divisors A, B with SNC support, and setting J ( a c · b e ) = µ ∗ O X (cid:48) (cid:0) K X (cid:48) /X − [ cA + eB ] (cid:1) . (cid:3) The subadditivity theorem compares these mixed ideals to the multiplier ideals of thetwo factors.
Theorem 3.11 (Subadditivity Theorem) . One has an inclusion J ( a c · b e ) ⊆ J ( a c ) · J ( b e ) . Similarly, J ( X, D + D ) ⊆ J ( X, D ) · J ( X, D ) for any two effective Q -divisors D , D on X Sketch of Proof of Theorem 3.11.
Consider the product X × X with projections p , p : X × X −→ X. The first step is to show via the K¨unneth formula that(*) J ( X × X, ( p ∗ a ) c · ( p ∗ b ) e ) = p ∗ J ( X, a c ) · p ∗ J ( X, b e ) . The one simply restricts to the diagonal ∆ = X ⊆ X × X using Corollary 3.4. Specifically, J ( X, a c · b e ) = J (∆ , (( p ∗ a ) c · ( p ∗ b ) e ) | ∆ ) ⊆ J ( X × X, ( p ∗ a ) c · ( p ∗ b ) e ) | ∆ = J ( X, a c ) · J ( X, b e ) , the last equality coming from (*). (cid:3) Exercise 3.12.
Let f : Y −→ X be a morphism, and let D be a Q -divisor on X whosesupport does not contain the image of Y . Then J ( Y, f ∗ D ) ⊆ f − J ( X, D ) . (cid:3) SHORT COURSE ON MULTIPLIER IDEALS 21
Skoda’s Theorem.
We now discuss Skoda’s theorem, which computes the multiplier idealsassociated to powers of an ideal.Let X be a smooth variety of dimension d , and let a , b ⊆ O X be ideal sheaves. In itssimplest form, Skoda’s theorem is the following: Theorem 3.13 (Skoda’s Theorem) . Assume that m ≥ d . Then J ( a m · b c ) = a · J ( a m − · b c ) for any c ≥ . Note that it follows that J ( a m b c ) = a m +1 − d J ( a d − b c ).The algebraic proof of the theorem – as in [7] or [21, § Lemma 3.14.
For any (cid:96), k ≥ there is an inclusion a k · J ( a (cid:96) · b c ) ⊆ J ( a k + (cid:96) · b c ) . (cid:3) Theorem 3.13 will follow from the exactness of certain “Skoda complexes.” Specifically,assume that X is affine, fix any point x ∈ X , and choose d general elements f , . . . , f d ∈ a . Theorem 3.15.
Still supposing that m ≥ d = dim X , the f i determine a Koszul-type complex −→ Λ d O d ⊗ J ( a m − d b c ) −→ Λ d − O d ⊗ J ( a m +1 − d b c ) −→ . . .. . . −→ Λ O d ⊗ J ( a m − b c ) −→ O d ⊗ J ( a m − b c ) −→ J ( a m b c ) −→ that is exact in a neighborhood of x . The homomorphism O d ⊗ J ( a d − b c ) −→ J ( a d b c ) on the right is given by multiplication bythe vector ( f , . . . , f d ). The surjectivity of this map implies that J ( a m b c ) = ( f , . . . , f d ) · J ( a m − b c ) . But thanks to the Lemma one has( f , . . . , f d ) · J ( a m − b c ) ⊆ a · J ( a m − b c ) ⊆ J ( a m b c ) , so Skoda’s theorem follows. Proof of Theorem 3.15.
Let µ : X (cid:48) −→ X be a log resolution of a and b as in Definition3.10. We keep the notation of that definition, so that in particular a · O X (cid:48) = O X (cid:48) ( − A ). The d general elements f , . . . .f d ∈ a determine sections f (cid:48) i ∈ H (cid:0) X (cid:48) , O X (cid:48) ( − A ) (cid:1) , and an elementary dimension count shows that after possibly shrinking X one can supposethat these sections actually generate O X (cid:48) ( − A ) (cf. [21, 9.6.19]). The f (cid:48) i then determine anexact Koszul complex(3.4) 0 −→ Λ d O d ⊗ O X (cid:48) ( − ( m − d ) A ) −→ Λ d − O d ⊗ O X (cid:48) ( − ( m − d − A ) −→ . . .. . . −→ Λ O d ⊗ O X (cid:48) ( − ( m − A ) −→ O d ⊗ O X (cid:48) ( − ( m − A ) −→ O X (cid:48) ( − mA ) −→ X (cid:48) . Now twist (3.4) through by O X (cid:48) (cid:0) K X (cid:48) /X − [ cB ] (cid:1) . The higher directimages of all the terms of the resulting complex vanish thanks to Theorem 2.3. This impliesthe exactness of the complex on X obtained by taking direct images of the indicated twistof (3.4), which is exactly the assertion of the Theorem. (cid:3) Remark 3.16.
The functions f , . . . , f r ∈ a occuring in Theorem 3.15 do not necessarilygenerate a . Rather (after shrinking X ) they generate an ideal r ⊆ a with the property that r · O X (cid:48) = a · O X (cid:48) , which is equivalent to saying the r and a have the same integral closure. Such an ideal r iscalled a “reduction” of a . See [21, § (cid:3) Following [22], we use Theorem 3.15 to show that multiplier ideals satisfy some unex-pected syzygetic conditions. By way of background, it is natural to ask which ideals d ⊆ O X can be realized as a multiplier ideal d = J ( b c ) for some b and c ≥
0. It follows from thedefinition that multiplier ideals are integrally closed, meaning that membership in a multi-plier ideal is tested by order of vanishing along some divisors over X . However until recently,multiplier ideals were not known to satisfy any other local properties. In fact, Favre–Jonsson[13] and Lipman–Watanabe [23] showed that in dimension d = 2, any integrally closed idealis locally a multiplier ideal.The next theorem implies that the corresponding statement is far from true in dimensions d ≥
3. We work in the local ring ( O , m ) of X at a point x ∈ X . Theorem 3.17.
Let j = J ( b c ) x ⊆ O be the germ at x of some multiplier ideal, and choose minimal generators h , . . . , h r ∈ j of j . Let b , . . . , b r ∈ m be functions giving a minimal syzygy (cid:88) b i h i = 0 among the h i . Then there is at least one index i such that ord x ( b i ) ≤ d − . To say that the h i are minimal generators means by definition that they determine a basis ofthe O / m = C -vector space j / m · j , the hypothesis on the b i being similar. Note that there areno restrictions on the order of vanishing of generators of a multiplier ideal, since for instance m (cid:96) = J ( m (cid:96) + d − ) for any (cid:96) ≥
1. On the other hand, the Theorem extends to statements forthe higher syzygies of j , for which we refer to [22]. SHORT COURSE ON MULTIPLIER IDEALS 23
Example 3.18.
Assume that d = dim X ≥
3, choose two general functions h , h ∈ m p vanishing to order p ≥ d at x , and consider the complete intersection ideal d = ( h , h ) ⊆ O . Since d ≥
3, the zeroes of d will be a reduced algebraic set of codimension 2, and therefore d is integrally closed. On the other hand, the only minimal syzygy among the h i is the Koszulrelation ( h ) · h + ( − h ) · h = 0 . In particular, the conclusion of the Theorem does not hold, and so d is not a multiplier ideal.(When d = 2, the ideal d is not integrally closed.) (cid:3) Idea of Proof of Theorem 3.17.
The plan is to apply Theorem 3.15 with a = m . Specifically,assume for a contradiction that each of the b i vanishes to order ≥ d , and choose localcoordinates z , . . . , z d at x , so that m = ( z , . . . , z s ). We can write b = z b + . . . + z d b d , · · · , b r = z b r + . . . + z d b rd , for some functions b ij vanishing to order ≥ d − x . Now put G = b h + . . . + b r h r , · · · , G d = b d h + . . . + b rd h d . Then G j ∈ J ( m d − b c ) thanks to Lemma 3.14, and(*) z G + . . . + z d G d = 0by construction. The relation (*) means that ( G , . . . , G d ) is a cycle for the Skoda complex(**) O ( n ) ⊗ J ( m d − b c ) −→ O d ⊗ J ( m d − b c ) −→ J ( m d b c )and using the fact that the b i are minimal one can show via some Koszul cohomology argu-ments that it gives rise to a non-trivial cohomology class in (**). But this contradicts theexactness of (**). (cid:3) Example 3.19 (Skoda’s theorem and the effective Nullstellensatz) . One can combine Skoda’stheorem with jumping numbers to give statements in the direction of the effective Nullstel-lensatz. Specifically, let a ⊆ O X be an ideal. Then there is an integer s > √ a ) s ⊆ a , and it is interesting to ask for effective bounds for s : see for instance [19] and [7], or [21, § x ∈ X , and consider the jumpingnumbers ξ i = ξ i ( a ; x ) of a at x (Proposition/Definition 1.21). Let σ = σ ( a ; x ) be the leastindex such that ξ σ ≥ d . Then J ( a ξ σ ) x ⊆ J ( a d ) x ⊆ ( a ) x , thanks to Skoda’s theorem, so it follows from Execise 1.23 that( √ a ) σ ( a ; x ) ⊆ a in a neighborhood of x . An analogous global statement holds using non-localized jumpingnumbers. It would be interesting to know whether one can recover or improve the results of[19] or [7] by using global arguments to bound σ . (The arguments in [7] also revolve aroundSkoda’s theorem, but from a somewhat different perspective.) (cid:3) Asymptotic Constructions
In this lecture we will study asymptotic constructions that can be made with multiplierideals. It is important in many geometric problems to be able to analyze for example thelinear systems | mL | associated to arbitrarily large multiples of a given divisor. Unfortunatelyone cannot in general find one birational model of X on which these are all well-behaved.By contrast, it turns out that there is some finiteness built into multiplier ideals, and theconstructions discussed here are designed to exploit this. Aymptotic Multiplier Ideals.
In this section we construct the asymptotic multiplierideals associated to a big divisor. For the purposes of motivation, we start by defining theirnon-asymptotic parents, which we have not needed up to now.Let X be a smooth projective variety, and L a divisor on X such that the completelinear series | L | is non-trivial. Given c > J ( c · | L | ) ⊆ O X as follows. Take µ : X (cid:48) −→ X to be a log resolution of | L | : this means that µ is a projectivebirational morphism, with µ ∗ | L | = | M | + F, where | M | is a basepoint-free linear series, and F + Exc( µ ) has SNC support. (This is thesame thing as a log-resolution of the base-ideal b (cid:0) | L | (cid:1) ⊆ O X of | L | .) One then defines J ( c · | L | ) = µ ∗ O X (cid:48) (cid:0) K X (cid:48) /X − [ cF ] (cid:1) . One can think of these ideals as measuring the singularities of the general divisor A ∈ | L | .The multiplier ideals attached to a linear series enjoy a Nadel-type vanishing: Theorem 4.1.
Assume that B is a nef and big divisor on X . Then H i (cid:0) X, O X ( K X + L + B ) ⊗ J ( | L | ) (cid:1) = 0 for i > . A proof is sketched in the following Exercise.
Exercise 4.2.
Choose k > c general divisors A , . . . , A k ∈ | L | , and let D = 1 k · (cid:0) A + . . . + A k ) . Then J ( c · | L | ) = J ( cD ) . In particular, Theorem 4.1 is a consequence of Theorem 2.4. (cid:3)
Exercise 4.3.
Let b = b (cid:0) | L | (cid:1) ⊆ O X be the base-ideal of | L | . Then J ( c · | L | ) = J ( b c ) . (cid:3) SHORT COURSE ON MULTIPLIER IDEALS 25
Remark 4.4 (Incomplete linear series) . Starting with a non-trivial linear series | V | ⊆ | L | ,one constructs in the similar manner a multiplier ideal J ( c · | V | ). The analogue of Theorem4.1 holds for these, as do the natural extensions of Exercises 4.2 and 4.3. The reader mayconsult [21, § (cid:3) Exercise 4.5. (Adjoint ideals for linear series). With X as above, suppose that S ⊆ X is asmooth irreducible divisor not contained in the base locus of | L | . Then, as in the previousLecture, one constructs for c > S ( X, c · | L | ) ⊆ O X . This sits in an exact sequence0 −→ J ( X, c · | L | ) ⊗ O X ( − S ) −→ Adj S ( X, c · | L | ) −→ J ( S, c · | L | S ) −→ , where the term | L | S on the right involves the (possibly incomplete) linear series on S obtainedas the restriction of | L | from X to S . (One can use Exercise 4.2 to reduce to the case ofdivisors.) (cid:3) The ideals J ( c · | L | ) reflect the geometry of the linear series | L | . But a variant of thisconstruction gives rise to ideals that involve the asymptotic geometry of | mL | for all m (cid:29) Proposition/Definition 4.6.
Assume that L is big, and fix c > . Then for p (cid:29) themultiplier ideals J ( X, cp · | pL | ) all coincide. The resulting ideal, written J ( X, c · (cid:107) L (cid:107) ) , is the asymptotic multiplier ideal of | L | with coefficient c .Idea of Proof. It follows from the Noetherian property that as p varies over all positiveintegers, the family of ideals J ( cp · | pL | ) has a maximal element. But one checks that J ( cp · | pL | ) ⊆ J ( cpq · | pqL | )for all q ≥
1, and therefore the family in question has a unique maximal element J ( c · (cid:107) L (cid:107) ),which coincides with J ( cp · | pL | ) for all sufficiently large and divisible p . For the fact thatthese agree with J ( cp · | pL | ) for any sufficiently large p (depending on c ), see [21, Proposition11.1.18]. (cid:3) The asymptotic multiplier ideals associated to a big divisor L are the algebro-geometricanalogues of the multiplier ideals associated to metrics of minimal singularities in the analytictheory. It is conjectured – but not known – that the two sorts of multiplier ideals actuallycoincide provided that L is big. (See [5]).The following theorem summarizes the most important properties of asymptotic multi-plier ideals. Theorem 4.7 (Properties of asymptotic multiplier ideals) . Assume that L is a big divisoron the smooth projective variety X . (i). Every section of O X ( L ) vanishes along J ( (cid:107) L (cid:107) ) , i.e. the natural inclusion H (cid:0) X, O X ( L ) ⊗ J ( (cid:107) L (cid:107) ) (cid:1) −→ H (cid:0) X, O X ( L ) (cid:1) is an isomorphism. Equivalently, b (cid:0) | L | (cid:1) ⊆ J ( X, (cid:107) L (cid:107) ) . (ii). If B is nef, then H i (cid:0) X, O X ( K X + L + B ) ⊗ J ( (cid:107) L (cid:107) ) (cid:1) = 0 for i > . (iii). J ( (cid:107) mL (cid:107) ) = J ( m · (cid:107) L (cid:107) ) for every positive integer m . (iv). ( Subadditivity. ) J ( (cid:107) mL (cid:107) ) ⊆ J ( (cid:107) L (cid:107) ) m for every m ≥ . (v). If f : Y −→ X is an ´etale covering, then J ( Y, (cid:107) f ∗ L (cid:107) ) = f ∗ J ( X, (cid:107) L (cid:107) ) . Concerning the vanishing in (ii), note that it is not required that B be big. In particular,one can take B = 0. Exercise 4.8.
Give counterexamples to the analogues of properties (ii) – (v) if one replacesthe asymptotic ideals by the multiplier ideals J ( | L | ) attached to a single linear series. (Forexample, suppose that L is ample, but that | L | is not free. Then J ( m · | L | ) (cid:54) = O X for m (cid:29)
0, whereas J ( | mL | ) = O X for large m .) (cid:3) Exercise 4.9.
In the situation of the Theorem, one has J ( c · (cid:107) L (cid:107) ) ⊆ J ( d · (cid:107) L (cid:107) )whenever c ≥ d . (cid:3) Indications of Proof of Theorem 4.7.
We prove (ii) and (iii) in order to give the flavor. For(ii), choose p (cid:29) J ( (cid:107) L (cid:107) ) = J ( p | L | ), and let µ : X (cid:48) −→ X be a log-resolution of | pL | , with µ ∗ | pL | = | M p | + F p , where | M p | is free. We can suppose that M p is big (since L is). Arguing as in the proof ofTheorem 2.4, one has H i (cid:0) X, O X ( K X + L + B ) ⊗ J ( (cid:107) L (cid:107) ) (cid:1) = H i (cid:0) X (cid:48) , O X (cid:48) ( K X (cid:48) + µ ∗ ( L + B ) − [ p F p ]) (cid:1) . But µ ∗ ( L + B ) − p F p ≡ num µ ∗ ( B ) + p M p is nef and big, so the required vanishing follows from the theorem of Kawamata and Viehweg.For (iii), the argument is purely formal. Specifically, for p (cid:29) J ( (cid:107) mL (cid:107) ) = J ( p · | pmL | )= J ( mmp · | pmL | )= J ( m · (cid:107) L (cid:107) ) , SHORT COURSE ON MULTIPLIER IDEALS 27 where in the last equality we are using mp in place of p for the large index computing theasymptotic multiplier ideal in question. For the remaining statements we refer to [21, 11.1,11.2]. (cid:3) Exercise 4.10 (Uniform global generation) . Theorem 2.11 extends to the asymptotic set-ting. Specifically, there exists a divisor B on X with the property that O X ( L + B ) ⊗ J ( (cid:107) L (cid:107) )is globally generated for any big divisor L on X . (cid:3) Exercise 4.11 (Characterization of nef and big divisors) . Let L be a big divisor on X . Then L is nef if and only if(*) J ( (cid:107) mL (cid:107) ) = O X for all m > . (Suppose (*) holds. Then it follows from the previous exercise that O X ( mL + B ) is globallygenerated for all m > B . This implies that (cid:0) ( mL + B ) · C (cid:1) ≥ C and every m >
0, and hence that ( L · C ) ≥ (cid:3) Variants.
We next discuss some variants of the construction studied in the previous section.To begin with, one can deal with possibly incomplete linear series. For this, recall that a graded linear series W • = { W m } associated to a big divisor L consists of subspaces W m ⊆ H (cid:0) X, O X ( mL ) (cid:1) , with W m (cid:54) = 0 for m (cid:29)
0, satisfying the condition that R ( W • ) = def ⊕ W m be a graded subalgebra of the section ring R ( L ) = ⊕ H (cid:0) X, O X ( mL ) (cid:1) of L . This lastrequirement is equivalent to asking that W (cid:96) · W m ⊆ W (cid:96) + m , where the left hand side denotes the image of W (cid:96) ⊗ W m under the natural map H (cid:0) X, O X ( (cid:96)L ) (cid:1) ⊗ H (cid:0) X, O X ( mL ) (cid:1) −→ H (cid:0) X, O X (( (cid:96) + m ) L ) (cid:1) . Then just as above one gets an asymptotic multiplier ideal by taking J ( c · (cid:107) W • (cid:107) ) = J ( cp · | W p | )for p (cid:29)
0. Analogues of Properties (i) – (iv) from Theorem 4.7 remain valid in this setting;we leave precise statements and proofs to the reader.
Example 4.12 (Restricted linear series) . An important example of this construction occurswhen Y ⊆ X is a smooth subvariety not contained in the stable base locus B ( L ) of the bigline bundle L on X . In this case, one gets a graded linear series on Y by taking W m = Im (cid:16) H (cid:0) X, O X ( mL ) (cid:1) −→ H (cid:0) Y, O Y ( mL Y ) (cid:1)(cid:17) . We denote the corresponding multiplier ideals by J ( Y, c · (cid:107) L (cid:107) Y ). Note that there is aninclusion J ( Y, c · (cid:107) L (cid:107) Y ) ⊆ J ( Y, c · (cid:107) L Y (cid:107) ) , but these can be quite different. For example if the restriction L Y is ample on Y , then theideal on the right is trivial. On the other hand, if for all m (cid:29) | mL | on X have base-loci that meet Y , then the ideals on the left could be quite deep. (cid:3) One can also extend the construction of adjoint ideals to the asymptotic setting. Assumethat S ⊆ X is a smooth irreducible divisor which is not contained in the stable base locus B ( L ) of a big divisor L . Then one defines an asymptotic adjoint ideal Adj S ( X, (cid:107) L (cid:107) ) thatfits into an exact sequence:(4.1) 0 −→ J ( X, (cid:107) L (cid:107) ) ⊗ O X ( − S ) −→ Adj S ( X, (cid:107) L (cid:107) ) −→ J ( S, (cid:107) L (cid:107) S ) −→ . The ideal on the right is the asymptotic multiplier ideal associated to the restricted linearseries of L from X to S , as in the previous example. This sequence will play a central rolein our discussion of extension theorems in the next Lecture.Finally, we discuss the asymptotic analogues of the ideals J ( a c ). A graded family ofideals a • = { a m } on a variety X consists of ideal sheaves a m ⊆ O X satisfying the propertythat a (cid:96) · a m ⊆ a (cid:96) + m . We will also suppose that a = O X and that a m (cid:54) = (0) for m (cid:29)
0. Then one defines: J ( a c • ) = J ( c · a • ) = def J (( a p ) c/p )for p (cid:29) Example 4.13.
The prototypical example of a graded system of ideals is the family ofbase-ideals b m = b (cid:0) | mL | (cid:1) associated to multiples of a big divisor L when X is projective. In this case J ( b c • ) = J ( c · (cid:107) L (cid:107) ) . (cid:3) Example 4.14 (Symbolic powers) . A second important example involves the symbolic pow-ers of a radical ideal q = I Z defining a (reduced) subscheme Z ⊆ X . Assuming as usual that X is smooth, define q ( m ) = (cid:8) f ∈ O X (cid:12)(cid:12) ord x ( f ) ≥ m for general x ∈ Z (cid:9) . (If X is reducible, we ask that the condition hold at a general point of each irreduciblecomponent of X .) Observe that by construction, membership in q ( m ) is tested at a generalpoint of Z , i.e. this is a primary ideal. The inclusion q ( (cid:96) ) · q ( m ) ⊆ q ( (cid:96) + m ) being evident, these form a graded family denoted q ( • ) . (cid:3) Exercise 4.15.
Let a • be a graded family of ideals. Then for every m ≥ a m ⊆ J ( a m • );(ii). J ( a m • ) ⊆ J ( a • ) m . (cid:3) SHORT COURSE ON MULTIPLIER IDEALS 29 ´Etale Multiplicativity of Plurigenera.
As a first illustration of this machinery, we provea theorem of Koll´ar concerning the behavior of plurigenera under ´etale covers.Given a smooth projective variety X , recall that the m th plurigenus P m ( X ) of X is thedimension of the space of m -canonical forms on X : P m ( X ) = def h (cid:0) X, O X ( mK X ) (cid:1) . These plurigeneral are perhaps the most basic birational invariants of a variety.Koll´ar’s theorem is that these behave well under ´etale coverings:
Theorem 4.16.
Assume that X is of general type, and let f : Y −→ X be an unramifiedcovering of degree d . Then for m ≥ , P m ( Y ) = d · P m ( X ) . Koll´ar was led to this statement by the observation that it would (and now does) follow fromthe minimal model program.
Exercise 4.17.
Show that the analogous statement can fail when m = 1. (Consider thecase dim X = 1.) (cid:3) Proof of Theorem 4.16.
We use the various properties of asymptotic multiplier ideals givenin Theorem 4.7. To begin with, 4.7 (i) yields: H (cid:0) X, O X ( mK X ) (cid:1) = H (cid:0) X, O X ( mK X ) ⊗ J ( (cid:107) mK X (cid:107) ) (cid:1) = H (cid:0) X, O X ( mK X ) ⊗ J ( (cid:107) ( m − K X (cid:107) ) (cid:1) , the second equality coming from the includion J ( (cid:107) ( m − K X (cid:107) ) ⊆ J ( (cid:107) mK X (cid:107) ). But since X is of general type, when m ≥ H i (cid:0) X, O X ( mK X ) ⊗ J ( (cid:107) ( m − K X (cid:107) ) (cid:1) = 0when i >
0. Therefore P m ( X ) = χ (cid:16) X, O X ( mK X ) ⊗ J ( (cid:107) ( m − K X (cid:107) ) (cid:17) , and similarly P m ( Y ) = χ (cid:16) Y, O Y ( mK Y ) ⊗ J ( (cid:107) ( m − K Y (cid:107) ) (cid:17) . On the other hand, f ∗ K X ≡ lin K Y f ∗ J ( X, (cid:107) ( m − K X (cid:107) ) = J ( Y, (cid:107) ( m − K Y (cid:107) )thanks to 4.7 (v). The Theorem then follows from the fact that Euler characteristics aremultiplicative under ´etale covers. (cid:3) Remark 4.18.
It was suggested in [21, Example 11.2.26] that a similar argument handlesadjoint bundles of the type O X ( K X + mL ). However this – as well as the reference giventhere – is erroneous. (cid:3) Exercise 4.19.
Assume that X is of general type. Then for m ≥ i > H (cid:0) X, O X ( mK X ) (cid:1) ⊗ H i (cid:0) X, O X ( K X ) (cid:1) −→ H i (cid:0) X, O X (( m + 1) K X ) (cid:1) defined by cup product are zero. (Argue as in the proof of Koll´ar’s theorem that the mapfactors through H i (cid:0) X, O X (( m + 1) K X ⊗ J ( (cid:107) mK X (cid:107) ) (cid:1) .) (cid:3) A Comparison Theorem for Symbolic Powers.
We start with a statement that followsformally from the subadditivity theorem in the form of Exercise 4.15. It shows that if agraded system of ideals has any non-trivial multiplier ideal, then that system must grow likethe power of an ideal.
Proposition 4.20.
Let a • be a graded family of ideals, and fix an index (cid:96) . Then for any m : a m(cid:96) ⊆ a (cid:96)m ⊆ J ( a (cid:96)m • ) ⊆ J ( a (cid:96) • ) m . In particular, if J ( a (cid:96) • ) ⊆ b for some ideal b , then a (cid:96)m ⊆ b m for every m ≥ . (cid:3) In spite of the rather formal nature of this result, it has a surprising application tosymbolic powers. Specifically, recall from Example 4.14 that if Z ⊆ X is a reduced subschemewith ideal q ⊆ O X , then the symbolic powers of q are defined to be: q ( m ) = (cid:8) f ∈ O X (cid:12)(cid:12) ord x ( f ) ≥ m for general x ∈ Z (cid:9) . Clearly q m ⊆ q ( m ) , and if Z is non-singular then equality holds. However in general theinclusion is strict: Example 4.21.
Let Z ⊆ C = X be the union of the three coordinate axes, so that q = ( xy , yz , xz ) ⊆ C [ x, y, z ] . Then xyz ∈ q (2) , but xyz (cid:54)∈ q .A result of Swanson [27] (holding in much greater algebraic generality) states that thereis an integer k = k ( Z ) with the property that q ( km ) ⊆ q m for every m . It is natural to suppose that k ( Z ) measures in some way the singularities of Z ,but in fact it was established in [8] that there is a uniform result. Theorem 4.22.
Assume that Z has pure codimension e in X . Then q ( em ) ⊆ q m for every m . In particular, for any reduced Z one has q ( dm ) ⊆ q m for every m , where as usual d = dim X . SHORT COURSE ON MULTIPLIER IDEALS 31
Proof.
Consider the graded system q ( • ) of symbolic powers. Thanks to the previous Propo-sition, it suffices to show that J ( q e ( • ) ) ⊆ q . As q is radical, membership in q is tested at a general point of (each component of) Z , sowe can assume that Z is non-singular. But in this case J ( q e ( • ) ) = J ( q e ) = q , as required. (cid:3) Example 4.23.
Let T ⊆ P be a finite set, viewed as a reduced scheme, and denote by I ⊆ C [ X, Y, Z ] the homogeneous ideal of T . Let F be a form with the property thatmult x ( F ) ≥ m for all x ∈ T. Then F ∈ I m . (Apply the Theorem to the affine cone over T .) (cid:3) Extension Theorems and Deformation Invariance of Plurigenera
The most important recent applications of multiplier ideals have been to prove extensiontheorems. Originating in Siu’s proof of the deformation invariance of plurigenera, extensiontheorems are what opened the door to the spectacular progress in the minimal model pro-gram. Here we will content ourselves with a very simple result of this type, essentially theone appearing in Siu’s original article [25] (see also [17], [18]). As we will see, the use ofadjoint ideals renders the proof very transparent. The statement for which we aim is the following.
Theorem 5.1.
Let X be a smooth projective variety, and S ⊆ X a smooth irreducibledivisor. Set L = K X + S + B, where B is any nef divisor, and assume that L is big and that S (cid:54)⊆ B + ( L ) . Then for every m ≥ the restriction map H (cid:0) X, O X ( mL ) (cid:1) −→ H (cid:0) S, O S ( mL S ) (cid:1) is surjective. Recall that by definition B + ( L ) denotes the stable base-locus of the divisor kL − A for A ample and k (cid:29)
0, this being independent of A provided that k is sufficiently large. Thehypothesis that S (cid:54)⊆ B + ( L ) guarantees in particular that L S is a big divisor on S .Before turning to the proof of Theorem 5.1, let us see how it implies Siu’s theorem onplurigenera in the general type case: The utility of adjoint ideals in connection with extension theorems became clear in the work [14], [28]of Hacon-McKernan and Takayama on boundedness of pluricanonical mappings. The theory is furtherdeveloped in [15] and [11].
Corollary 5.2 (Siu’s Theorem on Plurigenera) . Let π : Y −→ T be a smooth projective family of varieties of general type. Then for each m ≥ , the pluri-genera P m ( Y t ) = def h (cid:0) Y t , O Y t ( mK Y t ) (cid:1) are independent of t .Proof. We may assume without loss of generality that m ≥ T is a smooth affinecurve, and we write K t = K Y t . Fixing 0 ∈ T , one has P m ( Y ) ≥ P m ( Y t ) for generic t bysemicontinuity, so the issue is to prove the reverse inequality(*) h (cid:0) Y , O Y ( mK ) (cid:1) ≤ h (cid:0) Y t , O Y t ( mK t ) (cid:1) for t ∈ T in a neighborhood of 0. For this, consider the sheaf π ∗ O Y ( mK Y/T ) on T . This is atorsion-free (and hence locally free) sheaf, whose rank computes the generic value of the m th -plurigenus P m ( Y t ). Moreover, the fibre of π ∗ O Y ( mK Y/T ) at 0 consists of those pluri-canonicalforms on Y that extend (over a neighborhood of 0) to forms on Y itself. So to prove (*),it suffices to show that any η ∈ H (cid:0) Y , O Y ( mK ) (cid:1) extends (after possibly shrinking T ) tosome ˜ η ∈ H (cid:0) Y, O Y ( mK Y/T ) (cid:1) . We will deduce this from Theorem 5.1.Specifically, we start by completing π to a morphism π : Y −→ T of smooth projective varieties, where T ⊆ T and Y = π − ( T ). We view Y ⊆ Y as a smoothdivisor on Y . Let A be a very ample divisor on T , let B = π ∗ ( A − K T ), and set L = K Y + Y + B ≡ lin K Y /T + Y + π ∗ A. We can assume by taking A sufficiently positive that B is nef, and we assert that we canarrange in addition that L is big and that Y (cid:54)⊆ B + ( L ). For the first point, it is enough toshow that if D is an ample divisor on Y , and if A is sufficiently positive, then kL − D iseffective for some k (cid:29)
0. To this end, since Y t is of general type for general t , we can choose k sufficiently large so that kK Y /T − D is effective on a very general fibre of π . By taking A very positive, we can then guarantee that π ∗ O Y ( kL − D ) = π ∗ O Y ( kK Y /T − D ) ⊗ O T ( kA + k h (cid:0) Y , O Y ( kL − D ) (cid:1) (cid:54) = 0. For theassertion concerning B + , observe first that since | rY | = π ∗ | r · | is free for r (cid:29)
0, thedivisor Y cannot not lie in the base locus of | kL − D + qY | for arbitrarily large q . On theother hand, by making A more positive, we are free to replace L by L + pY , i.e. we maysuppose that Y (cid:54)⊆ B + ( L ), as claimed. SHORT COURSE ON MULTIPLIER IDEALS 33
We may therefore apply Theorem 5.1 with X = Y and S = Y to conclude that therestriction map(*) H (cid:0) Y , O Y ( mL ) (cid:1) −→ H (cid:0) Y , O Y ( mL ) (cid:1) is surjective for every m ≥
2. So provided that we take T sufficiently small so that O T ( A +0) | T is trivial, then O Y ( mL ) ∼ = O Y ( mK Y/T ) , and surjectivity of (*) implies the surjectivity of H (cid:0) Y, O Y ( mK Y/T ) (cid:1) −→ H (cid:0) Y , O Y ( mK ) (cid:1) , as required. (cid:3) The proof of Theorem 5.1 basically follows [25] (and [21]), except that as we havementioned the use of adjoint ideals substantially clarifies the presentation. Two pieces ofvocabulary will be useful. First, if A is a divisor on a projective variety V , we denote by b (cid:0) | A | (cid:1) ⊆ O V the base ideal of the complete linear series determined by A . Secondly, givenan ideal a ⊆ O V , we say that a section s ∈ H (cid:0) V, O V ( A ) (cid:1) vanishes along a if it lies in theimage of the natural map H (cid:0) V, O V ( A ) ⊗ a (cid:1) −→ H (cid:0) V, O V ( A ) (cid:1) .We also recall from the previous lecture a couple of facts about the asymptotic multiplierideals associated to L and its restriction to S : Remark 5.3.
Assume that L is big. Then:(i). For every m ≥ J ( X, (cid:107) ( m + 1) L (cid:107) ) ⊆ J ( X, (cid:107) mL (cid:107) ) , J ( S, (cid:107) ( m + 1) L (cid:107) S ) ⊆ J ( S, (cid:107) mL (cid:107) S )(ii). (Nadel vanishing). If P is nef, then H i (cid:16) X, O X ( K X + mL + P ) ⊗ J ( X, (cid:107) mL (cid:107) ) (cid:17) = 0 for i > m ≥ . (iii). (Subadditivity). For any m, q > J ( X, (cid:107) mqL (cid:107) ) ⊆ J ( X, (cid:107) mL (cid:107) ) q , J ( S, (cid:107) mqL (cid:107) S ) ⊆ J ( S, (cid:107) mL (cid:107) S ) q . (cid:3) Returning to the situation and notation of the Theorem, fix m ≥
2. Our analysis ofextension questions revolves around the adjoint exact sequence: (5.1) 0 → J ( X, (cid:107) ( m − L (cid:107) ) ⊗ O X ( − S ) → Adj S ( X, (cid:107) ( m − L (cid:107) ) → J ( S, (cid:107) ( m − L (cid:107) S ) → . We summarize what we will use in
Lemma 5.4. (i).
In order to prove the Theorem, it suffices to establish the inclusion b (cid:0) | mL S | (cid:1) ⊆ J ( S, (cid:107) ( m − L (cid:107) S ) . (ii). If a ⊆ J ( S, (cid:107) ( m − L (cid:107) S ) is an ideal such that O S ( mL S ) ⊗ a is globally generated,then a ⊆ J ( S, (cid:107) mL (cid:107) S ) . Proof.
For both statements, we twist through in (5.1) by O X ( mL ). Noting that mL − S ≡ lin ( m − L + K X + B, it follows from Nadel vanishing and the hypotheses that H (cid:16) X, O X ( mL − S ) ⊗ J ( X, (cid:107) ( m − L (cid:107) ) (cid:17) = 0provided that m ≥
2. (Note that this is where it is important that we use asyptotic multiplierideals: we do not assume that B is more than nef, and so there is no “excess positivity” inthe required vanishing.) Therefore the exact sequence (5.1) gives the surjectivity of the map H (cid:16) X, O X ( mL ) ⊗ Adj S ( (cid:107) ( m − L (cid:107) (cid:17) −→ H (cid:16) S, O S ( mL S ) ⊗ J ( (cid:107) ( m − L (cid:107) S ) (cid:17) . The group on the left being a subspace of H (cid:0) X, O X ( mL ) (cid:1) , this means that any section of O S ( mL S ) vanishing along J ( S, (cid:107) ( m − L (cid:107) S ) lifts to a section of O X ( mL ). So if we knowthe inclusion in (i), this implies that all sections of O S ( mL S ) lift to X .For (ii), remark that if M is a (Cartier) divisor on a projective variety V , and if a , b ⊆ O V are ideals such that O V ( M ) ⊗ a is globally generated, then a ⊆ b if and only if H (cid:0) V, O V ( M ) ⊗ a (cid:1) ⊆ H (cid:0) V, O V ( M ) ⊗ b ) (cid:1) , both sides being viewed as subspaces of H (cid:0) V, O V ( M ) (cid:1) . So in our situation it is enough toshow that H (cid:16) S, O S ( mL S ) ⊗ a ) (cid:17) ⊆ H (cid:16) S, O S ( mL S ) ⊗ J ( S, (cid:107) mL (cid:107) S ) (cid:17) . Suppose then that t ∈ H (cid:0) S, O S ( mL S ) ⊗ a (cid:1) is a section of O S ( mL S ) vanishing along a . Since a ⊆ J ( S, (cid:107) ( m − L (cid:107) S ), it follows asabove from (5.1) that t lifts to a section t ∈ H (cid:0) X, O X ( mL ) (cid:1) . By definition t vanishes along b (cid:0) | mL | (cid:1) ⊂ O X , and hence its restriction t vanishes along b (cid:0) | mL | (cid:1) · O S ⊆ J ( S, (cid:107) mL (cid:107) S ) , as required. (cid:3) In order to apply the statement (i) of the Lemma, the essential point is a comparisonbetween the multiplier ideals of the restricted divisor pL S and those of the restricted linearseries of pL from X to S : Proposition 5.5.
There exist a very ample divisor A on X , a positive integer k > , anda divisor D ∈ | k L − A | meeting S properly, such that (5.2) J ( S, (cid:107) pL S (cid:107) ) ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) ( p + k − L (cid:107) S ) for every p ≥ . SHORT COURSE ON MULTIPLIER IDEALS 35
Granting this for the moment, we complete the proof of the Theorem. In fact, fix m ,and apply equation (5.2) with p = qm for q (cid:29)
0. One finds: b (cid:0) | mL S | (cid:1) q ⊗ O S ( − D S ) ⊆ b (cid:0) | mqL S | (cid:1) ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) mqL S (cid:107) ) ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) ( mq + k − L (cid:107) S ) ⊆ J ( S, (cid:107) mqL (cid:107) S ) ⊆ J ( S, (cid:107) mL (cid:107) S ) q , the last inclusion coming from the subadditivity theorem. But we assert that having theinclusion(+) b (cid:0) | mL S | (cid:1) q ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) mL (cid:107) S ) q for all q (cid:29) b (cid:0) | mL S | (cid:1) ⊆ J ( S, (cid:107) mL (cid:107) S ) and hence also the inclusion in Lemma 5.4(i). In fact, it follows from the construction of multiplier ideals that there are finitely manydivisors E α lying over S , together with integers r α >
0, such that a germ f ∈ O S lies in J ( S, (cid:107) mL (cid:107) S ) if and only if ord E α ( f ) ≥ r α for every α . But (+) implies that if f ∈ b (cid:0) | mL S | (cid:1) ,then q · ord E α ( f ) + ord E α ( D S ) ≥ q · r α for each α , and letting q → ∞ one finds that ord E α ( f ) ≥ r α , as required. Thus Lemma 5.4applies, and this completes the proof of the Theorem granting Proposition 5.5.It remains to prove the Proposition. Here the essential point is statement (ii) of theLemma. Proof of Proposition 5.5.
By Nadel vanishing and Castelnuovo-Mumford regularity, we canfind a very ample divisor A so that for every q ≥ O S ( qL S + A S ) ⊗ J ( S, (cid:107) qL (cid:107) S )is globally generated (cf Exercise 4.10). Next, since S (cid:54)⊆ B + ( L ), for k (cid:29) D ∈ | k L − A | that does not contain S . We will show by induction on p that (5.2)holds with these choices of the data.For p = 0 the required inclusion holds by virtue of the fact that D + A ≡ lin k L , whichyields O S ( − D S ) ⊆ J ( S, (cid:107) k L (cid:107) S ) ⊆ J ( S, (cid:107) ( k − L (cid:107) S ) . Assuming that (5.2) is satisfied for a given value of p , we will show that it holds also for p + 1. To this end, observe first that O S (cid:0) ( p + k ) L S − D S (cid:1) ⊗ J ( S, (cid:107) pL S (cid:107) )is globally generated thanks to our choice of A . Applying Lemma 5.4 (ii) with m = p + k and a = J ( S, (cid:107) pL S (cid:107) ) ⊗ O S ( − D S ), it follows using the inductive hypothesis J ( S, (cid:107) pL S (cid:107) ) ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) ( p + k − L (cid:107) S )that in fact J ( S, (cid:107) pL S (cid:107) ) ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) ( p + k ) L (cid:107) S ) . Therefore also J ( S, (cid:107) ( p + 1) L S (cid:107) ) ⊗ O S ( − D S ) ⊆ J ( S, (cid:107) ( p + k ) L (cid:107) S ) , which completes the induction. (cid:3) References [1] Aldo Andreotti and Alan Mayer, On period relations for abelian integrals on algebraic curves, Ann.Scuola Norm. Sup. Pisa (1967), pp. 189 – 238.[2] Manuel Blickle and Robert Lazarsfeld, An informal introduction to multiplier ideals, in Trends inCommutative Algebra , MSRI Publ. , Cambridge Univ. Press, 2004, pp. 87 – 114.[3] Caucher Birkar, Paolo Cascini, Christopher Hacon and James McKernan, Existence of minimal modelsfor varieties of log general type, to appear.[4] Tommaso de Fernex and Christopher Hacon, Singularities on normal varieties, to appear.[5] Jean-Pierre Demailly, Lawrence Ein and Robert Lazarsfeld, A subadditivity property of mulitplier ideals,Michigan Math. J. (2000), pp. 137 – 156.[6] Lawrence Ein and Robert Lazarsfeld, Singularities of theta divisors and the birational geometry ofirregular varieties, J. Amer. Math. Soc. (1997), pp. 243 – 258.[7] Lawrence Ein and Robert Lazarsfeld, A geometric effective Nullstellensatz, Invent. Math. (1999),pp. 427–488.[8] Lawrence Ein, Robert Lazarsfeld and Karen Smith, Uniform bounds and symbolic powers on smoothvarieties, Invent. Math. (2001), pp. 241 – 252.[9] Lawrence Ein, Robert Lazarsfeld, Karen Smith and Dror Varolin, Jumping coefficients of multiplierideals, Duke Math. J. (2004), pp. 469 – 506.[10] Lawrence Ein and Mircea Mustat¸˘a, Invariants of singularities of pairs, Int. Cong. Math., Zurich 2006,Vol. II, pp. 583–602.[11] Lawrence Ein and Mihnea Popa, Extension of sections via adjoint ideals, to appear.[12] H´el`ene Esnault and Eckart Viehweg, Logarithmic DeRham complexes and vanishing theorems, Invent.Math. (1986), pp. 161 – 194.[13] Charles Favre and Mattias Jonsson, Valuations and multiplier ideals, J. Amer. Math. Soc. (2005),pp. 655–684.[14] Christopher Hacon and James McKernan, Boundedness of pluricanonical maps of varieties of generaltype, Invent. Math. (2006), pp. 1 – 25.[15] Christopher Hacon and James McKernan, Existence of minimal models for varieties of log general type,II, preprint.[16] Jason Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. (2001), pp. 2665–2671.[17] Yujiro Kawamata, Deformations of canonical singularities, J. Amer. Math. Soc. (1999), pp. 85 – 92.[18] Yujiro Kawamata, On the extension problem of pluricanonical forms, in Algebraic Geometry: Hirzebruch70 , Amer. Math. Soc., Providence, 1999, pp. 193 – 207.[19] J´anos Koll´ar, Sharp effective Nullstellensatz, J.Amer. Math. Soc. (1988), pp. 963 – 975.[20] J´anos Koll´ar and Shigefumi Mori, Birational Geometry of Algebraic Varieties , Cambridge Tracts inMathematics, vol. 134, Cambridge University Press, Cambridge, 1988.[21] Robert Lazarsfeld,
Positivity in Algebraic Geometry, I & II , Ergebnisse der Mathematik und ihrerGrenzgebiete, Vols. 48 & 49, Springer Verlag, Berlin, 2004.[22] Robert Lazarsfeld and Kyungyong Lee, Local syzygies of multiplier ideals, Invent. Math. (2007),pp. 409 – 418.
SHORT COURSE ON MULTIPLIER IDEALS 37 [23] Joseph Lipman and Kei-ichi Watanabe, Integrally closed ideals in two dimensional regular local ringsare multiplier ideals, Math. Res. Lett. (2003), pp. 423 – 434.[24] Jihun Park and Youngho Woo, A remark on hypersurfaces with isolated singularities, ManuscriptaMath. (2006), pp. 451 – 456.[25] Yum-Tong Siu, Invariance of plurigenera, Invent. Math. (1998), pp. 661 – 673.[26] Yum-Tong Siu, Multiplier ideal sheaves in complex and algebraic geometry, Sci. China Ser. A (2005),suppl., pp. 1 – 31.[27] Irena Swanson, Linear equivalence of ideal topologies, Math. Z. (2000), pp. 755 – 775.[28] Shigeharu Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. (2006), pp. 551 – 587.[29] A. N. Varchenko, Semicontinuity of the complex singularity exponent, Izv. Akad. Nauk SSSR Ser. Mat. (1981), pp. 540 – 591.[30] Oscar Zariski, On the irregularity of cyclic multiple planes, Ann. of Math. (1931), pp. 485 – 511. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
E-mail address ::