PLURICANONICAL SYSTEMS ON PROJECTIVE VARIETIES OF GENERAL TYPE

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1 PLURICANONICAL SYSTEMS ON PROJECTIVE VARIETIES OF GENERAL TYPE GIANLUCA PACIENZA Lecture notes on the work of Hacon-McKernan, Takayama, Tsuji, prepared for the Summer School "Geometry of complex projective varieties and the MMP". Grenoble, June 18th July 6th Preliminary version Contents 1. Introduction 1 2. Interesting consequences of the main result Proof of Corollaries 2.1 and Local constancy of pluricanonical maps for images of a given variety 4 3. The basic idea (and the basic difficulty) 7 4. Point separation for big pluricanonical systems 9 5. The missing piece: bounding the restricted volumes from below Proof of Theorem 4.4: the case codim X (V ) = Proof of Theorem 4.4: the case codim X (V ) Further work 15 Appendix A. 16 A.1. Multiplier ideals 16 A.2. Singularities of pairs and Non-klt loci 16 A.3. Complements to the proof of Theorem A.4. Weak positivity of direct images 19 References Introduction A basic invariant of any irreducible variety X is its Kodaira dimension κ(x). This (birational) invariant is defined to be the maximum of the dimension of the images of the rational pluricanonical maps ϕ m := ϕ mkx : X PH 0 (X, O X (mk X )) Date: July 23,

2 provided that a multiple of K X has non-zero sections. If not, we set κ(x) =. When κ(x) = dim(x), the variety X is said to be of general type 1. Thanks to the work of Iitaka, it is well-known that, if κ(x) > 0, for m large enough and such that h 0 (X, mk X ) 0, the images of the pluricanonical maps ϕ m yield a fibration into Kodaira dimension zero subvarieties. Namely, there exists a fiber space ϕ : X Iitaka(X) and a commutative diagram (1.1) X u ϕ m Im(ϕ m ) v m X ϕ Iitaka(X) where the horizontal maps are birational (hence dim(iitaka(x)) = κ(x)). Moreover, if we set K = u K X, and F is the very general fiber of ϕ, we have κ(f, K F ) = 0. This fibration is called the Iitaka fibration of X 2 (notice that as a consequence of the finite generation of the canonical ring proved in [BCHM] we have that, for large m, the images of the pluricanonical maps ϕ m are all isomorphic to Proj( m 0 H0 (X, mk X ))). The multiple that we have to take in order to get the Iitaka fibration a priori depends on X. It is natural to ask whether it depends only on certain invariants of X. For curves the answer is clear : thanks to the Riemann-Roch theorem we know that 3K X is very ample. For surfaces of Kodaira dimension 1 it follows from Kodaira s formula (see e.g. [FM]), while the birationality of 5K X for surfaces of general type is due to Bombieri [Bo] (and later recovered in a much simpler way by Reider in [Rei]). For 3-folds of Kodaira dimension 1, as a by product of their canonical bundle formula, Fujino and Mori have proved in [FM] the existence of an effectively computable number m 3 such that ϕ m3 yields the Iitaka fibration. Following a strategy indicated by Tsuji in [Ts1] and [Ts2], Hacon and McKernan [HM], and Takayama [T] have independently proved that for varieties of general type the integer m only depends on the dimension. Theorem 1.1 (Hacon-McKernan, Takayama, Tsuji). For any positive integer n, there exists an integer m n such that for any smooth complex projective variety X of general type of dimension n, the pluricanonical map ϕ m : X PH 0 (X, O X (mk X )) is birational onto its image, for all m m n. Notice that the integer m n is not effective. We will point out which part of the argument is ineffective (cf. Problem 4.3). If K X is also nef, then thanks to the important work of Angehrn and Siu [AS] (which actually represents the ancestor of Theorem 1.1) it is known that m n can be taken quadratic in n (cf. Exercise 4.8). In the case of 3-folds of general type, there is a large literature determining the integer m 3 that gives the birationality under additional hypotheses on X (cf. 5.3), and a very recent preprint by J.A. Chen and M. Chen [ChCh2] establishes the birationality of mk X, for m 77, for any 3-fold of general type. 1 Question/Exercise : Why was that equivalent to asking the birationality of ϕ m, for m 0? 2 If X is singular, the Kodaira dimension and the Iitaka fibration of X are, by definiton, those of a non-singular model of X. 2

3 The purpose of these lecture notes is to present the proof of this beautiful result, as well as of its consequences. The difference with respect to O. Debarre s very clear Bourbaki text [D] concerns the form, rather than the mathematical content. The presentation here seemed more appropriate for a graduate students short course. Also, we decided to focus directly on the big case, leaving the ample one as an exercise, to underline the importance of the restricted volume. 2. Interesting consequences of the main result Recall that a divisor D is big if and only if its volume 3 is positive. Nevertheless, there exist examples of integral big divisors with arbitrarily small volume (cf. [L1, Example 2.3.7]). A highly non-trivial consequence of Theorem 1.1 is that this cannot occur for canonical divisors. Corollary 2.1. For any positive integer n, any smooth complex projective variety X of dimension n and of general type we have : where m n is as in Theorem 1.1. vol(k X ) m n n A fundamental problem in algebraic geometry is to study parameter spaces for a given class of varieties. On the other hand, it is well known that the parameter spaces for varieties of general type are not bounded, 4 unless one fixes some invariants. To find such an example we do not have to look for exotic objects: take algebraic curves of genus 2, and their (unbounded, unless we bound g) moduli g 2 M g. A very interesting consequence of Theorem 1.1 is the following, in which the rôle of the genus in the previous example will be played by the (canonical) volume. Corollary 2.2. Let n be a positive integer and C a positive constant. Then the family of smooth n-dimensional complex projective varieties of general type having canonical volume vol(k X ) C, is birationally bounded. It was classically proved by Severi that an algebraic curve has only finitely many surjective morphisms onto curves of genus 2. (This is no longer true if we allow morphisms onto an elliptic curve E, as Aut(E) is infinite). Maehara noticed in [Mae] that from a uniformity statement for pluricanonical maps for varieties of general type it is possible to deduce the corresponding finiteness property in higher dimension, as conjectured by Iitaka. Corollary 2.3. Let X be a fixed smooth complex projective variety. Then there exist only finitely many dominant rational maps f : X Y (modulo birational equivalence) onto varieties of general type Y. 3 Recall that the volume of an integral divisor is the number vol X (D) := lim sup m + h 0 (X, md) m n. /n! One can show that vol X (ad) = a dim(x) vol X (D), and then extend the definition to Q-divisors (see [L1, 2.2.C] for a detailed account on the properties of this invariant). By the canonical volume of a variety X we mean the volume vol(k X ) of its canonical divisor. 4 Recall that a set T of algebraic varieties {X t : t T } is bounded (respectively birationally bounded) if there exists a morphism of algebraic varieties f : X B such that every X t is isomorphic (respectively birational) to a fiber of f. 3

4 In the rest of the section we will prove the corollaries above. Apart from their own interest, their proofs have two advantages : as for Corollaries 2.1 and 2.2, they allow us to get acquainted with volumes, while in the proof of Corollary 2.3 it appears a very simple form of a positivity-type result for the direct image of the relative dualizing sheaf. A generalization of such a result will play a key rôle in the proof of Theorem 1.1. Before getting started, we fix our : Notation and conventions. We work over the field of complex numbers. Unless otherwise specified, a divisor will be integral and Cartier. If D and D are Q-divisors on a smooth variety X we write D Q D, and say that D and D are Q-linearly equivalent, if an integral multiple of D D is linearly equivalent to zero. We write D D when they are numerically equivalent, that is when they have the same degree on every curve. The notation D D means that D D is effective. If D = a i D i, we denote by [D] the integral divisor [a i ]D i, where, as usual, [a i ] is the largest integer which is less than or equal to a i. We denote by {D} the difference D [D]. A log-resolution of a divisor D X is a proper birational morphism of smooth varieties µ : X X, such that the support of Exc(µ) + µ D has simple normal crossings. The existence of log-resolutions is insured by Hironaka s theorem. Given a surjective morphism f : X Y of smooth algebraic varieties we set K X/Y := K X f K Y Proof of Corollaries 2.1 and 2.2. Knowing what the volume of a nef divisor is, these two corollaries follow essentially from the obvious fact that the degree of a subvariety of a projective space is at least one. Proof of Corollary 2.1. Let m n be as in Theorem 1.1. Let µ : X X be the blow-up along the base locus of m n K X. Then we can write µ m n K X = G + F, where G is the base-point-free part, and F is the fixed part. In particular G is nef, so vol(g) = G n. In conclusion we have : (2.1) vol(k X ) = vol(µ m n K X ) m n n = 1 G n = m n n 1 vol(g) m n n 1 deg ϕ m n G (X ) n Proof of Corollary 2.2. By (2.1) we have in particular that all such varieties X are birational to subvarieties of degree deg C m n n contained in a fixed projective space (which, after projection, we may assume to be P 2n+1 ). These subvarieties are parametrized by a subset of points of a Chow variety, and we are done Local constancy of pluricanonical maps for images of a given variety. Given any projective variety X, we consider the set (2.2) F (X, T ) := {(f t, Y t ) : f t : X Y t is a family of dominant maps 1. m n n onto smooth projective varieties Y t of general type}. 4

5 Corollary 2.3 can be reduced to prove that the set F (X, T ) modulo birational equivalence, is finite. The precise set-up is the following. Let T 0 be a smooth affine scheme of finite type. Let q : Y T 0 be a smooth scheme projective over T 0, and of relative dimension n = dim(x). Consider p : X T 0 T 0, where p is the projection onto the second factor. Suppose there exists a dominant T 0 -rational map (2.3) f : X T 0 Y inducing dominant rational maps on each fiber f t : X Y t. We will assume that for a fixed positive integer m and all i 0 the sheaf R i q (O Y (mk Y/T 0)) is locally free, as well as the cokernel Q of the injective homomorphism given by the pull-back of forms : (2.4) q (O Y (mk Y/T 0)) H 0 (X, mk X ) O T 0 (the homomorphism exists since the relative dimension of Y equals n). So we have a short exact sequence (2.5) 0 q (O Y (mk Y/T 0)) H 0 (X, mk X ) O T 0 Q 0. Suppose moreover that (2.6) all the m-th pluricanonical maps ϕ mkyt are birational. Under the above conditions we have the following. Proposition 2.4. The maps f t are all birationally equivalent. Proof. Let r := h 0 (X, mk X ), and let s be the rank of q (O Y (mk Y/T 0). Consider the Grassmanniann G := Grass(s, r) of s-planes in C r = H 0 (X, mk X ) and the short exact sequence : (2.7) 0 S C r O G Q 0, where S and Q are, respectively, the universal sub-bundle and quotient bundle on G. The injection (2.4) induces a morphism (2.8) h : T 0 G t H 0 (Y t, mk Yt ) so that the exact sequence (2.5) is obtained by pulling back (2.7) to T 0 via h : (2.9) 0 h S C r O T 0 h Q 0. On the other hand, it is easy to check that we have the following commutative diagram (2.10) X T 0 Y f ϕ mkx PH 0 (X, mk X ) π ϕ mky /T 0 P(q (O Y (mk Y/T 0))) where the restriction of ϕ mky /T 0 to a fiber Y t is the rational map ϕ mkyt, and π is simply defined by the family of projections induced by the inclusion (2.4). Therefore, 5

6 since h S = q (O Y (mk Y/T 0)), the images of the pluricanonical maps ϕ mkyt are all the same, thanks to Lemma 2.5. Lemma 2.5. The morphism h defined in (2.8) is constant. Proof. We use the above notation. It is of course sufficient to prove the statement in the case T 0 has dimension 1. Let B be the normalization of a compactification of the affine curve T 0. Consider a desingularization of the Zariski closure of the graph of q : Y T 0 inside Y B. By abuse of notation we denote it by Y, as well as we still call f the dominant rational map from X B to Y induced by (2.3), and Q the cokernel of q (O Y (mk Y/B )) H 0 (X, mk X ) O B. Notice that since q (O Y (mk Y/B )) is a torsion-free sheaf on a smooth curve, then it is locally free. Let A be the open subset of B on which Q is locally free. As B is smooth we have a short exact sequence of locally free sheaves : inducing a morphism 0 Ker H 0 (X, mk X ) O B Q/Torsion(Q) 0, g : B G whose restriction to A coincides with (2.8). If g is not the constant morphism, then (2.11) deg O B ( det(ker)) = deg(g O G (1)) > 0, where O G (1) is the line bundle on G giving the Plücker embedding. On the other hand, by a theorem due to Fujita [Fu], (2.12) the sheaf q (O Y (mk Y/B )) is nef, and so is its determinant. But q (O Y (mk Y/B )) coincides with Ker on the open set A, and is a subsheaf of Ker. Therefore (2.13) deg O B (det(ker)) deg(detq (O Y (mk Y/B ))) 0. Equations (2.11) and (2.13) contradict, so g, and a fortiori h, is constant. The proof of Corollary 2.3 is now straightforward. Proof of Corollary 2.3. One can show, as in Maehara s paper [Mae, 2 and 3], the existence of a (constructible) parameter space for dominant maps from a fixed n-dimensional smooth projective variety X onto smooth Y s of general type. Notice that given a family of dominant maps f t : X Y t, by successively cutting X with hyperplane sections, we can always find a subvariety X X such that dim(x ) = dim(y t ), and X dominates the Y t s. Moreover condition (2.6) holds thanks to Theorem 1.1. Therefore by Proposition 2.4 (and arguing similarly to the way we do in the proof of Lemma 4.2) we are done. Exercise 2.6. What would be the consequence of a uniformity result for the Iitaka fibration of varieties of Kodaira dimension {1,..., n 1}? 6

7 We start with an exercise. 3. The basic idea (and the basic difficulty) Exercise 3.1. Let X be a variety of general type. Prove that a subvariety of X passing through a general point is also of general type. As mentioned in the introduction, Theorem 1.1 is known in low dimension. The natural idea, motivated by Exercise 3.1, is to prove the theorem by induction : given two general points x 1 and x 2 in X take a k-dimensional subvariety V joining them. The subvariety V is also of general type by Exercise 3.1 and then, by the inductive hypothesis, we have the existence of an integer m k and of a global section m k K V separating the two points. If we could lift such a section to X we would be done. The main problem is that "lifting" our section to a section lying in H 0 (mk X ) may be impossible, without imposing further conditions on V. Therefore we can reformulate the problem by saying that we are looking for a subvariety V containing x 1 and x 2 and for which the sections in Im ( r V : H 0 (X, mk X ) H 0 (V, m(k V det N 1 V/X )) H 0 (V, mk V ) are sufficient to separate the two points. The actual proof follows an apparently different path. Nevertheless we will end up lifting pluricanonical sections anyhow. The starting point is the following easy application of Nadel s vanishing theorem, which clearly suggests which class of subvarieties V we must consider. Lemma 3.2. Let X be a smooth projective variety. Let A (respectively E) be a Q-ample (resp. a Q-effective) divisor on X, and set B := A + E. Let x 1, x 2 be two points outside the support of E. Suppose there exists a positive rational multiple D 0 t 0 B of B such that: (i) x 1, x 2 Non-klt(X, D 0 ); 5 (ii) x 1 is an isolated point in Non-klt(X, D 0 ). Then, for all integer m > t 0 such that mb is integral, there exists a section s H 0 (X, K X + mb) such that s(x 1 ) 0 and s(x 2 ) = 0. Proof. Take any integer m > t 0 such that mb is integral. Notice that mb (D 0 + (m t 0 )E) = (m t 0 )A is a Q-ample divisor, hence by Nadel s vanishing theorem A.1 we have: Set H 1 (X, I (X, D 0 + (m t 0 )E) O X (K X + mb)) = 0. V 0 := Non-klt(X, D 0 + (m t 0 )E) and consider the short exact sequence of V 0 X : 0 I (X, D 0 + (m t 0 )E) O X O V Recall that Non-klt(X, D 0 ) is defined to be the (reduced) support of the sheaf O X /I (X, D 0 ), where I (X, D 0 ) is the multiplier ideal sheaf associated to D 0 (see A.1). Notice that, if D 0 is reduced, then Non-klt(X, D 0 ) is properly contained in Supp(D 0 ). 7

8 Tensoring it with O X (K X + mb), and taking cohomology, we thus get a surjection: (3.1) H 0 (X, K X + mb) H 0 (V 0, (K X + mb) V0 ). Notice that as the points x 1, x 2 lie outside the support of E, around them we have Non-klt(X, D 0 ) = Non-klt(X, D 0 + (m t 0 )E), that is, V 0 still contains x 1 and x 2, the former as an isolated point. By (3.1) we are done. There is an easy way (which is unfortunately the only known way) to construct a rational multiple of the big divisor K X satisfying the condition (i) above and depending only on vol(k X ). Indeed we have: Lemma 3.3. Let X be a variety of dimension n. Let x 1, x 2 X be two smooth points and B a big divisor on X. Then, for any m > n ( 2/ vol(b) ) 1/n, there exists a divisor B Q mb such that x 1, x 2 Non-klt(X, B ). Proof. We simply use the following fact (cf. [L2, Proposition 9.3.2]) (3.2) mult x (B) dim(x) I (X, B) x O X,x and count parameters. Indeed having multiplicity r at x 1 and x 2 imposes 2 ( ) n+r 1 n 2 rn conditions. On the other hand, for s 0, the dimension of H 0 (X, msb) grows as n! vol(b) (ms)n so we have n! vol(b) (ms)n n! > 2sn n n n! as soon as m is a rational number > n ( 2/ vol(b) ) 1/n. Hence there exists a divisor D msb having multiplicity sn at x 1 and x 2, and then B := 1 s D has multiplicity n at both points. By (3.2) we are done. Remark 3.4. The main difficulty in applying Lemma 3.3 is that the Non-klt locus of the divisor produced in the previous lemma may well have positive dimension at x 1. Of course, we want to use induction on the dimension to conclude. To understand what can go wrong suppose we are in a two steps situation, i.e. X is a variety of general type, x 1, x 2 X two general points in it, D ak X a divisor such that Non-klt(X, D) is a smooth subvariety V (necessarily of general type, by Exercise 3.1) passing through x 1 and x 2. and suppose given a divisor D V bk V such that Non-klt(V, D V ) satisfies both (i) and (ii). Again, we still do not know whether (the section defining) this divisor lies in Im(H 0 (X, bk X ) H 0 (V, b(k X ) V ) H 0 (V, bk V ). Keeping the previous discussion in mind, if V is a subvariety of X and D an integral divisor on X, we are lead to consider the restricted volume, which, following [ELMNP2], is defined as : where vol X V (D) := lim sup m + h 0 (X V, md) m d /d! h 0 (X V, md) := dim Im(H 0 (X, md) H 0 (V, md V )). 8

9 Again, it is a limit, and we have that vol X V (md) = m dim(v ) vol X V (D) (see [ELMNP2, Cor and Lemma 2.2] ). The following should be by now a straightforward, though useful, exercise. Exercise 3.5. Let x 1, x 2 be smooth points of a d-dimensional subvariety V of X. Let B be a divisor on X such that vol X V (B) > 0. Then for any m > d ( 2/ vol X V (B) ) 1/d there exists a divisor B Q mb such that mult xi (B V ) > d ( x 1, x 2 Non-klt(V, B V )). Remark 3.6. Notice the the smoothness assumption at x 1, x 2 is necessary to perform the local parameter count. Remark 3.7. The conclusion we would like to draw in order to cut down the dimension of V and eventually achieve condition (ii) of Lemma 3.2 would rather be that x 1 and x 2 belong to the Non-klt locus of the pair (X, B ). Note that this is false in general. Take for instance a cuspidal hyperplane section B of a smooth projective surface S: the hyperplane B is of course klt, while its restriction to S is non-klt at the cusp. The result we need is the object of Lemma A.4 but requires a deeper analysis of the restrictions of multiplier ideals (see [L2, Section 9.5.A]). Precisely, if X is smooth, D is an effective Q-divisor on X and H X a smooth hypersurface not contained in Supp(D), we will use that we have (3.3) I (H, D H ) I (X, D + (1 t)h) H 6 I (H, sd H ), for all 0 < s < 1 and 0 < t 1. The right-hand-side inclusion is often called inversion of the adjunction. Restricted volume is the right object to study for our problem. On the other hand our optimism should be moderated be the following : Exercise/Warning 3.8. (i) Find an example of a big divisor B on a smooth projective variety X, such that vol(b V ) = 0 (hence a fortiori vol X V (B) = 0), for some subvariety V X. (ii) Find an example of a big divisor B on a smooth projective variety X, such that vol X V (B) = 0 but vol(b V ) 0, for some subvariety V X. 4. Point separation for big pluricanonical systems Having in mind the discussion of the previous section, following [HM],[T] and [Ts1] we will now proceed by descending induction in order to cut down the dimension of the Non-klt locus at x 1 and end up with a divisor D 0 t 0 (K X ), with t 0 < a+b/(vol(k X )) 1/n and satisfying both hypotheses of Lemma 3.2. Precisely, we will prove : Theorem 4.1. Let n be a positive integer. Suppose there exists a positive constant v such that, for any smooth projective variety V of general type and of dimension < n we have vol(k V ) v. Then, there exist two positive constants a := a n,v and b := b n,v such that, for any smooth projective variety X of general type and of dimension n the pluricanonical map ϕ m : X PH 0 (X, O X (mk X )) 6 If I is an ideal sheaf and V X a subvariety, the restriction I V is the image of I O X O V. 9

10 is birational onto its image, for all m a + b vol(k X ) 1/n. The dependence on vol(x) is not annoying, and we can get rid of it thanks to a nice trick due to Tsuji. Proof of Theorem 1.1. The proof is by induction on the dimension of the varieties. Theorem 1.1 holds for n = 1. Suppose it holds for n 1. From Corollary 2.1, applied in dimension n 1, we deduce the existence of a positive lower bound vol(k V ) v n 1 for all smooth projective variety V of dimension n 1. In particular the hypotheses of Theorem 4.1 are fullfilled. Consider smooth projective varieties X of general type. For those such that volume of K X is bounded from below, say vol(k X ) 1, then Theorem 4.1 implies that mk X separates points, for all m > a + b a + b/ vol(k X ) 1/n. For those such that vol(k X ) < 1, then a priori the quantity a + b/ vol(k X ) 1/n may still be arbitrarily large. This does not occur. We argue as in the proof of Corollary 2.2 : using Theorem 4.1 and projecting down, we have that the variety X is birational to a subvariety of P 2n+1 of degree : ( b ) n a + vol(kx ) vol(k X ) ( 1/n n = a vol(k X ) 1/n + b) (a + b) n =: d n. Therefore these varieties are parametrized by points of a union of Chow varieties Chow := d dn Chow n,dn (P 2n+1 ). Let T = i T be the Zariski closure of those points in Chow. We have the following Lemma 4.2. For any i, we have inf{vol(k Xt ) : t T i and X t is of general type} > 0. Proof. It is of course sufficient to prove the statement for one T = T i. We argue by induction on dim(t ). If the dimension is zero, there is nothing to prove. Suppose dim(t ) > 0. Let U T be the universal family and Ũ U a desingularization. Let T o T be the open subset over which the induced map p : Ũ T is smooth. Let S T be the Zariski dense subset whose points correspond to varieties of general type. The S T o is also dense. By construction, for any s S T o, the fiber X s := p 1 (s) is a smooth variety of general type. By the upper semicontinuity of the h 0, the same is true for every fiber over T o, and moreover we have inf{vol(k Xt ) : t T o } > 0. On the other hand, as for the complement S (T \ T o ), we invoke the inductive hypothesis and are done. By the previous lemma the canonical volumes of n-dimensional varieties of general type are bounded from below by a positive constant c n (which is not effective!). Hence we may take m n := [1 + a + b/c 1/n n ] and we conclude that the pluricanonical system mk X separates two general points of X, for all m m n. 10

11 In light of the final part of the proof of Theorem 1.1 (where the effectivity of m n was lost) it seems interesting to think on the following : Problem 4.3. Find an effective lower bound c n,d for the canonical volume of n-dimensional degree d subvarieties Y P 2n+1 of general type. The proof of Theorem 4.1 follows the approach adopted by Angehrn and Siu [AS] in their study of the Fujita conjecture (see also [Ko2, Theorem 5.9]), with the crucial variations appearing in [HM] and [T] to make it work for big divisors. Its structure is transparent, though several technical complications may make it appear somehow messy. To avoid this we confine all the needed technical auxiliary statements (together with their proofs) in the Appendix. It was certainly well-known to the experts (and now, after the discussion in the previous section, it should also seem reasonable to the reader) that a positive lower bound to the restricted volumes of a big divisor L on a variety X allows to construct, along the lines of the Angehrn-Siu proof of the Fujita conjecture, a global section of K X + L separating two general points on X (cf. [T, Proposition 5.3] and [ELMNP1, Theorem 2.20]). Such a lower bound is the object of the following result. Theorem 4.4. Let X be a smooth projective variety, and V X be an irreducible subvariety. Let A (respectively E) be a Q-ample (resp. a Q-effective) divisor on X, and set L := A + E. Suppose that (4.1) V is a component of Non-klt(X, E) with (X, E) lc at the general point of V. Then : vol X V (K X + L) vol(k V ). The proof of the theorem is a fairly easy consequence of an extension result (see Theorem 5.1) when codim(v ) = 1. In the general case, it also requires, among other sophisticated things, the use of Campana s weak positivity result A.5 which generalizes Fujita s result (2.12). Assuming Theorem 4.4 for the moment, let us show how it yields effective points separation for big pluricanonical systems. Proof of Theorem 4.1. We will proceed by descending induction on d {0,..., n 1} to produce an effective Q-divisor D d t d K X such that : (1) x 1, x 2 Non-klt(X, D d ); (2) (X, D d ) is lc at a non-empty subset of {x 1, x 2 }, say at x 1 ; (3) Non-klt(X, D d ) has a unique irreducible component V d passing through x 1 and dim V d d; (4) t d < t d+1 + v d+1 with v d+1 = (d + 1)(2/v ) 1/(d+1) (t d+1 + 2) + ε, where v {v, vol(k X )} and ε > 0 is a rational that may be taken arbitrarily small. (We set t n = 0). Take two very general points x 1 and x 2 on X. Precisely, they must be outside (i) the augmented base locus 7 of K X, (ii) the support of the effective divisor E in the Kodaira decomposition of K X, and (iii) the union of all the subvarieties 7 Recall that given a Cartier divisor L on a variety X, its stable base locus (see [L1, pp ]) is B(L) := m 1 Base( ml ). The augmented base locus is defined in [ELMNP1], as B + (L) := B(mL H) for m 0 and H ample on X (the latter definition is independent of the choice of m and H). One checks that L is ample if, and only if, B + (L) =, and L is big if, and only if, B + (L) X. In the latter case, X \ B + (L) is the largest open set on which L is ample. 11

12 of X which are not of general type. 8 Thanks to Lemma 3.3, we can pick an effective Q-divisor D n 1 t n 1 K X such that Non-klt(X, D n 1 ) x 1, x 2, as soon as t n 1 = n2 1/n vol(k X ) 1/n + ε. (with ε > 0 arbitrarily small). Up to multiplying by a positive rational number 1, we can assume that (X, D n 1 ) is lc at one of the two points, say at x 1. Also, up to performing an arbitrarily small perturbation of D n 1, thanks to Lemma A.3, we may assume there exists a unique irreducible component of Non-klt(X, D n 1 ) through x 1. The base of the induction is therefore completed. For the inductive step (which is the troisième pas in [D]) we proceed as follows. Suppose that we have constructed an effective Q-divisor D d t d K X satisfying conditions (1), (2), (3) and (4) above. Simplifying assumption 4.5. We assume that x 1, x 2 V d and that D d is lc at both points. (The other possible cases can be treated in the same way, but render the discussion more complicated we discuss them in the appendix A.3). Also, we will make the following: Simplifying assumption 4.6. We assume that x 1, x 2 are non-singular points of V d. (If V d is not smooth at the points, then a limiting procedure described in A.3 allows to conclude). Since the points lie outside the support of E conditions (1), (2) and (3) also hold for (X, D d + te), where t := [t d ] + 1 t d. Now, we would like to add to D d an effective Q-divisor of the form v d K X which has multiplicity > d at x 1, chosen among those restricting to a non-zero divisor on V d, in order to cut down the dimension of the non-klt locus. This can be done thanks to Theorem 4.4 and Lemma A.4. By generality of x 1 the divisor K Vd is big, thanks to Exercise 3.1. Then, using the hypothesis, and applying Theorem 4.4 with L = ([t d ] + 1)K X, we have ([t d ] + 2) d vol X Vd (K X ) = vol X Vd (K X + ([t d ] + 1)K X ) vol(k Vd ) v. Therefore, by Exercise 3.5 and Lemma A.4, for small rational δ > 0, adding to (1 δ)d d a divisor G equivalent to (d(2/v) 1/d + ε))k X we get a divisor D d 1 t d 1 K X with t d 1 t d + d(2/v) 1/d (t d + 2) + ε and such that its Non-klt locus contains x 1, x 2. Moreover, the divisor G can be taken such that (4.2) I (X, D d + G) = I (X, D d ) around x 1, x 2 outside V d. To see this, take k 0 such that I Vd (kt d 1 K X ) is free outside Supp(E) (where E is the effective divisor in the Zariski decomposition of K X ). Choose a divisor F kt d 1 K X such that F Vd has multiplicity > k d at x 1 and x 2. The possible choices of F vary in the linear series I Vd (kt d 1 K X ), so by Lemma A.2 we can pick a F 0 such that I (X, D d + cf 0 ) = I (D d ) for every 0 c 1 (in particular for c = 1/k) away from 8 Actually, Takayama has proved in [T2] that any irreducible component of B + (K X ) is uniruled, so in particular condition (i) follows from (iii). See also [Br] for subsequent work in this direction. 12

13 Supp(E). Since x 1, x 2 Supp(E), the divisor G := 1 k F 0 satisfies (4.2). Therefore by the last assertion of Lemma A.4 we have that Non-klt(X, D d 1 ) has dimension at x 1, x 2 strictly lower than dim(v d ). Again, multiplying D d 1 by a rational number 1 and applying Lemma A.3, also conditions (2) and (3) are satisfied. The inductive step is thus proved. In conclusion, we have obtained an effective Q-divisor D 0 t 0 K X with t 0 < a n,v + b n,vt n 1 a n,v + b n,vn(2/ vol(k X )) 1/n whose Non-klt locus contains x 1 as an isolated point, and x 2. Therefore, by Lemma 3.2, we deduce the existence of a global section s H 0 (X, K X + m(k X )) separating the two points, for all m > a n,v + b n,vn(2/ vol(k X )) 1/n. Exercise 4.7. Compute explicitly a 3,v and b 3,v (actually, you can even drop v). Fujita conjectured that if A is an ample divisor on a smooth projective n-dimensional variety X, the linear system K X + ma is base-point-free for all m n + 1 and very ample for m n The conjecture is known only up to dimension 4 (work of Reider [Rei] for surfaces, and Ein Lazarsfeld [EL] and Kawamata [Ka2] for the basepoint-freeness in dimension 3 and 4 respectively). Having understood the proof above, it is instructive (and not too difficult) to do the next : Exercise 4.8. Prove the Angehrn-Siu theorem [AS], stating that, with the above notations, the rational map ϕ KX +ma is a morphism, for m 1 (n + 1)(n + 2), and injective 2 for m 1 (n + 2)(n + 3) (of course, you have the right to make the same simplifying 2 assumptions 4.5 and 4.6, but you don t have the right to copy the proof from Lazarsfeld s book!). 5. The missing piece: bounding the restricted volumes from below In M. Păun s lectures we have seen the following extension result. Theorem 5.1 (Takayama [T], Theorem 4.1). Let X be a smooth projective variety. Let H X be a smooth irreducible hypersurface. Let L Q A + E a big divisor on X with A a nef and big Q-divisor such that H B + (A); E an effective Q-divisor whose support does not contain H and such that the pair (H, E H ) is klt. Then the restriction is surjective for all integer m 0. H 0 (X, m(k X + H + L)) H 0 (H, m(k H + L H )) The analogous result in [HM] is Corollary For other extension results, all inspired by [S2] (see [Ka3] for an algebraic version of Siu s argument), the reader may look at [C], [Pă], [Var], and the very recent [BP]. In the following two subsections we derive Theorem 4.4 from the extension result above. 9 Question/Exercise: why can t we expect (conjectural) bounds better than m n+1 and m n+2? 13

14 5.1. Proof of Theorem 4.4: the case codim X (V ) = 1. Hypothesis (4.1) simply means that V appears with multiplicity 1 in E. We may therefore take a modification µ : X X such that the strict transform V of V is smooth, µ E = V + F has simple normal crossings and moreover (5.1) V Supp(F ). Take an integer m 0 > 0 such that m 0 (µ A + {F }) has integer coefficients. By (5.1) the support of this divisor does not contain V, so we have an inclusion (5.2) H 0 (V, mk V ) H 0 (V, m(k V + (µ A + {F }) V )) for any integer m > 0 divisible by m 0. Since the pair (X, {F }) is klt, applying the Extension Theorem 5.1 to the divisor µ A + {F }, and observing that µ L [F ] = V + µ A + {F }, we get a surjection (5.3) H 0 (X, m(k X + µ L [F ])) H 0 (V, m(k V + (µ A + {F }) V )). In conclusion we have h 0 (V, mk V ) = h 0 (V, mk V ) ( (5.2) + (5.3) ) h 0 (X V, m(k X + µ L [F ])) so Theorem 4.4 is proved in this case. h 0 (X V, m(k X + µ L)) = h 0 (X V, m(k X + L)) 5.2. Proof of Theorem 4.4: the case codim X (V ) 2. We follow almost word-byword Debarre s presentation [D, 6.2]. We may assume V smooth (see [T, Lemma 4.6]). Take a log-resolution µ = X X of E, and write µ E K X /X = F a F F. By hypothesis V is an irreducible component of Non-klt(X, E) such that (X, E) is lc at the general point of V. This means that if V is strictly contained in µ(f ), then a F < 1 ; if V = µ(f ), then a F 1, with equality for at least one F. Thanks to the so-called concentration method due to Kawamata and Shokurov (see [KMM, 3-1], and [T, Lemma 4.8]) one can further assume that there exists a unique divisor (which will be denoted by V ) among the F s such that µ(v ) = V. Therefore we have the commutative diagram of smooth varieties : (5.4) V X V f We set G := F V a F F, and write [G] as a difference of effective divisors without common components [G] = G 1 G 2 so that : G 2 is µ-exceptional ; V µ(supp(g 1 )). 14 µ X.

15 (We simply take G 1 := a F 1 [a F ]F, and G 2 := a F <0 a F F ). Hence, for any integer m > 0, the sheaf µ O X ( m[g]) is an ideal sheaf on X whose cosupport does not contain V, so that (5.5) H 0 (X, m(k X + L)) H 0 (X, µ O X ( m[g])(m(k X + L))) = H 0 (X, m(µ (K X + A + E) [G])) = H 0 (X, m(k X + V + {G} + µ A)). Since the pair (V, {G} V ) is klt, we can apply the Extension theorem 5.1 to the divisor L := µ L K X /X V [G] Q {G} + µ A and to the smooth hypersurface V X. Hence, for all m, we get a surjection H 0 (X, m(k X + V + {G} + µ A)) H 0 (V, m(k V + L V )). The last surjection together with the injection (5.5) and, again, the fact that the cosupport of µ O X ( m[g]) does not contain V, leads to the inclusion : (5.6) H 0 (V, m(k V + L V )) H 0 (X V, m(k X + L)). On the other hand, thanks to Campana s theorem A.5, one can show that for a suitable positive integer m we have : H 0 (V, m (K V /V + {G} V + f A V )) 0 (see [D, pages 17-18]). Hence we obtain, by multiplication, an injection H 0 (V, mm K V ) H 0 (V, mm (K V + L V )). The last inclusion, together with (5.6) and the fact that the restricted volumes are limits complete the proof of Theorem 4.4. Remark 5.2. The restricted volume function has been studied at length in the paper [ELMNP2], where among other things the authors prove that for any Q-divisor D, the irreducible components of B + (D) are the maximal (with respect to the inclusion) V X such that vol X V (D) = 0 (cf. [ELMNP2, Corollary 5.8]) Further work. As mentioned in the introduction, it is natural, though not easy, to try and obtain effective bounds at least in the case of 3-folds of general type. As already mentioned, the best result to date is due to J.A. Chen and M. Chen [ChCh2] who proved that 77K X is birational and vol(k X ) 1/2660, for any smooth 3-fold of general type. In any case, by examples due to Iano-Fletcher [Ia], it is necessary to have m Better bounds are available under some additional numerical hypotheses on the 3-fold. We recall a few of these results to give an idea of the type of hypotheses one adds. For instance, M. Chen has proved in [ChenM] that vol(k X ) 1/3, for 3-folds X such that h 0 (X, K X ) 2 (and the bound is sharp under the extra hypothesis). Another bound is vol(k X ) 1/30, obtained in [ChCh1], assuming χ(x, O X ) 0. For irregular 3-folds with χ(x, O X ) < 0, J.A. Chen and Hacon proved in [ChH] that mk X is birational, for all m 5. The birationality of mk X for all m 5 has been proved to hold for all but finitely many families of 3-folds of general type by Todorov [To]. People attracted by this kind of problems can start solving the following (non trivial) exercises : Exercise 5.3. Let X be a irregular 3-fold of general type. Assume that the image of the Albanese map X Alb(X) is a curve. Prove that vol(k X ) 2. 15

16 Exercise 5.4. Find an example of 3-fold of general type X such that 4K X is not birational. As for the problem of studying the uniformity of pluricanonical maps for varieties X of intermediate Kodaira dimension 0 < κ(x) < dim(x), apart from the work of Fujino- Mori [FM], not much seems to be known. In this direction, combining the canonical bundle formula of [FM] with the approach of Hacon-McKernan/Takayama/Tsuji we studied this problem under the hypotheses that the variation of the Iitaka fibration is maximal and the base is non-uniruled. In particular we obtained the uniformity of the Iitaka fibration for varieties verifying these extra hypotheses and of Kodaira dimension κ(x) = dim(x) 2 (see [Pa]). The hypothesis of the maximality of the variation excludes the possibility of dealing with the case κ(x) = dim(x) 1. Therefore we would like to conclude with the following: Problem 5.5. Study the uniformity of the Iitaka fibration for 3-folds of Kodaira dimension 2. Appendix A. A.1. Multiplier ideals. If D is an effective Q-divisor on X one defines the multiplier ideal as follows : I (X, D) := µ O X (K X /X [µ D]) where µ : X X is a log-resolution of (X, D). Notice that if D is a Q-divisor with simple normal crossings, then I (X, D) = O X ( [D]). If D is integral we simply have (A.1) I (X, D) = O X ( D). Again, we refer the reader to Lazarsfeld s book [L2] for a complete treatment of the topic. We now recall Nadel s vanishing theorem. Theorem A.1 (see [L2, Theorem 9.4.8]). Let X be a smooth projective variety. Let D be an effective Q-divisor on X, and L a divisor on X such that L D is big and nef. Then, for all i > 0, we have H i (X, I (X, D) O X (K X + L)) = 0. A.2. Singularities of pairs and Non-klt loci. Recall that, in the literature, a pair (X, D) is a normal variety together with a Q-Weil divisor such that K X + D is Cartier. In this paper the situation is much simpler : the variety will always be smooth and the divisor will be an effective Cartier divisor. A pair (X, D) is Kawamata log-terminal, klt for short, (respectively non-klt) at a point x, if I (X, D) x = O X,x (respectively I (X, D) x O X,x ). A pair is klt if it is klt at each point x X. A pair (X, D) is log-canonical, lc for short, at a point x, if I (X, (1 ε)d) x = O X,x for all rational number 0 < ε < 1. A pair is lc if it is lc at each point x X (The singularities of pairs in the framework of the MMP have been discussed in M. Mella s lectures. For a survey and many related 16

17 resuts, see also [Ko2]). Notice that, from the very definition of multiplier ideal, if D and D are two Q-effective divisors on X, then for any 0 < ε 1 we have : (A.2) I (X, D) = I (X, D + εd ). Another useful and easy result is the following Bertini-type lemma. Lemma A.2. Let E be a Q-divisor on a smooth variety X and M a general member of a base-point-free linear system. Then for any 0 < c < 1 we have I (X, E) = I (X, E + cm). We recall two fundamental results describing the effect of small perturbations of D on its Non-klt locus. Lemma A.3. Let X be a smooth projective variety, x 1 and x 2 two distinct points on X, and D an effective Q-divisor such that (X, D) is lc at x 1 and non-klt at x 2. Let V be an irreducible component of Non-klt(X, D) passing through x 1. Let B Q A + E be a big divisor on X, with A ample and E effective such that x 1, x 2 / Supp(E). There exists an effective divisor F Q B and, for any arbitrarily small rational δ > 0, there exists a unique rational number b δ > 0 such that: (1) (X, (1 δ)d + b δ F ) is lc at one of the two points, say x 1 ; (2) (X, (1 δ)d + b δ F ) is non-klt at x 2 ; (3) All the irreducible components of Non-klt(X, (1 δ)d + b δ F ) containing x 1 are contained in V. Moreover lim δ 0 b δ = 0. Proof. This proof is taken from [Dnotes, Prop. 4.3]. The reader may also have a look to [L2, Lemma ] or [Ko2, Th ]. Let V be an irreducible component of Non-klt(X, D) passing through x 1. Let m 0 be such that I V (ma) is generated by its global sections and G I V (ma) a general member. At x 1, by log-canonicity, the pair (X, (1 δ)d +bg) is klt for small b > 0 (here we use A.2), but non-lc for large b. If (X, D) is lc at x 2, then the same applies. If (X, D) is not lc at x 2, then a small perturbation (X, (1 δ)d + bg), for any δ and b sufficiently small, remains non-lc (hence non-klt) by (A.2). Therefore for δ > 0 sufficiently small, there is a unique b δ > 0 such that the pair (X, (1 δ)d + b δ G) is non-klt at x 1, x 2 and lc at one of the two points, say x 1. Moreover lim δ 0 b δ = 0. On the other hand, Lemma A.2 implies that for all 0 < δ, b < 1, the pair (X, (1 δ)d + bg) is klt outside V in a neighborhood of the points x 1, x 2. To conclude, set F = 1 G + E m Q A and b δ := m b δ. Since m is independent of δ we still have lim δ 0 b δ = 0. Since x 1, x 2 / Supp(E) and all the arguments were local around the points, the pair (X, (1 δ)d + b δ F ) verifies (1), (2) and (3). Lemma A.4. Let X be a smooth projective variety and D an effective Q-divisor. Let V be an irreducible component of Non-klt(X, D) of dimension d. There exists a dense subset U in the smooth locus of V and a rational number ε 0 : 0 < ε 0 < 1 such that, for any y U, any effective Q-divisor B whose support does not contain V and such that (A.3) mult y B V > d and any rational number ε : 0 < ε < ε 0, the locus Non-klt(X, (1 ε)d + B) contains y. If moreover (X, D) is lc at the generic poit of V and I (X, D + B) = I (X, D) away 17

18 from V, then Non-klt(X, (1 ε)d + B) is properly contained in V in a neighborhood of any y U. Proof. This proof is taken from [Dnotes, Prop. 4.4]. Again, for similar statements, see [L2, Lemma ] or [Ko2, Th ]. Let µ : X X be a log-resolution for the pair (X, D) such that there exists a smooth hypersurface E in X with µ(e) = V and (A.4) µ D = re + D, K X /X = (r 1)E + K where D and K are effective divisors whose supports do not contain E, and r > 0 is an integer. Let U be the set of smooth points of V, outside µ(supp(d + K )). Let U U be the subset where the morphism µ E : E V induced by µ is smooth. Take y U, and denote by F the fiber µ 1 E (y). Notice that F is smooth and has codimension d inside E. Let B be any effective Q-diviseur verifying (A.3). To determine Non-klt(X, (1 ε)d+b), we will use the birational transformation rule ([L2], Th ), which says that : (A.5) I (X, (1 ε)d + B) = µ (I (X, µ ((1 ε)d + B)) O X (K X /X)) By (A.4) the previous equality may be rewritten as I (X, (1 ε)d + B) = µ (X, I (µ ((1 ε)d + B (r 1)E) O X (K ))) and since D is effective we get the inclusion = µ (I (X, (1 εr)e + µ (B) + (1 ε)d ) O X (K ))) (A.6) I (X, (1 ε)d + B) µ (I (X, (1 εr)e + µ (B)) O X (K ))). By (A.3), the Q-diviseur s(µ B) E of E has multiplicity d at any point of F, for s close to 1. Hence by (3.2), it is non-klt at all points of F. By (3.3) the Q-divisor te + µ B has the same property for 0 < t < 1. Hence, for sufficiently small ε (i.e. for all ε smaller than a certain ε 0 ) we have I (X, (1 εr)e + µ B) I F, which, together with (A.6), yields ( I (X, (1 ε)d + B) µ IF O X (K ) ) Since F Supp(K ) and µ ( OX (K ) ) = O X, we finally obtain µ ( IF O X (K ) ) I y O X. Therefore the pair (X, (1 ε)d + B) is not klt at y. For the last assertion, observe that, by the hypotheses, outside V (and away from the other components of Non-klt(X, D)) we have O X,x = I (X, D) x = I (X, D + B) x I (X, (1 ε)d + B) x so that (X, (1 ε)d + B) is klt there. On the other hand, since (X, (1 ε)d) is klt at the general point of V, and B does not contain V, then (X, (1 ε)d + B) must be klt at the general point of V \ Supp(B V ). The assertion now follows. A.3. Complements to the proof of Theorem

19 A.3.1. On the simplifying assumption 4.5. In the course of the proof of Theorem 4.1 we have assumed that the pair (X, D d ) was lc at both points and an irreducible component V d of Non-klt(X, D d ) passes through x 1 and x 2. We treat here the cases where: (i) (X, D d ) is lc at both points and there exists an irreducible component V d of Non-klt(X, D d ) not passing through one of the two points, say x 2 ; (ii) (X, D d ) is lc at only one of the two points, say x 1. Case (i) : Write K X Q A + E, with A ample and E effective. Let m be an integer such that I V (ma) is globally generated. Let G I V (ma) be a general member of this sublinear system. By Lemma A.2, the pair (X, D d + εg) is lc, but non-klt at x 1, for all 0 < ε 1. Moreover, since the divisor G contains V and (X, D d ) is lc at x 2, the new pair (X, D d + εg) is not lc at x 2. Since all the arguments were local around x 1 and x 2, and the two points were taken outside Supp(E), the same conclusion holds for (X, D d + εf ), with F = ε ( 1 m G + E) Q εk X. So we are reduced to the case (ii). Case (ii) : Arguing as in the course of the proof of Theorem 4.1 we produce a new pair (X, D d 1 ) such that Non-klt(X, D d 1 ) contains x 1 and x 2 and is properly contained in V d around x 1. In order to get log-canonicity we are lead to multiply D d 1 by the log-canonical threshold of (X, D d 1 ) at x 1. The problem is that after that the new pair may well become klt at x 2! To avoid that, fix an ε (0, 1) such that (X, (1 ε)d d ) is still not lc at x 2 and set and t 0 := inf{t [0, 1] : (1 ε)d d + tg is non-klt at x 1 } D d 1 := (1 ε)d d + t 0 G, where G is taken as in the proof of Theorem 4.1 (that is, using Exercise 3.5) in order to have that G does not cointain V d and it is non-klt at x 1. The pair (X, D d 1 ) is now lc at x 1 and non-klt at x 2. By construction its non-klt locus is properly contained in V d around x 1 and we are done. A.3.2. On the simplifying assumption 4.6. If x 1, x 2 are singular points of V d, the idea (due to Angehrn and Siu) is to use a limiting process. Precisely, take a family of pairs of points u : Λ V d V d, such that u(0) = (x 1, x 2 ), for some point 0 Λ, and the points u(λ) = (y 1 (λ), y 2 (λ)) are smooth on V d for any λ 0 (actually, we suppose they belong to the open dense subset U appearing in the statement of Lemma A.4). Then, for each λ \0, we construct as in the proof of Theorem 4.1 a divisor G λ (d(2/v) 1/d + ε)k X such that y 1 (λ), y 2 (λ) Non-klt(X, (1 δ)d d + G λ ). By a semi-continuity result for multiplier ideals (see e.g. [L2, Corollary ]) the limiting locus Non-klt(X, (1 δ)d d + G 0 ) contains x 1 and x 2, and, up to small perturbations, all the properties (1),(2),(3),(4) are verified. A.4. Weak positivity of direct images. Viehweg introduced the notion of weak positivity for a torsion-free coherent sheaf E on a projective variety V : if V 0 is the largest open subset on which E is locally free, the sheaf E is weakly positive if there exists a open dense subset U of V 0 such that for any ample divisor H on V and any positive integer a, there exists a positive integer b such that the sheaf (Sym ab E V0 )(bh V0 ) is generated on U by its global sections on V 0. In the proof of Theorem 4.4 we used the following positivity result for direct images, due to Campana [Ca, Theorem 4.13], which 19

20 improves on previous results obtained by Kawamata [Ka1], Kollár [Ko1] and Viehweg [Vie1]. Theorem A.5 (Campana). Let f : V V be a morphism with connected fibres between smooth projective varieties. Let be an effective Q-divisor on V whose restriction to the generic fibre is lc and has simple normal crossings. Then, the sheaf f O V (m(k V /V + )) is weakly positive for all positive integer m such that m is integral. Notice that a locally free sheaf is weakly positive if, and only if, it is pseudoeffective (cf. [Vie2, Remark ]). When f is smooth, then f O V (mk V /V ) is locally free, and in this particular case (which unfortunately is not very useful in applications) it is nef (see [L2, Proposition ]). References [AS] U. Angehrn, Y.-T. Siu, Effective freeness and point separation, Invent. Math. 122 (1995), [BCHM] C. Birkar, P. Cascini, J. McKernan, C. Hacon, Existence of minimal models for varieties of log general type, preprint arxiv:math.ag/ [BP] B. Berndtsson, M. Păun, Bergman kernels and the pseudoeffectivity of relative canonical bundles, preprint arxiv:math/ [Bo] E. Bombieri, The pluricanonical map of a complex surface. In : Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970) pp Springer, Berlin. [Br] A. Broustet, Thèse de Doctorat, Grenoble [Ca] F. Campana, Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, [ChCh1] J.A. Chen, M. Chen, The canonical volume of 3-folds of general type with χ 0, preprint. [ChCh2] J.A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, preprint arxiv: [math.ag]. [ChH] J.A. Chen, Ch. Hacon, Pluricanonical systems on irregular 3-folds of general type, Math. Zeit. 255 (2007), [ChenM] M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann. 337 (2007), [C] B. Claudon, An extension theorem, preprint math.ag/ , to appear in the Ann. Inst. Fourier. [D] O. Debarre, Systèmes pluricanoniques sur les variétés de type général [d après Hacon-McKernan, Takayama, Tsuji], Séminaire BOURBAKI, n. 970, available at debarre/970.pdf. [Dnotes] O. Debarre, Systèmes pluricanoniques sur les variétés de type général [d après Hacon- [EL] McKernan, Takayama, Tsuji], notes du groupe de travail, version du 06/05/06. L. Ein and R. Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, J. Amer. Math. Soc., 6 (1993), [ELMNP1] L. Ein, R. Lazarsfeld, M. Mustață, M. Nakayame, M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier. 56 (2006), , also available as preprint math.ag/ [ELMNP2] L. Ein, R. Lazarsfeld, M. Mustață, M. Nakayame, M. Popa, Restricted volumes and asymptotic intersection theory, preprint math.ag/ [FM] O. Fujino, S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), [Fu] T. Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), [HM] [Ia] C. Hacon, J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1 25, also available as preprint math.ag/ A.R. Iano-Fletcher, Working with weighted complete intersections. Explicit birational geometry of 3-folds. Lond. Math. Soc. Lect. Note Series, vol. 281, Cambridge University Press (2000). 20