A PATHOLOGY OF ASYMPTOTIC MULTIPLICITY IN THE RELATIVE SETTING

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1 A PATHOLOGY OF ASYMPTOTIC MULTIPLICITY IN THE RELATIVE SETTING JOHN LESIEUTRE Abstract. We point out an example o a projective amily π : X S, a π-pseudoeective divisor D on X, and a subvariety V X or which the asymptotic multiplicity σ V (D; X/S) is ininite. This shows that the divisorial Zariski decomposition is not always deined or pseudoeective divisors in the relative setting.. Introduction We work throughout over C. Suppose that X is a smooth projective variety and D is a pseudoeective R-divisor on X. The asymptotic multiplicity o D along a subvariety V X, studied by Nakayama [] and Ein-Lazarseld-Mustaţă-Nakamaye-Popa [5], has proved to be a undamental tool in understanding the properties o the divisor D. For big divisors D, the deinition o the asymptotic multiplicity is straightorward: roughly, one considers the linear series md or larger and larger values o m, and takes σ V (D) = lim m mult m V md, where the multiplicity o a linear series along a subvariety is deined to be the multiplicity o a general member. Complications arise, however, in carrying out this construction or divisors D which are pseudoeective but not big, i.e. or divisors on the boundary o the pseudoeective cone E(X) N (X). Nakayama realized that σ V (D) can be extended to a lower semicontinuous unction on E(X) by setting σ V (D) = lim ɛ 0 σ V (D + ɛa), where A is a ixed ample divisor. In some applications (e.g. in the construction o Zariski decompositions), it is important to know that the limit in question takes a inite value. While it is clear that the quantity on the right is nondecreasing as ɛ is made smaller, it might a priori be unbounded in the limit. That this does not happen in the non-relative setting was observed by Nakayama. Our aim in this note is to demonstrate by example that when asymptotic multiplicity invariants are considered in the greater generality o divisors on a projective amily π : X S, this initeness need not hold: or a π-pseudoeective divisor, the limit deining σ V (D; X/S) can indeed be ininite. This answers a question o Nakayama [, pg. 33]. The example itsel is amiliar, a divisor on the versal deormation space o a iber o Kodaira type I 2, which has been considered in related contexts by Reid [2, 6.8] and Kawamata [8, Example 3.8(2)],[9, Example 9]. Theorem. There exists a projective amily π : X S, a π-pseudoeective divisor D, and a subvariety V X or which σ V (D; X/S) is ininite. An important use o asymptotic multiplicity invariants is in the construction o the divisorial Zariski decomposition, a higher-dimensional analog o the usual Zariski decomposition

2 2 JOHN LESIEUTRE on suraces. The example here shows that trouble arises i one generalizes this construction to pseudoeective classes in the relative setting: ater passing to a blow-up on which the valuation corresponding to V is divisorial, we obtain an example in which the decomposition is not deined. Corollary 2. Let π : X S be as in Theorem. I : W X is the blow-up along V with exceptional divisor E, then D = D has σ E ( D; W/S) = and N σ ( D; W/S) is not deined. Moreover, the divisor D does not admit any Zariski decomposition in a very strong sense: Corollary 3. There does not exist a birational model g : Z W or which g D admits a decomposition g D = P + N with P a g ( π)-movable divisor and N eective. In Section 2 we recall the basic deinitions and properties o the invariants σ V (D; X/S) and N σ (D; X/S) appearing in Theorem and Corollary 2, beore establishing the claims in Section 3. In Section 4, we describe a more general setting or making computations in a similar spirit. 2. Preliminaries Suppose that π : X S is a projective, surjective morphism with connected ibers, with X and S smooth (hereater, a nice amily). We will ind it convenient to allow the base S to be a surace germ, ollowing [7]. The proos o the results in this section hold either when S is a quasiprojective variety or a germ. In Section 3, it will be convenient to make computations with the base a germ. However, the germ we consider is algebraizable, and it ollows that the same pathology occurs when the base is extended to be an aine scheme; this is discussed in Remark. Two divisors D and D on X are said to be numerically equivalent over S, or π-numerically equivalent, i D C = D C or any curve C that is contracted by π; write D π D or the relation o numerical equivalence over S, and N (X/S) or the vector space o R-divisors on X, modulo this equivalence. The amiliar cones o positive divisors on a projective variety all have analogs in the relative setting: a divisor D on X is said to be () π-ample i D s is ample on every iber X s = π (s); (2) π-strongly movable i the support o the cokernel o O X (D) O X (D) has codimension at least 2; (3) π-big i the restriction o D to the generic iber is big. Corresponding to these classes o divisors are cones inside N (X/S): Amp(X/S) Mov(X/S) Big(X/S). The cones Amp(X/S) and Big(X/S) are both open. We will also consider the closures o these three cones inside N (X/S), which are, respectively: Ne(X/S). A divisor is π-ne i D s is ne on every iber X s (i.e. i D C 0 or every curve C contracted by π); Mov(X/S), the cone o π-movable divisors. E(X/S), the cone o π-pseudoeective divisors. A divisor D is π-pseudoeective i the restriction o D to the generic iber is pseudoeective.

3 RELATIVE ASYMPTOTIC MULTIPLICITY 3 The meaning o movable is unortunately not entirely uniorm in the literature, and we stress that here an R-divisor is called π-movable i it lies in the closed cone Mov(X/S); this is sometimes called π-ne in codimension. We note too that the cone E(X/S) is not necessarily a strictly convex cone, in that it might contain an entire line through the origin; this contrasts with the amiliar case when S is a point. For example, i D restricts to 0 on a general iber o π, then D and D are both π-pseudoeective. For simplicity, we will assume that the base space S is either aine or a germ. This is not really necessary, but the invariants under consideration can be computed when the base is projective simply by restricting to the preimage o a suitable aine open set; we reer to [, 3.2] or details. The existence o a π-ample divisor A on X is automatic in this setting. I D is a π-big divisor, then O X (md) 0 or suiciently large and divisible m, and so H 0 (X, O X (md)) = (O X (md)) is nonzero as well. Hence in these settings, any π-big class has an eective representative. Deinition. Suppose that X is smooth. Given an irreducible subvariety V X and a π-big R-divisor D, set σ V (D; X/S) = in D πd D 0 mult V (D ). Since D is π-big, there exists an eective R-divisor D that is π-numerically equivalent to D, and this inimum is taken over a nonempty set. When D is a big integral divisor, a sequence D m o eective R-divisors with multiplicities converging to the inimum can be ound by taking D m md, where we choose a general m element o the linear system md. We next extend the deinition o the asymptotic multiplicity rom π-big divisors to π- pseudoeective divisors. Deinition 2. Given a π-pseudoeective R-divisor D, set σ V (D; X/S) = lim ɛ 0 σ V (D + ɛa; X/S). This is evidently a nondecreasing unction as ɛ approaches 0, but it might have ininite limit. To show that it has a inite limit, it suices to bound σ V (D + ɛa; X/S) above, independent o ɛ. Nakayama gives several conditions under which this can be achieved. Theorem 4 ([], Lemmas 2..2, 3.2.6). I any o the ollowing holds, then σ V (D; X/S) is inite. () S = Spec C is a point; (2) D is numerically equivalent over S to an eective R-divisor ; (3) codim π(v ) < 2. We recall the proo in case (), perhaps the most important in practice. Case (2) is immediate rom the deinition, and we reer to [] or (3). Assume or a moment that V X is an irreducible divisor; that this implies the general statement will ollow rom Theorem 5(2) below. Proo o (). For any ɛ, (D + ɛa) σ V (D + ɛa)v is pseudoeective, and so ((D + ɛa) σ V (D + ɛa)v ) A n 0.

4 4 JOHN LESIEUTRE As long as ɛ < it ollows that σ V (D + ɛa) is bounded above as ɛ decreases to 0. (D + ɛa) An V A n (D + A) An V A n This argument relies in a crucial way on the properness o X to carry out intersection theory, and is not applicable in the relative setting in general. Proposition 5 ([], Lemmas 2..4, 2.2.2, 2..7). Suppose that π : X S is a nice amily and V X is an irreducible subvariety. () I F is any π-pseudoeective divisor on X, then lim σ V (D + ɛf ; X/S) = σ V (D; X/S). ɛ 0 (2) Let : W X be the normalized blow-up o X along V, and let E be the exceptional divisor over V. Then σ E ( D; W/S) = σ V (D; X/S). (3) The number o prime divisors Γ or which σ Γ (D; X/S) > 0 is inite. The irst o these shows that Deinition is independent o the choice o π-ample divisor A, while the second completes the proo o Theorem 4() in the case that V has codimension greater than. Deinition 3 ([], [4]). Suppose that π : X S is a nice amily and that D is a π- pseudoeective divisor such that σ Γ (D; X/S) is inite or every prime divisor Γ. Then set N σ (D; X/S) = Γ σ Γ (D; X/S) Γ, P σ (D; X/S) = D N σ (D; X/S). It ollows rom Proposition 5(3) that there are only initely many nonzero terms in the sum deining N σ (D; X/S). We reer to N σ (D; X/S) as the negative part o the divisorial Zariski decomposition, and P σ (D; X/S) as the positive part. The negative part is a rigid, eective divisor. The positive part might not be ne, but it lies in the cone Mov(X/S) o π-movable divisors. Corollary 2 shows that without the initeness hypothesis on σ Γ (D; X/S), the deinition is not always applicable in the relative setting. In the non-relative setting, the divisorial Zariski decomposition is deined or any pseudoeective class D, but it lacks certain useul properties o the classical Zariski decomposition in dimension 2: most importantly, the positive part P σ (D) is not ne in general. In many cases one may construct a birational model : W X on which P σ ( D) is actually ne, even i P σ (D) is not. However, a basic example o Nakayama shows that even this is not always possible [, Theorem 5.2.6]. Another variant on Zariski decomposition in higher dimensions, the weak Zariski decomposition o Birkar, imposes ewer conditions and so exists or a larger class o divisors, including that o Nakayama s example. In this decomposition, the positive part P is allowed to be any relatively ne divisor, not necessarily the positive part P σ ( D) o the divisorial Zariski decomposition.

5 RELATIVE ASYMPTOTIC MULTIPLICITY 5 Deinition 4 ([2]). Suppose that π : X S is a nice amily and that D is a pseudoeective divisor on X. We say that D admits a weak Zariski decomposition over S i there exists a birational map : Y X and a decomposition D = P + N, where P is ( π)-ne and N is eective. This condition is airly unrestrictive, but there nevertheless exist pseudoeective R-divisors on smooth threeolds which do not admit a weak Zariski decomposition [0]. Corollary 3 asserts that the divisor D provides another such example. Indeed, D admits no Zariski decomposition in a still stronger sense: even ater pulling back to a higher model, it cannot be decomposed as the sum o an eective divisor and a relatively movable divisor. The example is qualitatively rather dierent rom that o [0]: there, a certain pseudoeective divisor D λ has negative intersection with ininitely many curves; here, there is a single curve on which D is negative, but the multiplicity o D along this curve is ininite. 3. Main example The claimed pathology ollows rom a ew calculations on an example that has been studied by Reid [2, 6.8] and Kawamata [8, Example 3.8(2)]. Let π : X S be the versal deormation space o a iber o Kodaira type I 2. The base S is smooth, 2-dimensional germ. The iber over the central point 0 S is consists o two smooth rational curves C and C 2, meeting transversely at two points p and p 2. Let C = π (0) be the union o these two curves. There are two divisors Γ, Γ 2 S corresponding to the smoothings o the two nodes o C. The iber o π over a general point o Γ i is a nodal rational curve, while the iber over a general point o S is a smooth curve o genus. C C 2 Γ Γ 2 Figure. The amily π : X S Remark. The computations that ollow will give an example in which some σ V (D; X/S) is ininite, in the case where S is a germ. The calculations rely on the act that π : X S is a versal deormation space. However, the local analytic results imply that the same pathological behavior occurs even when the base S is an aine surace. Indeed, we will see in

6 6 JOHN LESIEUTRE Lemma 7 below that there is a projective amily π : X S where S is an aine surace, such that the restriction o π to the germ at a point 0 S coincides with the map π : X S. I Ḡ is a π-big divisor, with restriction G to the germ, then σ C (Ḡ; X/ S) σ C (G; X/S): indeed, i Ḡ is an eective divisor on X which is π-numerically equivalent to Ḡ, its restriction to the central germ is an eective divisor on X which is π-numerically equivalent to G. Thus the inimum deining σ (Ḡ; X/ S) C in is taken over a subset o the inimum deining σ C (G; X/S) in Deinition, giving the claimed inequality. It ollows that in the limit at the pseudoeective boundary, σ C ( D; X/ S) σ C (D; X/S), and it must be that σ C ( D; X/ S) is ininite as well. The claims about Zariski decomposition ollow in the same way. Lemma 6. The normal bundle N Ci /X is isomorphic to O P ( ) O P ( ). Proo. Suppose or instance that i =. There is an exact sequence 0 N C /X (N C/X ) C T C2,p T C2,p 2 0 with the property that a irst-order deormation, determined by a section s H 0 (C, N C/X ) smooths the node at p i i and only i s has nonzero image in T C2,p i [6, Lemma 2.6]. The shea in the middle is the trivial O C O C. In one direction p is smoothed, and in another p 2 is, so the map sends (, 0) to (, 0) and (0, ) to (0, ) with respect to the direct sum decompositions. It ollows that the kernel is O P ( ) O P ( ). Lemma 6 shows in particular that K X has intersection 0 with both C and C 2 and so is relatively numerically trivial. Lemma 7. There exists a lop τ : X X + /S with lopping curve C. Let C + X + be the lopped curve, and C 2 X + be the strict transorm o C 2. There exists an isomorphism σ : X + X/S which sends C + to C and C 2 to C 2. Furthermore, there exists an involutive automorphism ı : X X/S which exchanges the two curves C and C 2. Proo. The arguments here are due to Kawamata [8, Example 3.8(2)]. We make some aspects o the proo explicit by working with local deining equations given by Reid [2]. In what ollows, we use the notation to denote objects on a amily π : X S over an aine base, while objects with no bar will be the restrictions to a certain germ. Let S = A 2, with coordinates t and t 2. Fix two distinct complex numbers a and a 2 and deine X 0 (A A ) S by the equation x 2 = ((x 2 a ) 2 t )((x 2 a 2 ) 2 t 2 ). The closure X (P P ) S is smooth, and the second projection π : X S is proper. The iber o π over a general point (t, t 2 ) is a smooth curve o genus. I exactly one o t and t 2 is zero, the iber is nodal, while i t = t 2 = 0, the iber is given by x 2 = (x 2 a ) 2 (x 2 a 2 ) 2. This central iber has two components, the rational curves C deined by x = (x 2 a )(x 2 a 2 ) and C 2 deined by x = (x 2 a )(x 2 a 2 ). The restriction o π : X S to the germ at (0, 0) S is the versal deormation space π : X S considered above. The involution ı : X X/ S deined by ı(x, x 2 ) = ( x, x 2 ) exchanges the two components o the central iber. There is a section σ : S X given by x 2 (t, t 2 ) = a + a 2 2 t t 2 2(a a 2 ), x (t, t 2 ) = (x 2 (t, t 2 ) a ) 2 t.

7 RELATIVE ASYMPTOTIC MULTIPLICITY 7 ( ) This has σ(0, 0) = (a2 a ) 2, (a a 2 ), which lies on C and is disjoint rom C 2. Let Σ be the divisor σ( S). Since Σ C = and Σ C 2 = 0, the curves C and C 2 have distinct classes in N ( X/ S). Since all other ibers o π are irreducible, it must be that N ( X/ S) is has dimension 2. The divisor 2ı ( Σ ) Σ has positive degree on general ibers, and so is π-big. Since S is aine, there is an eective divisor representing this class. For suiciently small ɛ, the pair ( X, ɛ ) is klt. Since C < 0, there exists a (K X/ S + ɛ )-lip τ : X X+, which is a K X/ S-lop. The map π + : X+ S is a minimal model o X+. By Lemma 6, the map τ is the lop o a rational curve with normal bundle O P ( ) O P ( ). It ollows by looking at a resolution o τ (discussed in the proo o Lemma 8 below) that the map τ S simply blows up the point Σ C and that the strict transorm o Σ on X + is smooth. This strict transorm contains the curve C +, and satisies τ Σ C 2 = 2. Now, the varieties X + and S are smooth and π + : X+ S has all ibers -dimensional, so π + is lat. Since π : X S is a versal deormation space, there exists an isomorphism β : X + X over some automorphism o S. However, β might not be deined over the identity map on S. The divisor Σ 2 = β (τ (Σ )) is a smooth divisor on X, containing C, and meeting C 2 at two points. There is a translation on the smooth ibers o π sending Σ to Σ 2, which deines a birational sel-map γ : X X over the identity on S. The map π γ : X S must be isomorphic to some minimal model o X over S, and indeed must be to isomorphic to π + : X + S since the strict transorms o Σ under γ and τ have the same numerical classes. It ollows that there exists an isomorphism σ : X + X over the identity o S. Replacing σ with σ ı i necessary, we may assume that σ(c + ) = C and σ(c 2) = C 2, as required. Each o the maps σ τ and ı is a birational involution o X over S, but we will soon see that the composition φ = (σ τ) ı is o ininite order. Since ı(c 2 ) = C, the eect o repeatedly applying φ is to lop C, then C 2, then C again, and so on. We will denote by φ D the strict transorm o a divisor D under a birational map φ, and use the same notation or the induced map on numerical groups when conusion seems unlikely. C 2 C C+ C C 2 C + C 2 g τ Figure 2. Resolution o the lop τ

8 8 JOHN LESIEUTRE To an eective divisor D on X, associate the 4-tuples v D = (D C, D C 2, mult C (D), mult C2 (D)), σ D = (D C, D C 2, σ C (D; X/S), σ C2 (D; X/S)). Lemma 8. Suppose that D is a divisor on X, and let D denote the strict transorm o D under the lop τ : X X +. Then () D C + = D C, (2) D C 2 = D C 2 + 2(D C ), (3) mult C + ( D) = mult C (D) + D C, (4) mult C 2 ( D) = mult C2 (D). In matrix orm, we have v φ D = Mv D and σ φ D = Mσ D where M = Proo. Let W be the graph o the lop τ: X W g τ X + Since τ is the lop o a rational curve with normal bundle O P ( ) O P ( ) by Lemma 6, the map is simply the blow-up o X along C, while g is the blow-up o X + along C +. There is a single -exceptional divisor E on W, which is isomorphic to P P and has normal bundle o bidegree (, ). Let C be a ruling o E contracted by g, so that sends C isomorphically to C. Similarly, let C + be a a ruling o E contracted by, so that g maps C + isomorphically onto C +. Lastly, let C 2 be the strict transorm o C 2 on W, a curve which meets E transversely at 2 points. This resolution is illustrated in Figure 2. Then write D + ae = g D, or some constant a. Taking the intersection o both sides with C yields D C +a(e C ) = 0. Since E C =, we obtain a = D C. Intersecting with C +, we have a = D C +. Similarly, intersecting with C 2, we have D C 2 + a(e C 2) = D C 2, and since E C 2 = 2, we have (2). It is clear that mult C 2 ( D) = mult C2 (D), since τ is an isomorphism at the generic point o C 2. Finally, mult C + ( D) = mult E (g D) = multe ( D) + a = mult C (D) + a. These calculations immediately yield v φ D = Mv D, since the second map ı exchanges the two curves C and C 2. Write D m or a general divisor linearly equivalent to md, and then σ C (φ D) = lim m m mult C (φ D m ) = lim m m (mult C D m + D m C ) ( = lim m m mult C D m We are now in position to make the main computation. ) + D C = σ C (D) + a.

9 RELATIVE ASYMPTOTIC MULTIPLICITY 9 Theorem 9. Let π : X S be the versal deormation space o a singular iber o Kodaira type I 2, and let C be a component o the central iber. There exists a divisor D on the boundary o the cone E(X/S) with σ C (D; X/S) =. Proo. Fix a π-ample eective Q-divisor H = H 0 on X with H C = H C 2 =. Then σ Ci (H) = 0 or each i. For example, we might take H = Σ + i ( Σ ) in the notation o Lemma 7. Let H n = φ n (H) be the strict transorm o H on X under n applications o φ. Using the last part o Lemma 8, and the Jordan decomposition o M, which has a 3 3 block associated to the eigenvalue, we compute σ Hn = (2n +, 2n +, n(n )/2, n(n + )/2): n H n C H n C 2 σ C (H n ) σ C2 (H n ) n 2n + 2n + n(n ) 2 n(n+) 2 The key eature o the example is that while H n C grows linearly in n, the multiplicity σ C (H n ) grows quadratically. Let D be the divisor class on the boundary o E(X/S) with D C = and D C 2 =. Since C and C 2 span N (X/S), we see that H n π (2n)D + H 0. It ollows that H 2n n π D + H 2n 0 is a sequence o divisors converging to D, whose multiplicities along the curves is known. By Deinition, we compute σ C (D; X/S) = lim σ C (D + n 2n H 0) = lim n 2n σ n C H n = lim =. n 4 Note that codim π(c ) = 2, so there is no contradiction with Theorem 4(3). Corollary 0. I : W X is the blow-up along C with exceptional divisor E, then D = D has σ E ( D; W/S) = and N σ ( D; W/S) contains the divisor E with ininite coeicient. In particular, there does not exist a birational model g : Z W or which g D admits a decomposition g D = P + N with P a g ( π)-movable divisor and N eective. Proo. By Theorem 5(2), i : W X is the blow-up along C, with exceptional divisor E, we have σ E ( D; W/S) =. Now, suppose that g : Y W is any birational map, and that g D = P + N, where P is a (g π)-movable divisor and N is eective. Let Ẽ denote the strict transorm o E on Y. Then σẽ(g D; Y/S) σẽ(p ; Y/S) + σẽ(n; Y/S) = σẽ(n; Y/S). The last o these is inite since N is eective, while the irst is ininite, a contradiction. This completes the proo. 4. A general set-up The key eature that made possible the computation o the preceding example is that i the our numbers D C i and σ Ci (D) are all known, then the same our invariants can be computed or the strict transorm o D under φ using Lemma 8. In this section, we give an

10 0 JOHN LESIEUTRE C 2 Ne(X/S) D C D+ 2n H 0 E(X/S) Figure 3. Chambers in N (X/S) explanation or this, and describe how to make analogous computations in a more general setting. Suppose that X is normal and Q-actorial and that φ : X X is a pseudoautomorphism over S (i.e. a birational map or which neither φ nor φ contracts any divisors). We will say that a birational morphism : Y X rom a normal Q-actorial variety Y is a small lit o φ i the induced map ψ : Y Y is also a pseudoautomorphism. Y ψ Y X φ X Observe that i : Y X is a small lit, then the map ψ must permute the exceptional divisors o. Example. Suppose that φ : X X is a pseudoautomorphism and x is a point not contained in indet φ. The blow-up : Bl x X X is a small lit o φ i and only i x is a ixed point o φ. I x is not a ixed point, then the induced map ψ : Y Y contracts the exceptional divisor E, while i x is ixed, then ψ E : E E is an automorphism. The more interesting examples are those in which contracts a divisor lying over indet φ. Example 2. Next we construct a small lit o the map φ : X X/S rom Section 3. Let : W X be the blow-up along C as beore, with exceptional divisor E, and let h : Y W be the blow-up along C 2, with exceptional divisor E 2. The two exceptional divisors E and E 2 are swapped by the induced map ψ : Y Y, and h is a small lit. h Y X h ψ W φ The curves C and C 2 could have been blown up in the opposite order, yielding a dierent small lit : Y X. This makes no real dierence: the threeolds Y and Y dier only by g Y X h

11 RELATIVE ASYMPTOTIC MULTIPLICITY lops, and strict transorm induces an identiication N (Y ) N (Y ) with respect to which the maps ψ and ψ coincide. I : Y X is a small lit, it ollows rom the negativity lemma [, Lemma 3.6.2] and the Q-actoriality o X that there is a decomposition N (Y ) = N (X) V E, where V E = i R [E i]. I D is a divisor class on X, it is not necessarily true that φ D = ψ D. However, the dierence φ D ψ D is an -exceptional divisor, since ( φ D ψ D) = φ D ψ D = φ D φ D = φ D φ D = 0. Deine K : N (X) V E by K = φ ψ. The next lemma characterizes the action o the strict transorm ψ : N (Y ) N (Y ) with respect to this decomposition. Lemma 3. Suppose that X and Y are Q-actorial and : Y X is a small lit o a pseudoautomorphism φ : X X. With respect to the decomposition N (Y ) = N (X) V E, ψ is given in block orm as ( ) φ 0 ψ =, K P where P is the permutation matrix or the action o ψ on the E i. The eigenvalues o ψ are the union o those o φ and those o P, which are roots o unity. Its eigenvectors are () v i (λ i I P ) Kv i, where v i are the eigenvectors o φ, with eigenvalues λ i ; (2) E i, the exceptional divisors o, with eigenvalues that are roots o unity. Proo. For a divisor D on X, ψ D = φ D KD, while the exceptional divisors E i are simply permuted by ψ; this gives the block orm o the map. The eigenvectors ollow rom elementary linear algebra. Note that i ψ ixes the exceptional divisors, which can always be arranged by replacing φ and ψ by suitable iterates, the permutation matrix P is the identity. A rational map φ : X Y between Q-actorial varieties is said to be D-non-negative or an R-divisor D i on some common resolution p : W X, q : W Y, we have p D + E = q (φ D), where E is an eective q-exceptional divisor. I φ : X X is a pseudoautomorphism with a small lit : Y X, then we may consider a resolution o the orm W Y X p ψ I D is a divisor on X or which φ is D-non-negative, then we have p D + E = q φ D with E 0. Pushing orward both sides by q, this gives φ q Y X q p D + q E = φ D ψ D + E = φ D, where E is an eective -exceptional divisor. In particular, KD = φ D ψ D = E is eective.

12 2 JOHN LESIEUTRE Next we observe that i the divisorial Zariski decomposition o D is known or some divisor D, the decomposition o φ D can oten be computed. Although in earlier sections we assumed that X was smooth, in what ollows X need only be normal and Q-actorial, as we will only consider asymptotic multiplicities along divisors, rather than higher-dimensional subvarieties (which might lie entirely in the singular locus o X). We assume or simplicity that ψ ixes each o the -exceptional divisors E i ; this can always be arranged by replacing φ by a suitable iterate. This assumption implies that the permutation matrix P is the identity, and that ψ (KD) = KD since KD is -exceptional. Lemma 4. Suppose φ : X X is a pseudoautomorphism over S, and that D is a class in N (X/S) with σ Γ (D; X/S) inite or all divisors Γ. Then N σ (φ D; X/S) = φ N σ (D; X/S) and P σ (φ D; X/S) = φ P σ (D; X/S). I φ is D-non-negative and N σ ( D; Y/S), then P σ ( φ D; Y/S) = ψ P σ ( D; Y/S). Proo. Since φ neither contracts nor extracts any divisors, or any prime divisor E we have σ E (D; X/S) = σ φ E(φ D; X/S). The claim or N σ (φ D; X/S) ollows, and that or P σ (φ D; X/S) is immediate. Now, by the D-non-negativity hypothesis on φ, KD is an eective -exceptional divisor. By [, Lemma 3.5.], i E is an eective -exceptional divisor, we have N σ ( D + E) = N σ ( D) + E. This means that N σ ( φ D) = N σ (ψ D + KD) = N σ (ψ ( D + KD)) = ψ N σ ( D + KD) = ψ N σ ( D) + ψ KD = ψ N σ ( D) + KD. We have made use o the act that E is eective by the non-negativity hypothesis on D. It is now simple to compute the positive part o the decomposition: P σ ( φ D) = φ D N σ ( φ D) = φ D ψ N σ ( D) KD = ψ D N σ (ψ D) = P σ (ψ D) = ψ P σ ( D). Remark 2. The example o Section 3 can be interpreted as an instance o the calculations in this section. A small lit o the map φ is constructed in Example 2. Let F, F 2 be a basis or N (X/S) dual to C and C 2. A basis or N (Y/S) is given by the our classes (h ) F, (h ) F 2, E, and E 2. The vector v D gives the coeicients or the class o the strict transorm o D on Y with respect to this above basis. Lemma 8 is nothing more than the calculation o the induced map ψ o Lemma 3. The inal calculation in Theorem 9 can then be carried out as a repeated application o Lemma 4. Suppose now that S = Spec C and φ : X X is a pseudoautomorphism whose action φ : N (X) N (X) has a unique largest eigenvalue λ >, and that : Y X is a small lit o φ. Since S = Spec C, the pseudoeective cone is strictly convex, and by a version o the Perron-Frobenius theorem [3] there exists a λ-eigenvector D φ which is contained in the pseudoeective cone. Lemma 3 then provides the existence o a λ-eigenvector D ψ or ψ. We are then able to compute the Zariski decomposition o the divisor D φ using Lemma 4. Corollary 5. Let D φ be the dominant eigenvector o φ : N (X) N (X), and D ψ be the dominant eigenvector o ψ : N (Y ) N (Y ). I D is φ-non-negative, then P σ ( D φ ) = D ψ. Proo. I D is any pseudoeective divisor on X, then or every n we have P σ ( (λ n φ n D)) = λ n ψ n P σ ( D).

13 Take D = D φ + D φ, so that the above reduces to RELATIVE ASYMPTOTIC MULTIPLICITY 3 P σ ( (D φ + λ 2n D φ )) = λ n ψ n P σ ( D). The let hand side converges to P σ ( D φ ) by Proposition 5(). With a suitable choice o scaling, the right hand side converges to D ψ. 5. Acknowledgements I am grateul to James M c Kernan or some useul questions and suggestions. This material is based upon work supported by the National Science Foundation under agreement No. DMS Any opinions, indings and conclusions or recommendations expressed in this material are those o the author and do not necessarily relect the views o the National Science Foundation. Reerences. C. Birkar, P. Cascini, C.D. Hacon, and J. M c Kernan, Existence o minimal models or varieties o log general type, J. Amer. Math. Soc. 23 (200), no. 2, Caucher Birkar, On existence o log minimal models and weak Zariski decompositions, Math. Ann. 354 (202), no. 2, Garrett Birkho, Linear transormations with invariant cones, Amer. Math. Monthly 74 (967), Sébastien Boucksom, Divisorial Zariski decompositions on compact complex maniolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), no., Lawrence Ein, Robert Lazarseld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa, Asymptotic invariants o base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, Tom Graber, Joe Harris, and Jason Starr, Families o rationally connected varieties, J. Amer. Math. Soc. 6 (2003), no., Yujiro Kawamata, Crepant blowing-up o 3-dimensional canonical singularities and its application to degenerations o suraces, The Annals o Mathematics 27 (988), no., , On the cone o divisors o Calabi-Yau iber spaces, Internat. J. Math. 8 (997), no. 5, , Remarks on the cone o divisors, Classiication o algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 20, pp John Lesieutre, The diminished base locus is not always closed, Compos. Math. 50 (204), no. 0, Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 4, Mathematical Society o Japan, Tokyo, Miles Reid, Minimal models o canonical 3-olds, Algebraic varieties and analytic varieties (Tokyo, 98), Adv. Stud. Pure Math., vol., North-Holland, Amsterdam, 983, pp Institute or Advanced Study, Einstein Dr, Princeton, NJ 08540, USA address: johnl@math.ias.edu