K-semistability is equivariant volume minimization

Transcription

1 K-semistability is equivariant volume minimization Chi Li June 23, 27 Abstract This is a continuation to the paper [Li5a] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-instein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of V, K V is equivalent to the condition that the normalized volume is minimized at the canonical valuation ord V among all C -invariant valuations on the cone associated to any positive Cartier multiple of K V. In this case, it s shown that ord V is the unique minimizer among all C -invariant quasi-monomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V. Contents Introduction 2 2 Preliminaries 3 2. Normalized volumes of valuations K-semistability and Ding-semistability Statement of Main Results 7 4 K-semistable implies equivariant volume minimizing 9 4. A general volume formula A summation formula Case of C -invariant valuations Appendix: Interpolations in the C -invariant case Uniqueness among C -invariant valuations 7 6 quivariant volume minimization implies K-semistability 9 6. Valuation wt β on X Valuation w β on X w β vesus wt β Completion of the proof of Theorem An example of calculation Proof of Theorem Two examples of normalized volumes 34 9 Appendix: Convex geometric interpretation of volume formulas Integration formula by Okounkov bodies and coconvex sets xamples Postscript Note: 44 Acknowledgment 45

2 Introduction Valuation theory is a classical subject which has been studied in different fields of mathematics for more than a century. As a celebrated example of their applications in algebraic geometry, Zariski proved the resolution of surface singularities by proving a local uniformization theorem showing that every valuation of a surface can be resolved. Recently valuations have also become important in studying singularities of plurisubharmonic functions in complex analysis see for example [BFJ8]. In this paper, we use the tool of valuation to study K-stability of Fano varieties. The concept of K-stability was introduced by Tian [Tia97] in his study of Kähler-instein problem on any Fano manifold. It was introduced there to test the properness of so-called Mabuchi-energy. This energy is defined on the space of smooth Kähler potentials whose critical points are Kähler-instein metrics. Tian proved in [Tia97] that the properness of the Mabuchi-energy is equivalent to the existence of a Kähler-instein metric on a smooth Fano manifold without holomorphic vector field. The notion of K-stability was later generalized to a more algebraic formulation in [Don2]. By the work of many people on Yau-Tian- Donaldson conjecture spanning a long period of time see in particular [Tia97], [Ber5] and [CDS5, Tia5], we now know that, for a smooth Fano manifold V, the existence of Kähler-instein metric on V is equivalent to K-polystability of V. In [Li3], based on recent progresses on the subject, the author proved some equivalent characterizations of K- semistability. Roughly speaking, for smooth Fano manifolds, K-semistable is equivalent to almost Kähler-instein see [Li3] for details. In particular, it was proved there that K-semistability is equivalent to the lower boundedness of the Mabuchi-energy. Motivated by the work of [MSY8] in the study of Sasaki-instein metric, which is a metric structure closely related to the Kähler-instein metric, in [Li5a] the author proposed to understand K-semistability of Fano manifolds from the point view of volume minimizations. In this point of view, for any Fano manifold V, we consider the space of real valuations on the affine cone X = CV, K V with center being the vertex o we denote this space by Val X,o and study a normalized volume functional vol on Val X,o. Such space of valuations were studied in [JM, BFFU3] with close relations to the theory of Berkovich spaces. The volume functional volv of a real valuation v is defined to be A X v n volv, where A X v is the log discrepancy of v see [JM, BFFU3, Kol3] and volv is the volume of v see [LS3, Laz96, BFJ2]. In [Li5a], we conjectured that the Fano manifold V is K-semistable if and only if vol is minimized at the canonical valuations ord V. Here we prove one direction of this conjecture, i.e. volume minimizing implies K-semistability. Indeed we prove that any Q-Fano variety V is K-semistable if and only if vol is minimized among C -invariant valuations see Theorem 3.. The key idea to prove our result is to extend the calculations in [Li5a] to show a generalization of a result in [MSY8] in which they showed that the derivative of a volume functional on the space of normalized Reeb vector fields is the classical Futaki invariant originally defined in [Fut83]. Here we will show that the derivative of the normalized volume at the canonical valuation ord V is a variant of CM weight that generalizes the classical Futaki invariant. We achieve this by first deriving an integral formula for the normalized volumes for any real valuations centered at o. The derivation of this formula uses the tool of filtrations see [BC] and has a very concrete convex geometric meaning which allows us to use the theory of Okounkov bodies [Oko96, LM9] and coconvex sets [KK4] to understand it. Given this formula, then as in [Li5a] we can take its derivative and use the recent remarkable work of Fujita [Fuj5b] to show that Ding-semistability implies equivariant volume minimization, where Ding-semistability is a concept derived from Berman s work in [Ber5] and is shown very recently to be equivalent to K-semistability in [BBJ5, Fuj6]. Moreover, the volume formula shows that any C -invariant minimizer has a Dirac-type Duistermaat- Heckman measure. This together with some gap estimates allow us to conclude the uniqueness of C -invariant minimizing valuation that is quasi-monomial Theorem 3.4. For the other direction of implication, we directly relate the volume function on X i.e. on Val X,o to that on some degeneration of X which appears in the definition of K-stability. Recently it has been clear that the general case can be reduced to the equivariant case considered in this paper, see [LX6]. 2

3 The argument in this part also depends on the work of [BHJ5], which seems be the first to systematically use the tool of valuations to study K-stability. On the other hand, we will also use our earlier work in [LX4] which essentially helps us to reduce the test of K-stability to Tian s original definition. As a spin-off of our proof, we get the an equivalent characterization of K-semistability using divisorial valuations in Theorem 3.7. This generalizes a result about Fujita s divisorial stability in [Fuj5a] see also [Li5a]. We end this introduction by remarking that while the work here is a natural continuation of the idea initiated in [Li5a], the techniques and tools used here are mostly independent of those used in [Li5a] and so this paper can be read independent of [Li5a]. 2 Preliminaries 2. Normalized volumes of valuations In this section, we will briefly recall the concept of real valuations and their associated invariants see [ZS6] and [LS3] for details. We will also recall normalized volume functional defined in [Li5a]. Let X = SpecR be an n-dimensional normal affine variety. A real valuation v on the function field CX is a map v : CX R, satisfying:. vfg = vf + vg; 2. vf + g min{vf, vg}. In this paper we also require vc =, i.e. v is trivial on C. Denote by O v = {f CX; vf } the valuation ring of v. Then O v is a local ring. The valuation v is said to be finite over R, or on X, if O v R. Let Val X denote the space of all real valuations that are trivial on C and finite over R. For any v Val X, the center of v over X, denoted by center X v or c X v, is defined to be the image of the closed point of SpecO v under the map SpecO v SpecR = X. For any v Val X, let Γ v = vr denote its valuation semigroup and Γ v denote the abelian group generated by Γ v inside R. Denote by rat.rank v the rational rank of the abelian group Γ v. Denote by tr.deg v the transcendental degree of the field extension K v /C where K v is the residue field of O v. Abhyankar proved the following inequality in [Abh56]: tr.deg v + rat.rank v n. v is called Abhyankar if the equality in holds. A valuation is called divisorial if tr.deg v = n and hence rat.rank v = by. Divisorial valuations are Abhyankar. Since we are working in characteristic, it s well-known that Abhyankar valuations are quasi-monomial see [LS3, Proposition 2.8], [JM, Proposition 3.7]. v Val X,o is quasi-monomial if there exists a birational morphism µ : Y X, and a regular system of parameters z = {z,..., z r } for the local ring O Y,W of v on Y W is the center of v on Y such that vz,..., vz r freely generates the value group Γ v. In this case, we can assume that there exist r Q-linearly independent positive real numbers α,..., α r such that v is defined as follows. For any f R, we can expand µ f = m Z c r m z m with each c m either or a unit in Ô Y,W, and then: { r } vf = min m i α i ; c m. i= Following [JM], we will also use log smooth models in addition to algebraic coordinates for representing quasi-monomial valuations. A log smooth pair over X is a pair Y, D with Y regular and D a reduced effective simple normal crossing divisor, together with a proper birational morphism µ : Y X. We will denote by QM η Y, D the set of all quasi-monomial valuations v that can be described in the above fashion at the point η Y with respect to coordinates {z,..., z r } such that z i defines at η an irreducible component D i of D. We put QMY, D = η QM η Y, D. From now on in this paper, we assume that X has Q-Gorenstein klt singularity. Following [JM] and [BFFU3], we can define the log discrepancy for any valuation v Val X. This is achieved in three steps in [JM] and [BFFU3]. Firstly, for a divisorial valuation ord associated to a prime divisor over X, define A X = ord K Y/X +, where π : Y X is 3

4 a smooth model of X containing. Next for any quasi-monomial valuation v QM η Y, D where Y, D = N k= D k is log smooth and η is a generic point of an irreducible component of D D N, we define A X v = N k= vd ka X D k. Lastly for any valuation v Val X, we define A X v = sup A X r Y,D v Y,D where Y, D ranges over all log smooth models over X, and r Y,D : Val X QMY, D are contraction maps that induce a homeomorphism Val X lim QMY, D. For details, see Y,D [JM] and [BFFU3, Theorem 3.]. We will need one basic property of A X : for any proper birational morphism Y X, we have see [JM, Remark 5.6], [BFFU3, Proof 3.]: A X v = A Y v + vk Y/X. 2 From now on, we also fix a closed point o X with the corresponding maximal ideal of R denoted by m. We will be interested in the space Val X,o of all valuations v with center X v = o. If v Val X,o, then v is centered on the local ring R m see [LS3]. In other words, v is nonnegative on R m and is strictly positive on the maximal ideal of the local ring R m. For any v Val X,o, we consider its valuative ideals: a p v = {f R; vf p}. Then by [LS3, Proposition.5], a p v is m-primary, and hence is of finite codimension in R cf. [AM69]. We define the volume of v as: volv = lim p + dim C R/a p v p n. /n! By [LS3, Mus2, LM9, Cut2], the limit on the right hand side exists and is equal to the multiplicity of the graded family of ideals a = {a p }: volv = ea p lim p + p n =: ea, 3 where ea p is the Hilbert-Samuel multiplicity of a p. Now we can define the normalized volume for any v Val X,o see [Li5a]: volv = { AX v n volv, if A X v < +, +, if A X v = +. The following estimates were proved in [Li5a]. The second estimate motivates the definition of volv = + when A X v = +. Theorem 2. [Li5a, Corollary 2.6, Theorem 3.3]. Let X, o be a Q-Gorenstein klt singularity. The following uniform estimates hold:. There exists a positive constant K = KX, o > such that volv > K for any v Val X,o. 2. There exists a positive constant c = cx, o > that volv c A Xv vm, for all v Val X,o with A X v < +. From now on in this paper, we will always assume our valuation v satisfies A X v < +. Notice that volv is rescaling invariant: volλv = volv for any λ >. By Izumitype theorem proved in [Li5a, Proposition 2.3], we know that vol is uniformly bounded from below by a positive number on Val X,o. If vol has a global minimum v on Val X,o, i.e. volv = inf v ValX,o volv, then we will say that vol is globally minimized at v over X, o. In [Li5a], the author conjectured that this holds for all Q-Gorenstein klt singularities and proved that this is the case under a continuity hypothesis. 2 In this paper, we will be interested in the following C -invariant setting. 2 H.Blum recently confirmed this conjecture in [Blum6] 4

5 Definition 2.2. Assume that there is a C -action on X, o. We denote by Val X,o C the space of C -invariant valuations in Val X,o. We say that vol is C -equivariantly minimized at a C -invariant valuation v Val X,o C if volv = inf volv. v Val X,o C 2.2 K-semistability and Ding-semistability In this section we recall the definition of K-semistability, and its recent equivalence Dingsemistability. Definition 2.3 [Tia97, Don2], see also [LX4]. Let V be an n -dimensional Q-Fano variety.. For r > Q such that r K V = L is Cartier. A test configuration resp. a semi test configuration of X, L consists of the following data A variety V admitting a C -action and a C -equivariant morphism π : V C, where the action of C on C is given by the standard multiplication. A C -equivariant π-ample resp. π-semiample line bundle L on V such that V, L π C\{} is equivariantly isomorphic to V, r K V C with the natural C -action. A test configuration is called a special test configuration, if the following are satisfied V is normal, and V is an irreducible Q-Fano variety; L = r K V/C. 2. Assume that V, L C is a normal test configuration. Let π : V, L P be the natural equivariant compactification of V, L C. The CM weight of V, L is defined by the intersection formula see [Wan2, Oda3]: CMV, L = n r n n K V n Ln + nr n Ln K V/P. 3. V is called K-semistable if CMV, L for any normal test configuration V, L/C of V, r K V. V is called K-polystable if CMV, L for any normal test configuration V, L/C of V, r K V, and the equality holds if and only if V = V C. Remark 2.4. For any special test configuration, the C -action on V induces a natural action on K V by pushing forward the holomorphic n, -vectors. Because L = r K V, we can always assume that the C -action on L is induced from this natural C -action. Moreover, for special test configurations, CM weight reduces to a much simpler form: CMV, L = rn Ln n K V n = K V/P n. 4 n K V n We will need the concept of Ding-semistability, which was derived from Berman s work in [Ber5]. Definition 2.5 [Ber5, Fuj5b].. Let V, L/C be a normal semi-test configuration of V, r K V and V, L/P be its natural compactification. Let D V,L be the Q- divisor on V satisfying the following conditions: The support SuppD V,L is contained in V. The divisor r D V,L is a Z-divisor corresponding to the divisorial sheaf Lr K V/P. 2. The Ding invariant Ding invariant DingV, L of V, L/C is defined as: DingV, L := rn Ln n K V n lctv, D V,L ; V. 3. V is called Ding semistable if DingV, L for any normal test configuration V, L/C of V, r K V. 5

6 Notice that DingV, L = CMV, L for special test configurations. It was proved in [LX4] that to test K-semistability or K-polystability, one only needs to test on special test configurations. By the recent work in [BBJ5, Fuj6], the same is true for Ding-semistability. Moreover we have the following equivalence result: Theorem 2.6 [BBJ5, Fuj6]. For a Q-Fano variety V, V is K-semistable if and only if V is Ding-semistable. Remark 2.7. In [BBJ5] Berman-Boucksom-Jonsson outlined a proof of the above result when V is a smooth Fano manifold. They showed that it s sufficient to check Ding semistability on special test configurations. This was achieved by showing that the normalized Ding invariant is decreasing along a process that transforms any test configuration into a special test configuration. This process was first used in [LX4] for proving results on K-semistability of Q-Fano varieties. The equivalence of Ding-semistability and K-semistability for the general case of Q-Fano varieties has been proved in [Fuj6] by similar and more detailed arguments. We will need another result of Fujita, which was proved by applying a criterion for Ding semistability [Fuj5b, Proposition 3.5] to a sequence of semi-test configurations constructed from a filtration. We here briefly recall the relevant definitions about filtrations and refer the details to [BC] see also [BHJ5] and [Fuj5b, Section 4.]. Definition 2.8. A good filtration of a graded C-algebra R = + m= R m is a decreasing, multiplicative and linearly bounded R-filtrations of R. In other words, for each m Z, there is a family of subspaces {F x R m } x R of R m such that:. F x R m F x R m, if x x ; 2. F x R m F x R m F x+x R m+m, for any x, x R and m, m Z ; 3. e min F > and e max F < +, where e min F and e max F are defined by the following operations: e min R m, F = inf{t R; F t R m R m }; e max R m, F = sup{t R; F t R m }; e min R m, F e min F = e min R, F = lim inf ; m + m e max R m, F e max F = e max R, F = lim sup. i + m 5 Following [BHJ5], given any good filtration {F x R m } x R and m Z, the successive minima is the decreasing sequence where N m = dim C R m, defined by: λ m max = λ m λ m N m = λ m min λ m j = max { λ R; dim C F λ R m j }. Denote R t = + k= F kt R k. When we want to emphasize the dependence of R t on the filtration F, we also denote R t by FR t. From now on, we let R m = H V, ml. The following concept of volume will be important for us: vol R t = vol FR t := lim sup k + dim C F mt H V, ml m n. 6 /n! We need the following lemma by in [BHJ5] in our proof of uniqueness result. Proposition 2.9 [BC], [BHJ5, Corollary 5.4]. ν m := N m j δ m λ m j. The probability measure = d dim C F mt H V, ml dt N m converges weakly as m + to the probability measure: ν := V d vol R t = V d dt vol R t dt. 6

7 2. The support of the measure ν is given by supp ν = [λ min, λ max ] with { λ min := inf t R vol R t < L n } ; 7 λ max = lim m + λ m max m = sup m λ m max m. 8 Moreover, ν is absolutely continuous with respect to the Lebesgue measure, except perhaps for a Dirac mass at λ max. Following [Fuj5b], we also define a sequence of ideal sheaves on V : I F m,x = Image F x R m L m O V, 9 and define F x R m := H V, L m I F m,x to be the saturation of F x R m such that F x R m F x R m. F is called saturated if F x R m = F x R m for any x R and m Z. Notice that with our notations we have: vol FR t := lim sup k + dim C F kt H V, kl k n. /n! The following result of Fujita will be crucial for us in proving that Ding-semistability implies volume minimization. Theorem 2. [Fuj5b, Theorem 4.9]. Assume V, K V is Ding-semistable. Let F be a good R-filtration of R = + m= H V, ml where L = r K V. Then the pair V C, Ir t d is sub log canonical, where I m = Im,me F + + IF m,me + t + + Im,me F + tme+ e + t me+ e, r n e+ d = re + e + K V n vol FR t dt and e +, e Z with e + e max R, F and e e min R, F. Remark 2.. By Theorem 2.6, we can replace Ding-semistable by K-semistable in Fujita s result. 3 Statement of Main Results Let V be a Q-Fano variety. If L = r K V is an ample Cartier divisor for an r Q >, we will denote the affine cone by C r := CV, L = Spec + k= H V, kl. Notice that C r has Q-Gorenstein klt singularities at the vertex o see [Kol3, Lemma 3.]. C r, o has a natural C -action. Denote by v the canonical C -invariant divisorial valuation ord V where V is considered as the exceptional divisor of the blow up Bl o X X. The following is the main theorem of this paper, which in particular confirms one direction of part 4 of the conjecture in [Li5a]. Theorem 3.. Let V be a Q-Fano variety. The following conditions are equivalent:. V, K V is K-semistable; e 2. For an r Q > such that L = r K V is Cartier, vol is C -equivariantly minimized at ord V over C r, o ; 3. For any r Q > such that L = r K V is Cartier, vol is C -equivariantly minimized at ord V over C r, o. xample Over C n,, vol is globally minimized at the valuation v = ord, where = P n is the exceptional divisor of the blow up Bl o C n C n. This follows from an inequality of de-fernex-in-mustaţă as pointed out in [Li5a]. As a corollary we have a purely algebraic proof of the following statement: 7

8 Corollary 3.3. P n is K-semistable. As pointed out in [PW6], an algebraic proof of this result could also be given by combining Kempf s result on Chow stabilities of rational homogeneous varieties, and the fact that asymptotic Chow-semistability implies K-semistability. 2. Over n-dimensional A singularity X n = {z zn+ 2 = } C n+ with o =,...,, vol is minimized among C -invariant valuations at v = ord, where = Q the smooth quadric hypersurface is the exceptional divisor of the blowup Bl o X X. This follows from the above theorem plus the fact that Q has Kähler-instein metric and hence is K-polystable. In [Li5a], v was shown to minimize vol among valuations of the special form v x determined by weight x R n+ >. In the following paper [LL6], we will show that v = ord Q indeed globally minimizes vol. Under any one of the equivalent conditions of Theorem 3., we can prove the uniqueness of minimizer among C -invariant quasi-monomial valuations. Further existence and uniqueness results for general klt singularities will be dealt with in a forthcoming paper [LX6]. Theorem 3.4. Assume V is K-semistable. Then ord V is the unique minimizer among all C -invariant quasi-monomial valuations. Through the proof of the above theorem, we get also a characterization of K-semistability using divisorial valuations of the field CV. Definition 3.5. For any divisorial valuation ord F over V where F is a prime divisor on a birational model of V, we define the quantity: Θ V F = A V F r n K V n vol F F R t dt, 2 where L = r K V is an ample Cartier divisor for r Q > and F F R = {F F R k } is the filtration defined in 8 by ord F on R = + k= H V, kl. Remark It s easy to verify that the Θ V F does not depend on the choice of L. See Lemma 7.. In particular when K V is Cartier then we can choose L = K V and r =. 2. If F is a prime divisor on V, then A V F = and Θ V F is nothing but a multiple of Fujita s invariant ηf defined in [Fuj5a]. In particular, the following theorem can be seen as a generalization of result about Fujita s divisorial stability in [Fuj5a] see also [Li5a]. Theorem 3.7. Assume V is a Q-Fano variety. Then V is K-semistable if and only if Θ V F for any divisorial valuation ord F over V. Moreover, if Θ V F > for any divisorial valuation ord F over V, then V is K-stable. The rest of this paper is devoted to the proof of Theorem 3., Theorem 3.4 and Theorem 3.7. Before we go into details, we make some general discussion on idea of the proof of Theorem 3., from which the proof of Theorem 3.4 and the proof Theorem 3.7 will be naturally derived. In Section 4, we will prove that K-semistability, or equivalently Ding-semistability by Theorem 2.6, implies equivariant volume minimization. More precisely we prove 3. To achieve this, we will generalize the calculations in [Li5a] where we compared the volv, the normalized volume of the canonical valuation v = ord V, with volv for some special C -invariant divisorial valuation v. Here we will deal with general real valuations directly using the tool of filtrations. More precisely, we will associate an appropriate graded filtration to any real valuation v with A X v < +, and will be able to calculate volv using the volumes of associated sublinear series. Izumi s theorem is a key to make this work, ensuring that we get linearly bounded filtrations. Motivated by the formulas in the case of C -invariant valuations, we will consider a concrete convex interpolation between volv and volv. By the convexity, we just need to show that the directional derivative of the interpolation at v is nonnegative. The formulas up to this point works for any real valuation not necessarily C -invariant. 8

9 In the C -invariant case, same as for the case in [Li5a], it will turn out that the derivative matches Fujita s formula and his result in Theorem 2. [Fuj5b] gives exactly the nonnegavitity. As mentioned earlier, we will deal with the non-c -invariant case in [LL6]. In Section 6, we will prove the implication 2. Since 3 2 trivially, this completes the proof of Theorem 3.. This direction of the implication depends on the work [BHJ5] and [LX4]. We learned from [BHJ5] that test configurations can be studied using the point of view of valuations: the irreducible components of the central fibre give rise to C -invariant divisorial valuations, which however is not finite over the cone in general. In our new point of view, these C -invariant valuations should be considered as tangent vectors at the canonical valuation. By [LX4], special test configurations are enough for testing K-semistability. For any special test configuration, there is just one tangent vector and so it s much easier to deal with. Then again the point is that the derivative of vol along the tangent direction is exactly the Futaki invariant on the central fibre. So we are done. 4 K-semistable implies equivariant volume minimizing Assume that V is a Q-Fano variety and L = r K V is an ample Cartier divisor for a fixed r Q >. Define R = + i= H V, il =: + i= R i and denote by X = CV, L = SpecR the affine cone over V with the polarization L. Let v = ord V be the canonical valuation on R corresponding to the canonical C -action on X. Theorem 4.. If V is K-semistable, then vol is C -equivariantly minimized at the canonical valuation ord V over X = CV, L, o. The rest of this section is devoted to the proof of this theorem. 4. A general volume formula Let v be any real valuation centered at o with A X v < +. In this section, we don t assume v is C -invariant. We will derive a volume formula for volv with the help of an appropriately defined filtration associated to v. To define this filtration, we decompose any g R = + k= R k into homogeneous components g = g k + + g kp with g kj and k < k 2 < < k p and define ing to be the initial component g k. Then the filtration we will consider is the following: F x R k = {f R k ; g R such that v g x and ing = f} {}. 3 We notice that following properties of F x R.. For fixed k, {F x R k } x R is a family of decreasing C-subspace of R k. 2. F is multiplicative: v g i x i and ing i = f i R ki implies v g g 2 x + x 2, ing g 2 = f f 2 R k+k If Av < +, then F is linearly bounded. Indeed by Izumi s theorem in [Li3, Proposition.2] see also [Izu85, Ree89, BFJ2], there exist, c 2, + such that v v c 2 v. If v g x, then k := v g = v ing c 2 x. This implies that if x > c 2k, then F x R k =. So F is linearly bounded from above. On the other hand, if v f = k, then v f k. This implies that if x k, then F x R k = R k. So F is linearly bounded from below. Note that the argument in particular shows the following relation: inf m v v e min F e max F sup m For later convenience, from now on we will fix the following constant: := inf m v v. 4 v v >. 5 To state the following lemma, we first introduce a notation see [DH9, Section 2]. If Z is a proper closed subscheme of Y and v Val Y, let I Z be the ideal sheaf of Z and define: vz = vi Z := min {vφ; φ I Z U, U center Y v }. 6 9

10 Lemma 4.2. For any v Val X,o, we have the following identity: = v V = inf i> where v V is defined as in 6 and v R i = inf{vf; f R i }. v R i, 7 i Proof. Let µ : Y := Bl o X X be the blow up of o with exceptional divisor V. Then Y can be identified with the global space of the line bundle L V and we have the natural projection π : Y V. For any f H V, il, we have v f = v µ f iv V. To see this, we write f = h s i on any affine open set U of X where h O X U and s is a generator of O V LU. Then µ f = µ h s i can be considered as a regular function on π U = U C. Since V π U = {s = }, we have v s = v V and v µ f = v h s i = v h + iv s iv V. The identity holds if and only v h =, i.e. if and only f does not vanish on the center of v over Y which is contained in V. When m is sufficiently large, ml is globally generated on V so that we can find f H V, ml such that f does not vanish on the center of v on V. Then we have vf v = v f f/m = v µ f/m = v V. So from the above discussion we get: = inf m v v R i inf = v V. v i> i For any t >, there exists f R such that v f/v f < t. Decompose f into components: f = f i + + f ik, where f ij R ij = H V, i j L j k and i < < i k. Then there exists i {i,..., i k } such that v f v f = v f i + + f ik min{v f i,..., v f ik }} i i = v f i v f i v V. i i So we get t > v V for any t >. As a consequence v V. The reason why we can use filtration in 3 to calculate volume comes from the following observation. Proposition 4.3. For any m R, we have the following identity: + k= dim C R k /F m R k = dim C R/a m v. Notice that because of linear boundedness, the sum on the left hand side is a finite sum. More precisely, by the above discussion, when k m/, then dim C R k /F m R k =. Proof. For each fixed k, let d k = dim C R k /F m R k. Then we can choose a basis of R k /F m R k : { } [f k i ] k f k i R k, i d k, where [ ] k means taking quotient class in R k /F m R k. Notice that for k m/, the set is empty. We want to show that the set { } B := [f k i ] i d k, k m/, is a basis of R/a m v, where [ ] means taking quotient in R/a m v. We first show that B is a linearly independent set. For any nontrivial linear combination of [f k i ]: [ N d k N ] d k c k i [f k i ] = c k i f k i = [f k + + f kp ] =: [F ], k= i= k= i= where f kj R kj \ F m R kj and k < k 2 < < k p. In particular inf = f k F m R k. By the definition of F m R k, we know that which is equivalent to [F ] R/a m v. f k + + f kp a m v,

11 We still need to show that B spans R/a m v. Suppose on the contrary B does not span R/a m v. Then there is some k Z > and f R k a m v such that [f] R/a m v can not be written as a linear combination of [f k i ], i.e. not in the span of B. We claim that we can choose a maximal k such that this happens. Indeed, this follows from the fact that the set {v g g R a m v } is finite because v g < m implies v g c v g < m/. So from now on we assume that k has been chosen such that for any k > k and g R k, [g] is in the span of B. Then there are two cases to consider.. If f R k \ F m R k, then since {[f k i ] k } is a basis of R k /F m R k, we can write f = d k j= c jf k j + h k where h k F m R k. So there exists h a m v such that inh = h k and f = d k j= c jf k j + h. By the maximality of k, we know that [h k h] = [h k ] is in the span of B. Then so is [f] = d k j= c j[f k j ] R/a m v. This contradicts the condition that [f] is not in the span of B. 2. If f F m R k R k, then by the definition of F m R k, f +h a m v for some h R such that inf + h = f. Since we assumed that [f] R/a m v, we have h and k := v h > v f = k. Now we can decompose h into homogeneous components: h = h k + + h kp, with h kj R kj and k = k < < k p. Because k > k and the maximal property of k, we know that each [h kj ] in the span of B. So we have [f] = [f + h h] = [ h] is in the span of B. This contradicts our assumption that [f] is not in the span of B. The above proposition allows us to derive a general formula for volv with the help of F defined in 3. Indeed, we have: + n! dim C R/a m v = n! dim C R k /F m R k For the first part of the sum, we have: m/c n! k= k= m/c = n! dim C R k dim C F m R k. 8 k= dim C R k = mn c n L n + Om n. 9 For the second part of the sum, we use Lemma 4.5 in Section 4.. to get: lim m + m/c n! m n k= dim C F m R k = n vol R t dt. 2 tn+ Combining 8, 9 and 2, we get the first version of volume formula: volv = lim p + = c n L n n n! m n dim C R/a m v 2 vol R t dt t n+.

12 We can use integration by parts to get a second version: volv = c n L n + vol R t d t n [ = ] L n + vol R t + = c n t n dvol R t t n dvol R t t n. 22 Motivated by the case of C -invariant valuations see 37 in Section 4.2, we define a function of two parametric variables λ, s, + [, ]: Φλ, s = = λ s + s n Ln n vol R t λsdt s + λst n+ dvol R t s + λst n. 23 In Section 9, we will interpret and re-derive the above formulas 2-23 using the theory of Okounkov bodies and coconvex sets. The usefulness of Φλ, s can be seen from the following lemma. Lemma 4.4. Φλ, s satisfies the following properties:. For any λ, +, we have: Φ λ, = volλv = λ n volv, Φλ, = volv = L n. 2. For fixed λ, +, Φλ, s is continuous and convex with respect to s [, ]. 3. The derivative of Φλ, s at s = is equal to: Φ s λ, = nλl n λ L n vol R t dt. 24 Proof. The first item is clear. To see the second item, first notice that dvolr t has a positive density with respect to the Lebesgue measure dt since volr t is decreasing with respect to t R. Moreover it has finite total measure equal to volr x = L n for any x. So the claim follows from the fact that the function s / s + λst n is continuous and convex. The third item follows from direct calculation using the formula in 23. Roughly speaking, the parameter λ is a rescaling parameter, and s is an interpolation parameter. To apply this lemma to our problem, we let λ = r A X v =: λ such that Φλ, = A Xv n volv r n = volv r n. Recall that volv = r n L n. So our problem of showing volv volv is equivalent to showing that Φλ, Φλ,. By item 2-3 of the above lemma, we just need to show that Φ s λ, is non-negative. This will be proved in the C -invariant case in the next section. 4.. A summation formula Lemma 4.5. Let {F x R i } x R be a good filtration as in Definition 2.8. For any α R and β >, we have the following identity: αp/β+c n! lim p + p n dim C F αp βi R i 25 i= = n 2 vol R x α n dx β + x n+.

13 Proof. Define φy = dim C F αp βy R y. Then φy is an increasing function on [m, m + for any m Z and φy dim C R y Cy n. Moreover, because F x is decreasing in x, φy dim C F αp β y R y. So we have: αp/β+c i= We notice that: dim C F αp βi R i lim sup p + = φαp/β + x p n /n! αp/β+ i= dim C F αp βi R i + dim C R αp/β+c αp/β+c φydy + Op n p = lim sup p + dim C F x αp/β+x R αp/β+x lim sup p + αp/β + x n /n! = vol R x α n β + x n. αp αdx φ β + x β + x 2 + Op n. dim C F αpx/β+x R αp/β+x p n /n! αp/β + x n α n αp/β + x n β + x n The last identity holds by [LM9] and [BC] see also [BHJ5, Theorem 5.3]. So by Fatou s lemma, we have: lim sup p + n lim sup n n p + n! αp/β+c p n i= lim sup p + vol n! p n φ R x dim C F αp βi R i φαp/β + x p n /n! αp αdx β + x β + x 2 + Op αdx β + x 2 α n dx. 26 β + x n+ Similarly we can prove the other direction. Define ψy = dim C F αp βy R y. Then ψy is an increasing function on m, m + ] for any m Z and satisfies ψy dim C R y Cy n. Moreover, ψy dim C F αp β y R y. So we have: αp/β+c i= dim C F αp βi R i We can then estimate: = αp/β+ i= dim C F αp βi R i dim C R αp/β+c αp/β+c ψydy + Op n p αp αdx ψ β + x β + x 2 + Op n. ψαp/β + x dim C F αpx/β+x R αp/β+x lim inf p + p n = lim inf /n! p + p n /n! dim C F x αp/β+x R αp/β+x lim inf p + αp/β + x n /n! = vol R x α n β + x n. αp/β + x n α n αp/β + x n β + x n 3

14 Similar as before, by using Fatou s lemma, we get the other direction of the inequality: n! lim inf p + n lim inf p + n n αp/β+c p n i= lim inf p + vol n! p n ψ dim C F αp βi R i ψαp/β + x p n /n! R x αp αdx β + x β + x 2 + Op αdx β + x 2 Combining 26 and 27, we get the identity 25 we wanted: αp/β+c n! lim p + p n dim C F αp βi R i = n i= α n dx. 27 β + x n+ vol R x α n dx β + x n Case of C -invariant valuations v = ord V corresponds to the canonical C -action on R given by a f k = a k f k for f k R k and a C. From now on, v is assumed to be C -invariant: v a f = v f for any f R and a C. Let w := v CV. Then it s easy to see that v is a C -invariant extension of w in the following way. First choose an affine neighborhood U of center V w and a local generator s of O X LU. ach f k R k = H V, kl can be written as f k = h s k on the open set U. Define wf k := wh. Then for any f k R k = H V, kl, we have: v f k = wf k + k where = vv is the constant appearing in Lemma 4.2. More generally if f = k f k R = k R k, we have: { } v f = min wf k + k; f = k f k with f k. 28 By 28 it s clear that the valuative ideals a m v are homogeneous with respect to the N-grading of R = k N R k. Lemma 4.6. Assume that v is C -invariant. F = {F x } x R defined in 3 is equal to: F x R k = a x v R k = {f R k v f x}. 29 As a consequence, F is left-continuous: F x R m = x <x F x R m and saturated. Proof. The identity 29 follows easily from the C -invariance of v and the left continuity is clear from 29. We just need to show the saturatedness. Let π : Y = Bl o X X be the blow up of o on X. For any f H V, ml, locally we can write f = h s where s is a local trivializing section of L m on an open neighborhood of center Y v V. Then we have v f = v π hs = v π h + v V m. So we have v f x if and only if v π h = v h x m. From this we see that v Im,x F x m. If f H V, ml Im,x F, then locally f = h s with h IF m,x. So we have: v f = v h + m x and hence f F x H V, ml. Remember that we want to prove Φ s λ,. By 24 and λ = r/a X v, we have Φ s λ, = nλ L n λ L n vol R t dt 3 = nln A X v r r A X v L n vol R t dt. 3 4

15 Under the blow-up π : Y X, we have π K X = K Y r V see [Kol3, 3.]. So by the property of log discrepancy in 2, we have Φ s λ, = nln A X v A X v = A Y, rv v = A Y v + r v V = A Y v + r. by 7 This gives A X v r = A Y v. So, using also L = r K V, we get another expression for Φ s λ, in 3: A Y v vol FR t dt 32 = nln A X v A Y v + r n K V n r n K V n Next we bring in Fujita s criterion for Ding-semistability in [Fuj5b]. denote F x = F cx. By 4 and 5 we can choose e + with e + sup m v v inf m v v e maxf = e max F volfr ct dt. For simplicity, we and e =. Notice that by relation 4, e min F = e min F/. Denote by the I V the ideal sheaf of V as a subvariety of Y. Notice that Y is nothing but the total space of the line bundle L V so that we have a canonical projection denoted by ρ : Y V. Now similar to that in [Fuj5b], we define: I F m,x = Image F x R m L m O V Ĩ m = ρ I F m,me+ + ρ I F m,me+ I V + + ρ I F m,me + Ime+ e V + I me+ e V. 33 Then I F m,x resp. Ĩ m is an ideal sheaf on V resp. Y. Moreover, Ĩ := {Ĩm} m is a graded family of coherent ideal sheaves by [Fuj5b, Proposition 4.3]. By Lemma 4.2, we have v I V =. On the other hand, using Lemma 4.6, we get: v I F m,x = v ImageF x R m L m O V x m. 34 Combining 34 and the inclusion I me+ e V me + e v I V = m e + and hence Ĩm, we get v Ĩm = v I me+ e V = v Ĩ = inf m We define d as in Fujita s Theorem 2.: Then we get: d = re + e + v Ĩm m = e + e = e +. r n K V n v Ĩr I d V = re + e + d = + = + r n K V n e+ vol FR t dt. r n K V n vol FR ct dt e+ Comparing the above expression with that in 32 we get the estimate: Φ s λ, = nln A Y v v A X v Ĩr I d + nln A X v nln A X v r n K V n V A Y v v Ĩr I d V. 5 vol FR t dt volfr ct volfr ct dt

16 Note that the last inequality is actually an equality because F is saturated by Lemma 4.6. So to show Φ s λ,, we just need to show A Y v v Ĩr I d V. 35 We prove this by deriving the following regularity from Fujita s result. Lemma 4.7. Assume that V, K V is K-semistable. Then Y, Ĩr I d is sub log canonical. V Proof. By Fujita s Theorem 2. [Fuj5b, Theorem 4.9] and Remark 2., V C, I r t d is sub log canonical where I = {I m } is the ideal sheaf on V C defined in. Informally we get I on V C if in 33 ρ is replaced by the canonical fibration ρ : V C V. By choosing an affine cover of {U i } of V, we have Ĩr m I d V ρ U i = Im r t d ρ U i for any m. Since sub log canonicity can be tested by testing on all open sets of an affine cover, we get the conclusion. By the above Lemma, we get 35 using the same approximation argument as in [BFFU3, Proof of Theorem 4.]. Because the space of divisorial valuations is dense in Val X,o we want to use some semicontinuity properties to get the inequality 35 that already holds for divisorial valuations. More precisely, the sub log canonicality means that, for any divisorial valuation ord F over Y, we have: A Y F r ord F Ĩ d ord F I V. Consider the function φ : Val X,o R {+ } defined by: φv = A Y v rvĩ d vi V = A X v rvĩ r + d vi V. If we endow Val X,o with the topology of pointwise convergence, then v vi V is continuous by [BFFU3, Proposition 2.4]; v vĩ is upper semicontinuous by [BFFU3, Proposition 2.5]; v A Y v is lower semicontinuous by [BFFU3, Theorem 3.]. Combining these properties we know that φ is a lower semi-continuous function on Val X,o. Furthermore it was shown in [BFFU3] that φ is continuous on small faces of Val X,o and satisfies φ φ r τ for any τ a good resolution dominating the blow up τ : Y X we refer to [BFFU3] for the definition of the contraction r τ and the notions of small faces and good resolutions. This allows us to show that φ on the space of divisorial valuations implies φv for any valuation v Val X,o Appendix: Interpolations in the C -invariant case Assume v is C -invariant. Let w = v CV be its restriction as considered at the beginning of Section 4.2. We can connect the two valuations v and v by a family of C -invariant valuations v s for s [, ]. v s is defined as the following C -invariant extension of sw: for any f k R k define: v s f k = swf k + s + sk = sv f k + s v f k. 36 The second identity follows from 28. For f R, define { v s f = min v s f k ; f = k f k with f k }. Because >, it s easy to see that center X v s = o, i.e. v s Val X,o. From 36 v s is indeed an interpolation between v = ord V and the given valuation v. The valuation v s is C -invariant and its valuative ideals a m v s are homogeneous under the natural N-grading of R = k R k. Proposition 4.8. Assume that v is C -invariant. The volume of v s defined above is equal to Φ, s defined in 23. In other words, we have the formula: dvol R t volv s = s + st n. 37 6

17 Proof. Because a m v s is homogeneous and using 36 and Lemma 4.6, we have: dim C R/a m v s = = = + i= + i= + i= dim C R i / a m v s R i dimc R i dim C am si/s v R i dim C R i dim C F m si s R i. Because F x R i = R i if x i by the definition of see the discussion before Lemma 4.2, we deduce that: F m si s R i = R i if So we get the following identity: m si s i or equivalently i m s + s. m/s+ s n! dim C R/a m v s = n! dim C R i dim C F m si s R i. 38 For the first part of the sum, we have: m/s+ s n! i= dim C R i = i= For the second part of the sum, we use Lemma 4.5 to get: lim m + m/s+ s n! m n i= Combining 38, 39 and 4, we have: volv s = lim m + = m n s + s n Ln + Op n. 39 dim C F m si s R i 4 = n s + s n Ln n vol R t sdt s + st n+. n! m n dim C R/a m v s 4 vol R t sdt s + st n+. We can use integration by parts to further simplify the formula: + volv s = s + s n Ln + vol R t d s + st n = s + s n Ln + [ ] vol R t + + dvolr t st + s n t= st + s n dvol R t = st + s n Uniqueness among C -invariant valuations In this section, we prove Theorem 3.4. We first prove the result for divisorial valuations. Suppose 2 is a prime divisor centered at o such that volord 2 = volord V. We want to 7

18 show that 2 = V. The main observation is that the interpolation function Φλ, s in 23 must be a constant function independent of s [, ]. Indeed in this case Φs := Φλ, s is a convex function satisfying Φ = Φ = min s [,] Φs. So Φs Φ = Φ. Recall the expression in 23: Φs = λ s + s n Ln n c2 vol R t λ sdt. 43 s + λ st n+ Here we have changed the improper integral to a finite integral by choosing c 2 such that vol R t = for t c 2. Because vol R t is piecewise continuous on [, c 2 ], we easily verify that Φs in 43 is a smooth function of s [, ]. We can calculate its second order derivative: Φ s = n + nλ 2 λ s + s nn + Ln n+2 c2 c2 = nn + λ t 2 dvol λ st + s vol R t λ λ t 2 + nsλ t dt s + λ st n+3 R t n The second identity follows from integration by parts. By Proposition 2.9, ν = L dvol R t, n supported on the interval [λ min, λ max ], is absolutely continuous on [λ min, λ max with respect to the Lebesgue measure and possibly has a Dirac mass at λ max. We also know that Φ s for s [, ] since Φs is a constant. Using the last expression in 44, we see that λ max = λ min and ν = δ λmax. Otherwise, Φ s is not identically equal to zero on the nonempty open interval λ min, λ max. We will show that ν being a Dirac measure indeed implies ord 2 = ord V. The latter statement can be thought of a counterpart of the result in [BHJ5, Theorem 6.8]. Indeed, the argument given below is motivated the proof in [BHJ5, Lemma 5.3]. Let ord 2 denote the restriction of ord 2 under the inclusion CV CX. As before, v := ord 2 is a C -invariant extension of ord 2 in the following way. Choose an affine open neighborhood U of center V ord 2 and a trivializing section s O V LU, then any f H V, kl can be locally written as f = h s k. We define ord 2 f = ord 2 h such that ord 2 f = ord 2 h j + k ord 2 s j = ord 2 h j + k = ord 2 f + k. In particular we have, F kx R k = F kx c ord 2 R k = {f R k ; ord 2 f kx }. Let Z be the center of ord 2 on V. Then by general Izumi s theorem see [HS], we have: ord 2 h C 2 wh, for any h O V,Z, 45 where w is a Rees valuation of Z. Let V V be the normalized blow up of Z inside V and G = µ Z. The Rees valuations are given up to scaling by vanishing order along irreducible components of G see [Laz4, xample 9.6.3], [BHJ5, Definition.9]. By Izumi s theorem, the Rees valuations of Z are comparable to each other [Ree89, HS]. So for any x > and < δ, we have: F kx R k = F kx c R ord k H Bl Z V, kl δg. 2 Since µ L δg is ample on V for < δ, this implies: volfr x µ L δg n < L n for any x >. So λ min = = inf m ord 2 ord V by 7 in Lemma 2.9. For any x < λ max, volr x > by [BC, Lemma.6] see also [BHJ5, Theorem 5.3]. It s easy to see that 8

19 ord λ max = sup 2 m ord V =: c 2 by 8 and our definition of filtration FR in the C -invariant case. So if dvolr x is the Dirac measure, then = c 2 and hence ord 2 = ord V. It s clear that the above proof works well for a C -invariant valuation v as long as the general Izumi s inequality as in 45 holds for w := v CV. In particular, this holds for any C -invariant quasi-monomial valuation v. Indeed, if v is a C -invariant quasimonomial valuation. Then w = v CV is also a quasi-monomial valuation by Abhyankar- Zariski inequality see [BHJ5,.3]. Denote Z = center V w. We can find a log smooth model W, i i V such center V i = Z and w is a monomial valuation on the model W, i i. Since the general Izumi inequality holds for ord i, it also holds for w cf. [BFJ2]. Remark 5.. We expect that the generalized Izumi inequality holds for any valuation v of CV with A V v < +, i.e. there exists C = Cv such that v h Cwh for any h O V,Z where Z = center V v and w is a Rees valuation of Z. If Z is a closed point, this is indeed true see [Li5a]. The general case can probably be reduced to the case of closed point cf. [JM, Proof of Proposition 5.]. 6 quivariant volume minimization implies K-semistability In this section, we will prove the other direction of implication. Theorem 6.. Let V be a Q-Fano variety and L = r K V an ample Cartier divisor for a fixed r Q >. Assume that vol is minimized among C -invariant valuations at ord V on X := CV, L. Then V is K-semistable. As explained in Section 3, this combined with Theorem 4. will complete the proof of Theorem 3.. Indeed, Theorem 4. shows 3 and 3 2 trivially. The rest of this section is devoted to the proof of this theorem. By [LX4], to prove the K-semistability, we only need to consider special test configurations. As we will see, this reduction simplifies the calculations and arguments in a significant way. So from now on, we assume that there is a special test configuration V, L C such that L = r K V/C is a relatively very ample line bundle. Notice that r is in general not the same as r. However, for the sake of testing K-semistability which is of asymptotic nature see [Tia97, Don2, LX4], we can choose < r = N for N sufficiently divisible such that L is an integral multiple of L. For later convenience we will define this constant: σ := r r = r r Z >, such that L = σl. 46 The central fibre of V C is a Q-Fano variety, denoted by V or. Because the polarized family V, L C is flat, we can choose r = N sufficiently divisible such that L = r K V/C = NK V/C is a relatively very ample line bundle over C and H V, ml = H V, ml for every m Z. By taking affine cones, we then get a flat family X C with general fibre X = CV, L and the central fibre X = CV, L see [Kol3, Aside on page 98]. Notice that we can write: X = Spec + i= H V, il =: Spec R ; X = Spec + i= H V, il =: Spec R; X = Spec + i= H V, il =: Spec S; X = Spec + i= H V, il =: Spec R. Our plan is to define families of valuations wt β on X, w β on X and w β on X, and then study the relation between their normalized volumes. 9

20 6. Valuation wt β on X By the definition of special test configuration, there is an equivariant T = C action on V, L C which fixes the central fibre V. This naturally induces an equivariant T =: e Cη action on X C which fixes X. There is also another natural T := e Cξ = C -action which fixes each point on the base C and rescales the fibre. These two actions commute and generate a T = C 2 action on X. For any τ = τ, τ Z 2 and t = t, t C 2, we will denote t τ = t τ. Consider the weight decomposition of S under the T action: tτ S = τ Γ S τ, where S τ = {f S; t f = t τ f for all t C 2 } for any τ Z 2 and Γ = {τ Z 2 ; S τ {}} Z 2. Denote by t C the Lie algebra of T. The two generator {ξ, η} allows us to identity t C = N Z C for a lattice N = Z 2. Let t R = N Z R. Any element ξ t R is a holomorphic vector field of the form ξ = a ξ + a η. Any ξ = a ξ + a η t R determines a valuation wt ξ on CX : wt ξ f = min { τ, ξ ; f τ } for f = f τ S. τ This is clearly a T-invariant valuation. In particular it is C -invariant, i.e. T -invariant. Definition 6.2. The cone of positive vectors is defined to be: t + R = {ξ = a ξ + a η t R ; τ, ξ = a τ + a τ > for any τ Γ \, }. 47 t + R is essentially the same as the cone of Reeb vector fields considered in Sasaki geometry see [MSY8]. The following is a standard fact: Lemma 6.3. The cone t + R contains an open neighborhood of ξ. Proof. For a fixed τ Z >, H V, τ L = S τ = τ S τ,τ with S τ,τ is the weight space decomposition with respect to the action T = e Cη on V, L. Because S is finitely generated, there exists A > R such that Aτ < τ < Aτ for any τ, τ appearing in the above decomposition see [BHJ5, Proof of Corollary 3.4]. So for any a, a R 2 with a > and a a < A, we have: a τ + a τ a τ + a a τ a τ a a Aτ >. The lemma follows immediately by choosing a =. One sees directly that ξ t + R if and only if the center of wt ξ is the vertex o X, i.e. wt ξ Val. X,o For β R, let ξ β = ξ + βη t R. The associated valuation wt ξβ will be denoted by wt β. It s clear that ξ t + R. By Lemma 6.3, for β, ξ β t + R and hence the valuation wt β is in Val X,o. Moreover, from the definition, we know that 6.2 Valuation w β on X wt β f = k + β wt η f for any f S k = H V, kl. 48 For any f H V, ml, f determines a T -invariant section f H V C, πml. Since we have a T -equivariant isomorphism V\V, L = V C, πl, f can be seen as a meromorphic section of H V, ml. Recall that V will also be denoted by. We will define a valuation v on CX satisfying v f = ord V f for any f H V, ml and another valuation v on CX such that v is the restriction of v under inclusion CX CX that is induced by the natural embedding: R = + k= H V, kl = + k= H V, σkl R = 2 + m= H V, ml.