In this section we fix a fieldk, which is eitherCwith the usual absolute value, or a complete field equipped with a non-archimedean absolute value. Ifk=Cthen we will denote byCk the classC defined in §2.3.
4.1. Monomial bases and the Okounkov semi-group.— In this subsection we recall the construction of the Okounkov semi-group of a graded linear series. We refer the reader to [15, 14, 5, 13] for more details.
Consider an integral projective schemeX of dimensiond>1defined over the field k. We assume that the scheme X has a regular rational pointx: the local ringOX,x is then a regular local ring of dimensiond. We fix a regular sequence(z1, . . . , zd)in its maximal ideal mx. The formal completion of OX,x with respect to the maximal idealmxis isomorphic to the algebra of formal series in the parametersz1, . . . , zd (cf.
[11, proposition 10.16]).
If we choose a monomial ordering(6)6onNd we obtain a decreasingNd-filtration (called theOkounkov filtration)FonObX,xsuch thatFα(ObX,x)is the ideal generated by monomials of the formzβ such thatβ >α. This filtration is multiplicative: we have that
Fα(ObX,x)Fβ(ObX,x)⊂ Fα+β(ObX,x).
6. This is a total order6onNdsuch that06αfor anyα∈Ndandα6α′impliesα+β6α′+β for anyα, α′ andβinNd.
The filtrationF induces by grading an Nd-graded algebragr(OX,x)which is iso-morphic(7) to k[z1, . . . , zd]. In particular, for any α∈ Nd, grα(OX,x)is a rank-one vector space onk.
IfLis an inversibleOX-module then on taking a local trivialisation in a neighbour-hood ofxwe can identifyLx withOX,x. The filtrationF then induces a decreasing Nd-filtration onH0(X, L)which is independent of the choice of trivialisation. For any s ∈H0(X, L)we denote by ord(s)the upper bound of the set of α∈ Nd such that s∈ FαH0(X, L). We have that
∀s, s′ ∈H0(X, L), ord(s+s′)>min ord(s),ord(s′) .
Moreover, for anys∈H0(X, L)and anya∈k× we have thatord(s) = ord(as).
Let L be an invertible OX-module. We let V•(L) be the graded ring L
n>0H0(X, nL) (with the additive notation for the tensor product of invert-ible sheaves). By agraded linear system ofLwe mean a graded subalgebra ofV•(L).
Any graded linear systemV• ofLcan be identified, on choosing a local trivialisation ofLaroundx, with a graded subalgebra of the algebra of polynomialsOX,x[T]. The filtration F induces a decreasing Nd-filtration on each homogeneous piece Vn. We denote by gr(V•) the Nd+1-graded k-algebra induced by this filtration. This is an Nd+1-graded subalgebra ofgr(OX,x)[T]∼=k[z1, . . . , zd, T]. In particular, the elements (n, α) ∈Nd+1 such that gr(n,α)(V•) 6={0} form a sub-semigroup ofNd+1 which we denote by Γ(V•). For any n∈Nwe denote byΓ(Vn)the subset ofNd of elementsα such that(n, α)∈Γ(V•).
4.2. Monomial norms.— As above, we consider an integral projective schemeXof dimensiond>1overSpeck. We fix a regular rational point x∈X(k)(it is assumed that such a point exists), a system of parametersz= (z1, . . . , zd)atxand a monomial order onNd. LetLbe an invertibleOX-module and letV• be a graded linear system of L. We assume that every k vector spaceVn is equipped with two norms ϕn and ψn, which are ultrametric if k is non-archimedean. We assume moreover that these norms are submultiplicative - ie. that for any(n, m)∈N2and any(sn, sm)∈Vn×Vm
we have that
(28) ksn⊗smkϕn+m6ksnkϕn· ksmkϕm, ksn⊗smkψn+m 6ksnkψn· ksmkψm. In this subsection, we study the asymptotic behaviour of deg(Vd n, ϕn, ψn). As the Harder-Narasimhan filtration is not functorial in Ck we cannot study this problem directly using the method developped in [6]. We will avoid this problem by using Okounkov filtrations. The norms ϕn and ψn induce quotient norms on each of the sub-quotients grα(Vn) (α ∈ Γ(Vn)) which by abuse of notation we will continue to denote byϕn andψn. By the results of previous sections, notably (12) and (22), we have that
ddeg(Vn, ϕn, ψn)− X
α∈Γ(Vn)
deg(grd α(Vn), ϕn, ψn)
6Ak(rk(Vn)),
7. This follows from the fact thatOX,xis dense inObX,x.
where Ak(r) = rln(r) ifk =Cand Ak(r) = 0 ifk is non-archimedean. We deduce that
(29) lim
n→+∞
bµ(Vn, ϕn, ψn)
n − 1
n#Γ(Vn) X
α∈Γ(Vn)
ddeg(grα(Vn), ϕn, ψn) = 0.
Moreover, if we equip the space
gr(Vn) = M
α∈Γ(Vn)
grα(Vn)
with normsϕˆn andψˆn such that the rank1 subspaces grα(Vn)are orthogonal with the induced (subquotient) normsϕn andψn respectively then we have that
X
α∈Γ(Vn)
ddeg(grα(Vn), ϕn, ψn) =deg(gr(Vd n),ϕˆn,ψˆn).
The problem then reduces to the study of the k-algebra of the semi-group Γ(V•) (isomorphic to gr(V•)) with the appropriate norms. As the semi-group Γ(V•) is a multiplicative basis for the algebrak[Γ(V•)]we can construct a new normηn on each space gr(Vn) which is multiplicative: for any γ ∈ Γ(V•) we let sγ be the canonical image ofγ∈Γ(V•)in the algebrak[Γ(V•)]and we equip
gr(Vn) = M
α∈Γ(Vn)
ks(n,α)
with the norm ηn such that the vectorssn,α are orthogonal of norm1. Using these auxillary norms we can write the degreedeg(gr(Vd n),ϕˆn,ψˆn)as a difference
deg(gr(Vd n),ϕˆn, ηn)−ddeg(gr(Vn),ψˆn, ηn), or alternatively
X
α∈Γ(Vn)
deg(grd α(Vn), ϕn, ηn)− X
α∈Γ(Vn)
ddeg(grα(Vn), ψn, ηn).
It is easy to see that the real valued functions (n, α) 7→ ddeg(grα(Vn), ϕn, ηn) and (n, α)7→deg(grd α(Vn), ϕn, ηn)defined onΓ(V•)are superadditive, so their asymptotic behaviour can be studied using the methods developped in [6].
4.3. Limit theorem. — In this subsection we fix an integer d > 1 and a sub-semigroupΓinNd+1. For any integern∈Nwe denote byΓnthe set{α∈Nd|(n, α)∈ Γ}. We suppose that the semi-groupΓverifies the following conditions (cf. [14, §2.1]) : (a) Γ0={0},
(b) there is a finite subsetB in {1} ×Nd such thatΓis contained in the sub-monoid ofNd+1generated byB,
(c) the groupZd+1 is generated byΓ.
We let Σ(Γ)be the (closed) convex cone in Rd+1 generated byΓ. Under the above conditions the projection ofΣ∩({1} ×Rd)intoRd is a convex body inRd, denoted
∆(Γ). Moreover, we have that
n→+∞lim
#Γn
nd = vol(∆(Γ)),
wherevol(.)is Lesbesgue measure onRd (cf. [14, proposition 2.1]).
We say that a function Φ : Γ→Rissuperadditive ifΦ(γ+γ′)>Φ(γ) + Φ(γ′). In what follows, we study the asymptotic properties of super-additive functions.
Lemma 4.1. — LetΦbe a superadditive function defined onΓsuch thatΦ(0,0) = 0.
(1) For any real numbertthe setΓtΦ:={(n, α)∈Γ|Φ(n, α)>nt}is a sub-semigroup of Γ.
(2) Ift∈Ris a real number such that t < lim
n→+∞ sup
α∈Γn
1
nΦ(n, α), then ΓtΦ satisfies conditions (a)–(c) above.
Proof. — (1) AsΦis superadditive, for any(n, α)and(m, β)in ΓtΦwe have that Φ(n+m, α+β)>Φ(n, α) + Φ(m, β)>nt+mt= (n+m)t,
and hence (n+m, α+β)∈ΓtΦ.
(2) It is easy to check that (a) and (b) are satisfied byΓtΦ. We now prove (c). Let A be a finite subset of Γ generatingZd+1 as a group. By hypothesis, there exists a ε > 0 and a γ = (m, β) ∈Γ such that Φ(m, β)>(t+ε)m. It follows that for any (n, α)∈Γ, we have that
Φ(km+n, kβ+α)
km+n > Φ(n, α) +kΦ(m, β)
km+n > Φ(n, α) +km(t+ε) km+n >t for large enough k. There therefore exists ak0 > 1 such that kγ+ξ ∈ΓtΦ for any ξ∈A andk>k0, soΓtΦ generatesZd+1as a group.
Remark 4.2. — The superadditivity ofΦimplies that
∆(ΓεtΦ1+(1−ε)t2)⊃ε∆(ΓtΦ1) + (1−ε)∆(ΓtΦ2).
By the Brunn-Minkowski theorem, the function t 7→ vol(∆(ΓtΦ))1/d is concave on ]− ∞, θ[, where
θ= lim
n→+∞ sup
α∈Γn
1
nΦ(n, α),
so it is continuous on this interval. Moreover, as the set (dense in∆(Γ)) {α/n : (n, α)∈Γ, n>1}
is contained inS
t∈R∆(ΓtΦ), we get that vol(∆(Γ)) = lim
t→−∞vol(∆(ΓtΦ)).
The following result is a limit theorem for superadditive functions defined onΓ. It is a natural generalisation of [6, Theorem 1.11]
Theorem 4.3. — LetΦ : Γ→Rbe a superadditive function such that
(30) θ:= lim
n→+∞ sup
α∈Γn
1
nΦ(n, α)<+∞.
For any integer n>1, we consider Zn = Φ(n, .) as a uniformly distributed random variable onΓn. The sequence of random variables Zn/n
n>1then converges in law(8) to a limit random variable Z whose law is given by
P(Z>t) = vol(∆(ΓtΦ))
vol(∆(Γ)) , t6=θ.
Proof. — By Remark 4.2 the function F defined on t ∈ R \ {θ} by F(t) :=
vol(∆(ΓtΦ))/vol(∆(Γ)) is decreasing and continuous and lim
t→−∞F(t) = 1. Moreover, condition (30) implies thatF(t) = 0 for large enough positivet and it follows that if we extend the domain of definition ofF to Rby takingF(θ)to be the limit ofF(t) asttends toθfrom the left we get a (left continuous) probability function onR. For any integern>1and any real numbertwe have that
P(Zn>t) =#ΓtΦ,n is also empty, so equation (31) also holds for t > θ. Finally, if the function F is continuous at θ then since both t 7→ P(Zn > t) and F are decreasing we also have that lim
n→+∞P(Zn>θ) =F(θ). The result follows.
Remark 4.4. — The limit law in the above theorem can also be characterised as the pushforward of Lesbesgue measure on∆(Γ) by a function determined byΦ. Let GΦ : ∆(Γ) →R∪ {−∞}be the map sending xto sup{t∈ R : x ∈∆(ΓtΦ)}. This is a real concave function on(9) ∆(Γ)◦. The function GΦ is therefore continuous on ∆(Γ)◦. By definition, the limit law is equal to the pushforward of normalised Lesbesgue measure on ∆(Γ) by GΦ. In particular, if h is a continuous bounded function then we have that
8. We say that a sequence of random variables(Zn)n>1converges in law to a random variableZ if the law ofZnconverges weakly to that ofZ, i.e., for any continuous bounded functionhonRwe have that lim
n→+∞E[h(Zn)] =E[Z], or equivalently, the probability function ofZnconverges to that ofZat any pointx∈Rsuch thatP(Z=x) = 0.
9. The setS
t∈R∆(ΓtΦ)is convex and its volume is equal tovol(∆(Γ))so it contains∆(Γ)◦.
This enables us to realise the random variableZ as the functionGΦ defined on the convex body∆(Γ)equipped with normalised Lesbesgue measure.
In the rest of this section we apply these results to the situation described in §4.2.
We consider an integral projective scheme X of dimension d> 1 defined on a field k and an invertible OX-module L. We also choose a regular rational point (it is assumed that such a point exists)x∈X(k), a local system of parameters(z1, . . . , zd) and a monomial order onNd. LetV•be a graded linear system onLwhose Okounkov semi-group Γ(V•) satisfies(10) conditions (a)-(c) of section 4.3. For any n ∈ N let Vn be equipped with two normsϕn and ψn which are assumed to be ultrametric for non-archimedeank.
Theorem 4.5. — Assume the normsϕn andψn satisfy the following conditions:
(1) the system of norms (ϕn, ψn)n∈N is submultiplicative (i.e. satisfies (28));
(2) we have thatd(ϕn, ψn) =O(n)asn→+∞;
(3) there is a constant C >0 such that(11) infα∈Γ(Vn)lnks(n,α)kϕˆn >−Cn for any n∈N,n>1.
Then the sequence(n1µ(Vb n, ϕn, ψn))n>1 converges inR.
Proof. — We introduce auxillary monomial normsηn as in §4.2. LetΦ : Γ(V•)→R be the function that sends (n, α) ∈ Γ(V•) to ddeg(grα(Vn), ϕn, ηn). This function is superadditive and condition (3) implies that
n→+∞lim sup
α∈Vn
1
nΦ(n, α)<+∞.
LetZΦ,n= Φ(n, .)be a uniformly distributed random variable onΓ(Vn). By Theorem 4.3 the sequence of random variables (ZΦ,n/n)n>1 converges in law to a random variable ZΦ defined on ∆(Γ(V•)) (as in remark 4.4). Similarly, conditions (2) and (3) prove that (3) also holds for the norms ψˆn. Denote by Ψ : Γ(V•) → R the function sending (n, α) ∈ Γ(V•) to deg(grd α(Vn), ψn, ηn) and by ZΨ,n = Ψ(n, .) the random variable onΓ(Vn)such thatn∈N,n>1. The sequence of random variables (ZΨ,n/n)n>1 then converges in law to a random variable ZΨ defined on ∆(Γ(V•)).
Moreover, (2) implies that the function|ZΦ−ZΨ|is bounded on∆(Γ(V•))◦. By equation (29) and the equality
deg(grd α(Vn), ϕn, ψn) =ddeg(grα(Vn), ϕn, ηn)−deg(grd α(Vn), ψn, ηn),
10. Note that these three conditions are automatically satified whenever V• contains an ample divisor, ie. Vn6={0}for large enoughnand there is an integerp>1, an ampleOX-moduleAand a non-zero sectionsofpL−A, such that
Im H0(X, nA)−→·sn H0(X, npL)
⊂Vnp
for anyn∈N,n>1. We refer the reader to [14, lemma 2.12] for a proof.
11. See §4.2 for notation.
it will be enough to prove that the sequence (E[ZΦ,n/n]−E[ZΨ,n/n])n>1 converges inR. Condition (2) of the theorem implies that the functions n1|ZΦ,n−ZΨ,n|(n∈N) are uniformly bounded. LetA >0be a constant such that
∀n>1, |ZΦ,n−ZΨ,n|6An.
As(ZΦ,n/n)n>1and(ZΨ,n/n)n>1converge in law toZΦandZΨrespectively, for any ε >0there is aT0>0and a n0∈Nsuch that
∀T >T0, ∀n>n0, P(ZΦ,n6−nT)< εet P(ZΨ,n6−nT)< ε.
It follows that
E[ZΦ,n/n]−E[ZΨ,n/n]−E[max(ZΦ,n/n,−T)] +E[max(ZΨ,n/n,−T)]
62εE[|ZΦ,n/n−ZΨ,n/n|]62εA (33)
wheneverT >T0andn>n0. Moreover, as the random variablesZΦ,n/nandZΨ,n/n are uniformly bounded above and the sequences(ZΦ,n/n)n>1 and (ZΨ,n/n)n>1 con-verge in law it follows that
n→+∞lim E[max(ZΦ,n/n,−T)]−E[max(ZΨ,n/n,−T)] =E[max(ZΦ,−T)−max(ZΨ,−T)].
Moreover, as the function|ZΦ−ZΨ|is bounded, the dominated convergence theorem implies that
T→+∞lim E[max(ZΦ,−T)−max(ZΨ,−T)] =E[ZΦ−ZΨ].
Equation (33) then implies that lim sup
n→+∞
E[ZΦ,n/n]−E[ZΨ,n/n]−E[ZΦ−ZΨ]62εA.
Asεis arbitrary, we get that
n→+∞lim 1
nµ(Vb n, ϕn, ψn) =E[ZΦ−ZΨ].
Condition (3) in the above theorem holds wheneverϕn comes from a continuous metric on the invertibleOX-moduleL. This can be proved by considering a monomial order 6 on Nd such that(12) α1+· · ·+αd < β1+· · ·+βd implies (α1, . . . , αd) <
(β1, . . . , βd). Let Xan be the analytic space associated to the k-scheme X (in the Berkovich sense [1] ifkis non-archimedean) and letLanbe the pull-back ofLtoXan. LetCX0an be the sheaf of continuous real functions on Xan. Acontinuous metric on L is a morphism of set sheaves, k.k, from Lan⊗ CX0an to CX0an which in every point x∈Xaninduces a normk.k(x)on the fibreLan(x). Given a continuous metricϕon X we can equipH0(X, L)with the supremum normk.kϕ,supsuch that
∀s∈H0(X, L), kskϕ,sup:= sup
x∈Xankskϕ(x).
For any integer n ∈ N the metric ϕ induces by passage to the tensor product a continuous metricϕ⊗nonnL. Letϕn be the supremum norm onH0(X, nL)induced
12. Whenα1+· · ·+αd=β1+· · ·+βdwe may use the lexicographic order, for example.
byϕ⊗n (or its restriction toVn by abuse of language): the system of norms(ϕn)n>0
then satisfies condition (3) of Theorem 4.5. This follows from Schwarz’s (complex or non-archimedean) Lemma (cf. [8, pp.205-206]). This gives us the following corollary.
Corollary 4.6. — Let X be a projective integral scheme defined over a field k and letLbe an invertible OX-module equipped with two continuous metricsϕandψ. Let V• be a graded linear system of L such thatΓ(V•) satisfies conditions (a)–(c) above.
For any integern∈Nletϕn andψn be the supremum norms onVn associated to the metricsϕ⊗n andψ⊗nrespectively. The sequence(µ(Vb n, ϕn, ψn)/n)n>1then converges inR.
Proof. — The system of norms(ϕn)n>0is submultiplicative. Ifsands′are elements ofVn andVmrespectively we have that
ks⊗s′kϕn+m = sup
x∈Xanks⊗s′kϕ⊗(n+m)(x)
6
sup
x∈Xankskϕ⊗n(x)
· sup
x∈Xanks′kϕ⊗m(x)
=kskϕn· ks′kϕm.
Similarly, the system of norms(ψn)n>0 is also submultiplicative. Moreover, as the topological spaceXanis compact, we have that
sup
x∈Xan
d(k.kϕ(x),k.kψ(x))<+∞.
and it follows that d(ϕn, ψn) = O(n) as n → +∞. Finally, as the norms ϕn
satisfy condition (3) of theorem 4.5, the convergence of (µ(Vb n, ϕn, ψn)/n)n>1 as a consequence of this theorem.
Remark 4.7. — This result invites comparison with a result of Witt Nyström’s [17, théorème 1.4]. Both methods use the monomial basis to construct super or subadditive functions on the Okounkov semi-group. However, the method in [17] is based on a comparison between theL2 metric and theL∞ metric, whereas we use the Harder-Narasimhan formalism. This new approach is highly flexible and enables us to prove our result in the very general setting of a submultiplicatively normed linear system satisfying moderate conditions, in both the complex and non-archimedean cases.