Asymptotic polygon of a big line bundle

ˆℎ⁰(𝐸)−degˆ₊(𝐸) =

ˆℎ⁰(𝐸𝑗)−deg(𝐸ˆ 𝑗)

≤rk𝐸𝑗log∣Δ𝐾∣+𝐶0(rk𝐸𝑗)≤rk𝐸log∣Δ𝐾∣+𝐶0(rk𝐸).

4. Asymptotic polygon of a big line bundle Let𝑘be a field and𝐵 =⊕

𝑛≥0𝐵_𝑛 be an integral graded 𝑘-algebra such that, for 𝑛 sufficiently positive, 𝐵𝑛 is non-zero and has finite rank. Let 𝑓 : ℕ^∗ → ℝ+ be a mapping such that lim

𝑛→∞𝑓(𝑛)/𝑛= 0. Assume that each vector space𝐵𝑛 is equipped with an ℝ-filtration ℱ^(𝑛) such that 𝐵 is 𝑓-quasi-filtered (cf. [11] §3.2.1). In other words, we assume that there exists𝑛0 ∈ℕ^∗ such that, for any integer 𝑟≥2 and all homogeneous elements𝑥1,⋅ ⋅ ⋅ , 𝑥𝑟in𝐵 respectively of degree𝑛1,⋅ ⋅ ⋅, 𝑛𝑟inℕ≥𝑛₀, one has

𝜆(𝑥₁⋅ ⋅ ⋅𝑥_𝑟)≥

𝑟

∑

𝑖=1

(𝜆(𝑥_𝑖)−𝑓(𝑛_𝑖)) .

We suppose in addition that sup_𝑛≥1𝜆max(𝐵𝑛)/𝑛 < +∞. Recall below some results in [11] (Proposition 3.2.4 and Theorem 3.4.3).

Proposition 4.1. — 1) The sequence (𝜆_max(𝐵_𝑛)/𝑛)_𝑛≥1 converges inℝ. 2) If 𝐵 is finitely generated, then the sequence of measures (𝑇1

𝑛𝜈𝐵_𝑛)𝑛≥1 converges vaguely to a Borel probability measure on ℝ.

In this section, we shall generalize the second assertion of Proposition 4.1 to the case where the algebra 𝐵 is given by global sections of tensor power of a big line bundle on a projective variety.

4.1. Convergence of measures. — Let 𝑋 be an integral projective scheme of dimension 𝑑 defined over 𝑘 and 𝐿 be a big invertible 𝒪𝑋-module: recall that an invertible𝒪𝑋-module𝐿is said to be big if itsvolume, defined as

vol(𝐿) := lim sup sequence of measures(𝑇¹

𝑛𝜈𝐵𝑛)𝑛≥1 converges vaguely to a probability measure onℝ. Proof. — For any integer𝑛 ≥1, denote by 𝜈𝑛 the measure𝑇1

𝑛𝜈𝐵_𝑛. Since 𝐿 is big, for sufficiently positive𝑛,𝐻⁰(𝑋, 𝐿^⊗𝑛)∕= 0, and hence𝜈𝑛is a probability measure. In addition, Proposition 4.1 1) implies that the supports of the measures𝜈𝑛are uniformly bounded from above. After Fujita’s approximation theorem (cf. [13, 23], see also [18] Ch. 11), the volume functionvol(𝐿)is in fact a limit:

vol(𝐿) = lim

𝑛→∞

rk𝑘𝐻⁰(𝑋, 𝐿^⊗𝑛) 𝑛^𝑑/𝑑! .

Furthermore, for any real number𝜀,0< 𝜀 <1, there exists an integer𝑝≥1together with a finitely generated sub-𝑘-algebra 𝐴^𝜀 of 𝐵^(𝑝) = ⊕

The graded𝑘-algebra𝐴^𝜀, equipped with induced filtrations, is𝑓-quasi-filtered. There-fore Proposition 4.1 2) is valid for 𝐴^𝜀. In other words, If we denote by𝜈_𝑛^𝜀 the Borel measure 𝑇1

𝑛𝑝𝜈𝐴^𝜀_𝑛 = 𝑇1 𝑝(𝑇1

𝑛𝜈𝐴^𝜀_𝑛), then the sequence of measures (𝜈_𝑛^𝜀)_𝑛≥1 converges vaguely to a Borel probability measure 𝜈^𝜀 on ℝ. In particular, for any function ℎ ∈ 𝐶_𝑐(ℝ), the sequence of integrals ( ∫

ℎd𝜈_𝑛^𝜀)

𝑛≥1 is a Cauchy sequence. This as-sertion is also true when ℎis a continuous function on ℝwhose support is bounded from below: the supports of the measures𝜈_𝑛^𝜀 are uniformly bounded from above. The exact sequence 0 //𝐴^𝜀_𝑛 //𝐵_𝑛𝑝 //𝐵𝑛𝑝/𝐴^𝜀_𝑛 //0 implies that Therefore, for any bounded Borel functionℎ, one has

(11)

Hence, for any bounded continuous functionℎ satisfyinginf(supp(ℎ))>−∞, there exists𝑁ℎ,𝜀∈ℕsuch that, for any 𝑛, 𝑚≥𝑁ℎ,𝜀,

Let ℎ be a smooth function on ℝ whose support is compact. We choose two increasing continuous functionsℎ1andℎ2such thatℎ=ℎ1−ℎ2and that the supports

of them are bounded from below. Let𝑛0∈ℕ^∗be sufficiently large such that, for any induced filtrations. As the algebra 𝐵 is 𝑓-quasi-filtered, we obtain, by [11] Lemma 1.2.6,𝜈_𝑛,𝑟^′ ≻𝜏𝑎_𝑛,𝑟𝜈𝑛,𝑟 ≻𝜏𝑏_𝑛,𝑟𝜈𝑛𝑝, where

We deduce, from (12), (15) and (16), lim sup

According to (12), we obtain that there exists𝑁_ℎ,𝜀^′ ∈ℕ^∗ such that, for all integers𝑙 and𝑙^′ such that𝑙≥𝑁_ℎ,𝜀^′ ,𝑙^′≥𝑁_ℎ,𝜀^′ , one has

∫

ℎd𝜈_𝑙−

∫ ℎd𝜈_𝑙^′

≤4𝜀(∥ℎ1∥sup+∥ℎ2∥sup) + 2𝜀∥ℎ∥sup+ 6𝜀, which implies that the sequence(∫

ℎd𝜈𝑛)𝑛≥1 converges inℝ.

Let 𝐼 :𝐶₀^∞(ℝ) →ℝbe the linear functional defined by 𝐼(ℎ) = lim

𝑛→∞

∫

ℎd𝜈𝑛. It extends in a unique way to a continuous linear functional on 𝐶𝑐(ℝ). Furthermore, it is positive, and so defines a Borel measure 𝜈 on ℝ. Finally, by (11), ∣𝜈(ℝ)−(1− 𝜀)𝜈^𝜀(ℝ)∣ ≤𝜀. Since𝜀is arbitrary,𝜈 is a probability measure.

4.2. Convergence of maximal values of polygons. — If𝜈is a Borel probability measure on ℝ and𝛼 ∈ℝ, denote by 𝜈^(𝛼) the Borel probability measure on ℝ such that, for anyℎ∈𝐶𝑐(ℝ),

∫

ℎd𝜈^(𝛼)=

∫

ℎ(𝑥)11_[𝛼,+∞[(𝑥)𝜈(𝑑𝑥) +ℎ(𝛼)𝜈(]− ∞, 𝛼[).

The measure𝜈^(𝛼)is called thetruncationof𝜈 at𝛼. The truncation operator preserves the order “≻”.

Assume that the support of 𝜈 is bounded from above. The truncation of 𝜈 at 𝛼 modifies the “polygon” 𝑃(𝜈) only on the part with derivative < 𝛼. More precisely, one has

𝑃(𝜈) =𝑃(𝜈^(𝛼))on{𝑡∈[0,1[

𝐹_𝜈^∗(𝑡)≥𝛼}.

In particular, if𝛼≤0, then

(19) max

𝑡∈[0,1[𝑃(𝜈)(𝑡) = max

𝑡∈[0,1[𝑃(𝜈^(𝛼))(𝑡).

The following proposition shows that given a vague convergence sequence of Borel probability measures, almost all truncations preserve vague limit.

Proposition 4.3. — Let (𝜈_𝑛)_𝑛≥1 be a sequence of Borel probability measures which converges vaguely to a Borel probability measure𝜈. Then, for any𝛼∈ℝ, the sequence (𝜈𝑛^(𝛼))_𝑛≥1 converges vaguely to𝜈^(𝛼).

Proof. — For any𝑥∈ℝand𝛼∈ℝ, write𝑥∨𝛼:= max{𝑥, 𝛼}. Assume thatℎ∈𝐶𝑐(ℝ) and𝜂 is a Borel probability measure. Then

∫

ℎ(𝑥)𝜂^(𝛼)(d𝑥) =

∫

ℎ(𝑥∨𝛼)𝜂(d𝑥).

By [9] IV §5n^∘12Proposition 22, for any𝛼∈ℝ,

𝑛→∞lim

∫

ℎ(𝑥)𝜈_𝑛^(𝛼)(d𝑥) = lim

𝑛→∞

∫

ℎ(𝑥∨𝛼)𝜈_𝑛(d𝑥) =

∫

ℎ(𝑥∨𝛼)𝜈(d𝑥) =

∫

ℎ(𝑥)𝜈^(𝛼)(d𝑥).

Therefore, the proposition is proved.

Corollary 4.4. — Under the assumption of Theorem 4.2, the sequence ( max

𝑡∈[0,1]𝑃𝐵_𝑛(𝑡)/𝑛)

𝑛≥1

converges inℝ.

Proof. — For𝑛∈ℕ^∗, denote by𝜈_𝑛 =𝑇1

𝑛𝜈_𝐵𝑛. By Theorem 4.2, the sequence(𝜈_𝑛)_𝑛≥1 converges vaguely to a Borel probability measure𝜈. Let𝛼 <0be a number such that (𝜈𝑛^(𝛼))_𝑛≥1 converges vaguely to 𝜈^(𝛼). Note that the supports of 𝜈_𝑛^𝛼 are uniformly bounded. So𝑃(𝜈𝑛^(𝛼))converges uniformly to𝑃(𝜈^(𝛼))(see [11] Proposition 1.2.9). By (19),(

𝑡∈[0,1]max𝑃𝐵_𝑛(𝑡)/𝑛)

𝑛≥1 converges to max

𝑡∈[0,1]𝑃(𝜈)(𝑡).

If𝑉 is a finite dimensional filtered vector space over𝑘, we shall use the expression 𝜆+(𝑉) to denote max

𝑡∈[0,1]𝑃𝑉(𝑡). With this notation, the assertion of Corollary 4.4 becomes: lim

𝑛→∞𝜆+(𝐵𝑛)/𝑛exists inℝ.

Lemma 4.5. — Assume that 𝜈₁ and 𝜈₂ are two Borel probability measures whose supports are bounded from above. Let𝜀∈]0,1[and𝜈 =𝜀𝜈₁+ (1−𝜀)𝜈₂. Then

(20) max

𝑡∈[0,1]𝑃(𝜈)(𝑡)≥𝜀 max

𝑡∈[0,1]𝑃(𝜈1)(𝑡).

Proof. — After truncation at 0 we may assume that the supports of 𝜈1 and 𝜈2

are contained in [0,+∞[. In this case 𝜈 ≻ 𝜀𝜈1 + (1−𝜀)𝛿0 and hence 𝑃(𝜈) ≥ 𝑃(𝜀𝜈1+ (1−𝜀)𝛿0). Since

𝑃(𝜀𝜈₁+ (1−𝜀)𝛿₀)(𝑡) =

{𝜀𝑃(𝜈₁)(𝑡/𝜀), 𝑡∈[0, 𝜀], 𝜀𝑃(𝜈1)(1), 𝑡∈[𝜀,1[, we obtain (20).

Theorem 4.6. — Under the assumption of Theorem 4.2, one has

𝑛→∞lim 𝜆₊(𝐵_𝑛)/𝑛 >0 if and only if lim

𝑛→∞𝜆_max(𝐵_𝑛)/𝑛 >0.

Furthermore, in this case, the inequality lim

𝑛→∞𝜆+(𝐵𝑛)/𝑛≤ lim

𝑛→∞𝜆max(𝐵𝑛)/𝑛 holds.

Proof. — For any filtered vector space𝑉,𝜆max(𝑉)>0if and only if𝜆+(𝑉)>0, and in this case one always has𝜆max(𝑉)≥𝜆+(𝑉). Therefore the second assertion is true.

Furthermore, this also implies

𝑛→∞lim 1

𝑛𝜆₊(𝐵_𝑛)>0 =⇒ lim

𝑛→∞

1

𝑛𝜆_max(𝐵_𝑛)>0.

It suffices then to prove the converse implication. Assume that 𝛼 > 0 is a real number such that lim

𝑛→∞𝜆max(𝐵𝑛)/𝑛 >4𝛼. Choose sufficiently large 𝑛0∈ℕsuch that 𝑓(𝑛)< 𝛼𝑛for any 𝑛≥𝑛0 and such that there exists a non-zero𝑥0∈𝐵𝑛₀ satisfying 𝜆(𝑥0)≥4𝛼𝑛0. Since the algebra𝐵 is 𝑓-quasi-filtered,𝜆(𝑥^𝑚₀ )≥4𝛼𝑛0𝑚−𝑚𝑓(𝑛)≥ 3𝛼𝑚𝑛0. By Fujita’s approximation theorem, there exists an integer𝑝divisible by𝑛0

and a subalgebra 𝐴 of 𝐵^(𝑝) =⊕

𝑛≥0𝐵𝑛𝑝 generated by a finite number of elements in 𝐵𝑝 and such that lim inf

𝑛→∞ rk𝐴𝑛/rk𝐵𝑛𝑝 > 0. By possible enlargement of 𝐴 we

may assume that 𝐴 contains 𝑥^𝑝/𝑛₀ ⁰. By Lemma 4.5, lim

Furthermore, by Noether’s normalization theorem, we may assume that 𝐵 = 𝑘[𝑥₁,⋅ ⋅ ⋅ , 𝑥_𝑞] is an algebra of polynomials, where 𝑥₁ coincides with the element in

>0, which implies lim

𝑛→∞

1

𝑛𝜆+(𝐵𝑛)> 0 by Lemma 4.5.