4.1. Adelic vector bundles. — The comparison principle (Proposition 2.3) actually applies to general number fields and allows to compare several arithmetic invariants.
Let K be a number field and let MK be the set of all places of K. For v ∈ MK, we denote by |.|v the absolute value in the class v which extends either the usual absolute value or a certainp-adic absolute onQ(in the former case, v is said to be Archimedean, while in the latter case, v is said to be non-Archimedean). We denote byKv andQv the completion ofK andQwith respect to the absolute value|.|v, respectively. Note that the product formula shows that
(4.1) ∀a∈K\ {0}, X
v∈MK
[Kv :Qv] ln|a|v = 0,
As adelic vector bundle on SpecK (cf. [14, §3]), we refer to the datum E consisting of a finite-dimensional vector space E over K, equipped with a family (k.kv)v∈MK of norms, indexed by the places of K, which satisfies the following conditions:
(1) for any v∈MK,k.kv is a norm onEKv =E⊗KKv, (2) there exists a basis(ei)ni=1 of E such that the relation
∀(λ1, . . . , λn)∈Kvn, kλ1e1+· · ·+λnenk= max
i∈{1,...,n}|λi| · keik holds for all but finitely many places v.
Ifk.kv is induced by an inner product when the placev is Archimedean, and is ultrametric when v is non-Archimedean, we say that the adelic vector bundle E isHermitian. If the normk.kv is pure (namely its image is contained in that of|.|v) for any non-Archimedean place v, we say that the adelic vector bundle E is pure. In the case where E is a one-dimensional K-vector space, we say that E is an adelic line bundle on SpecK. Note that an adelic line bundle is necessarily Hermitian.
LetE be a Hermitian adelic vector bundle onSpecK andK0/K be a finite extension. We denote by E ⊗K K0 the Hermitian adelic vector bundle on
SpecK0 consisting of the K0-vector space E⊗KK0 and norms (k.kw)w∈MK0, such that, for anyv∈MK and any w∈MK0 lying overv,
(a) in the case where v is non-Archimedean, k.kw is the largest ultrametric norm onEK0 0
w
∼=EKv⊗KvKw0 extending k.kv,∗∗,
(b) in the case where v is Archimedean, k.kw is the Hilbert-Schmidt tensor product of k.kv and the absolute value|.|w onKw0 .
We refer the readers to [11, §1.3] for more details about the extension of norms.
Let E = (E,(k.kv)v∈MK) be a Hermitian adelic vector bundle on SpecK.
For any non-zero elementsof E, we define degdE(s) :=− X
v∈MK
[Kv :Qv]
[K:Q] lnkskv. Note that the product formula (4.1) shows that
∀a∈K\ {0}, ddegE(as) =degdE(s).
If K0 is a finite extension of K and if s is a non-zero element of EK0, we use the simplified notationdegdE(s) to denote ddegE⊗
KK0(s). Since the sum of local degrees of a finite separable extension is equal to the global degree (see for example [23] Chapter II, Corollary 8.4), ifK00/K0/K are successive finite extensions and ifsis a non-zero element of EK0, then one has
ddegE⊗
KK0(s) =degdE⊗
KK00(s),
where on the right-hand side of the formula, we considersas a non-zero element of E ⊗K K00. This observation allows to consider ddegE(.) as a function on E⊗KKa, whereKa is a fixed algebraic closure of K.
Given an adelic vector bundleE= (E,(k.kv)v∈MK)onSpecK, we construct an adelic line bundle det(E) as follows. The vector space part of det(E) is give by the maximal exterior power of E. For any v ∈ MK, we equip det(EKv)∼= det(E)⊗KKv with the determinant normk.kv,det, defined as
∀η∈EKv, kηkv,det= inf
(xi)ni=1∈EnKv η=x1∧···∧xn
kx1kv· · · kxnkv.
The Arakelov degree of E is defined as ddeg(E) :=− X
v∈MK
[Kv :Qv]
[K :Q] lnkηkv,det,
where η is a non-zero element of det(E). Note that this definition does not depend on the choice of the non-zero determinant vector η (this is a consequence of the product formula (4.1)). Note that the Arakelov degree function thus constructed satisfies the inequalities in the form of (1.2) and
(1.3) for Hermitian adelic vector bundles. Hence an analogue of Harder-Narasimhan’s theorem is valid and leads to the notion of successive slopes µbi(E) for a Hermitian adelic vector bundleE.
The notion of minima can also be naturally generalised to the setting adelic vector bundles. There actually exist several versions of successive minima. We recall below some of them. We fix a non-zero adelic vector bundleE and letn be the rank ofE over K and ibe an arbitrary element of{1, . . . , n}.
(1) The Bombieri-Vaaler ith minimum of E is defined as the supremum νbiBV(E) of the set of t∈R such that there exists a family ofK-linearly independent vectors{s1, . . . , si}inE which satisfies the following condi-tion: for any j ∈ {1, . . . , i}, ksjkv 61 for any non-Archimedean placev and ksjkσ 6e−t for any Archimedean placesσ.
(2) TheRoy-Thunder ith minimum ofE is defined as the supremumbνiRT(E) of the set of t∈Rsuch that there exists aK-linearly independent family {s1, . . . , si}of vectors inE satisfyingdegdE(sj)>tfor anyj∈ {1, . . . , i}.
(3) The absolute ith minimum of E is defined as the supremum bνiabs(E) of the set of t ∈ R such that there exists a Ka-linearly independent family {s1, . . . , si}of vectors inE⊗KKa satisfyingddegE(sj)>tfor any j∈ {1, . . . , i}.
Note that, if s is a non-zero element of E such that ksk 61 for any non-Archimedean place v and kskσ 6e−t for any Archimedean place σ, then one has
ddegE(s)>− X
σ∈MK,∞
[Kσ :Qσ]
[K :Q] lnkskσ>t,
whereMK,∞ denotes the set of Archimedean places of K. Therefore, for any i∈ {1, . . . , n}, the following inequalities hold
bνiBV(E)6bνiRT(E)6bνiabs(E).
Moreover, ifK0is a finite extension ofK, and{s1, . . . , sn}is a basis ofE⊗KK0, then Hadamard’s inequality (see for example [11, Proposition 1.1.66] for a proof) shows that
deg(E) =d ddeg(E⊗KK0) =− X
v∈MK0
[Kv0 :Qv]
[K0:Q] lnks1∧ · · · ∧snkv
>−
n
X
i=1
X
v∈MK0
[Kv0 :Qv]
[K0 :Q] lnksikv =
n
X
i=1
ddegE(si),
which shows thatµ(E)b >bνminabs(E). Applying this inequality to quotient adelic vector bundles ofE leads to
bµmin(E)>bνminabs(E).
Note that the analogue of Propositions 2.4 and 2.6 (for any of the above minima and for the Harder-Narasimhan filtration respectively) in the setting of Hermitian adelic vector bundle is still true. We obtain that, for any non-zero pure Hermitian adelic vector bundle E and any i∈ {1, . . . ,rkZ(n)}, one has
06µbi(E)−bνiBV(E)6δBV(n), 06µbi(E)−bνiRT(E)6δRT(n), where
δBV(n) := sup
F
(µbmin(F)−bνminBV(F)), δRT(n) := sup
F
(bµmin(F)−bνminRT(F)),
with F running over the set of pure Hermitian adelic vector bundles of rank n over SpecK. Similarly, for any (non-necessarily pure) non-zero Hermitian adelic vector bundleE and anyi∈ {1, . . . ,rkZ(E)}, one has
06µbi(E)−bνiabs(E)6δabs(n), where
δabs(n) := sup
F
(µbmin(F)−νbminabs(F)),
withF running over the set of Hermitian adelic vector bundles of ranknover the spectrum of a number field.
Given a Hermitian adelic vector bundle E over SpecK, we construct a Euclidean lattice as follows. We consider the real vector space
M
σ∈MK,∞
E⊗KKσ,
which is of dimension dimK(E)[K :Q]. We equip this vector space with the orthogonal direct sum of the normsk.kσ. Then
E:={s∈E : ∀v∈MK\MK,∞, kskv 61}
forms an Euclidean lattice inside L
σ∈MK,∞E ⊗K Kσ. It can be shown that one has (see for example [10] Proposition 4.6)
(4.2) νminBV(E)>νmin(E).
Moreover, the inequality
(4.3) µbmin(E)6µbmin(E) +ln|∆K| [K:Q]
holds, provided that E is pure. This can for example be deduced from [3, (2.1.13)]. Therefore Banaszczyk’s transference theorem applied to E leads to
(see [10] Corollary 4.8) the following inequality for any pure Hermitian adelic vector bundle
µbmin(E)−νbminBV(E)6ln([K :Q] rkK(E)) +ln|∆K| [K :Q].
In the case where the Hermitian adelic vector bundle E is not pure, the inequality (4.2) remains true. However, the inequality (4.3) should be corrected by the “default of purity” introduced in [15, §2.1.1].
4.2. Arakelov geometry method. — LetE be a Hermitian adelic vector bundle on SpecK. The Arakelov theory permits to consider the function
−ddegE(.) on (E ⊗K Ka)\ {0} as an Arakelov height function on the set of algebraic points of P(E∨). Denote byOE∨(1) be the tautological line bundle on P(E∨).
For any place v ∈ MK, let P(E∨)anv be the analytic space (in the sense of Berkovich ifv is non-Archimedean) associated with
P(E∨)×SpecKSpecKv ∼=P(EK∨v).
Recall that each point x of P(E∨)anv corresponds to a pair(P(x),|.|x), where P(x)is a scheme point ofP(EK∨
v), and |.|x is an absolute value on the residue field of P(x) which extends the absolute value|.|v on Kv. We denote bybκ(x) the completion of the residue field of P(x) with respect to the absolute value
|.|x. By abuse of notation, the continuous extension of |.|x on bκ(x) is still denoted by |.|x. By the functorial interpretation of the scheme P(EK∨
v), the scheme pointP(x) corresponds to a surjectivebκ(x)-linear map
EK∨v⊗Kvbκ(x)−→P(x)∗(OE∨(1)Kv)⊗Kvbκ(x).
The extension of the dual norm k.kv,∗ induces by quotient a norm on the one-dimensional vector space
OE∨(1)(x) :=P(x)∗(OE∨(1)Kv)⊗Kvbκ(x),
denoted by|.|FSv(x)and called theFubini-Study normonOE∨(1)(x)associated with k.kv. The Fubini-Study norms |.|FSv(x)
x∈P(E∨)anv form a continuous metric onOE∨(1)Kv, called theFubini-Study metric associated withk.kv. The line bundle OE∨(1)equipped with the family of Fubini-Study metrics indexed by v∈MK forms anadelic line bundle on P(E∨) in the sense of Zhang [26].
LetXbe an integral projective scheme over SpecK andL= (L,(ϕv)v∈MK) be an adelic line bundle on X. Recall that L is an invertible OX-module, and each ϕv is a continuous metric on Lv, the pull-back of L on Xv = X×SpecKSpecKvby the projection morphism. For any closed pointyofX, the metric structure ofLinduces a structure of adelic line bundle ony∗(L)(viewed as a one-dimensional vector space over the residue field of y). The Arakelov height ofy with respect toLis defined as the Arakelov degree of y∗(L)and is
denoted byhL(y). Moreover, the linear seriesH0(X, L) is naturally equipped with supremum normsk.kϕv, where
s∈H0(X, L)⊗KKv =H0(Xv, Lv), kskϕv := sup
x∈Xvan
|s|ϕv(x).
The datum
H0(X, L) = (H0(X, L),(k.kϕv)v∈MK) forms actually an adelic vector bundle onSpecK.
The following conjecture relates the essential minimum of the height function to an invariant of the metrised graded linear series ofL.
Conjecture 4.1. — Let X be an integral projective scheme overSpecK and L = (L,(ϕv)v∈MK) be an adelic line bundle on X. Assume that L is big and the metrics ϕv are semi-positive, then the following equality holds:
(4.4) sup
Y(X
y∈(X\Yinf)(Ka)hL(y) = lim
N→+∞
µbmax(H0(X, L⊗N),(k.kϕ⊗N
v )v∈MK)
N ,
where Y runs over the set of all strict closed subscheme of X, and for any non-zero adelic vector bundleE onSpecK, bµmax(E) denotes the supremum of slopes of non-zero adelic vector subbundles of E.
Remark 4.2. — The left-hand side of the formula (4.4) is a classic invariant of height function, called the essential minimum of L, which is denoted by ess.min.(L). The limit on the right-hand side of the formula (4.4) is called the asymptotic maximal slope of L (see [7, §4.2] for the proof of its existence). It can be shown that the essential minimum is always bounded from below by the asymptotic maximal slope. The equality between these invariants is closely related to the equidistribution problem of algebraic points of small height in an arithmetic projective variety. We refer the readers to [8, §5.2] for more details.
Let y be a closed point of P(E∨), which corresponds to a one-dimensional quotient space of E∨ ⊗K Ka, or, by duality, to a one-dimensional vector subspace ofE⊗KKa. Note that theheight ofywith respect to the adelic line bundleOE∨(1)is equal to−ddeg(sy), wheresy is an arbitrary non-zero element ofE⊗KKa which spans the one-dimensional vector subspace corresponding to y. Therefore−νminabs(E)identifies with the essential infimum of heights of closed points ofP(E∨)avoiding linear closed subschemes of codimension1spanned by non-zero vectors ofE⊗KKa of height >−νn−1abs(E), where nis the dimension of E over K. Therefore, one has
(4.5) ess.min.(OE∨(1))>−νminabs(E).
In [3], the supremum norm of the Fubini-Study metric has been compared with the symmetric product norm. For anyv ∈MKand anyN ∈N, we denote
by k.kFS⊗N
v the supremum norm on
H0(P(E∨),OE∨(N))⊗KKv ∼= SymN(EK∨v)
associated with the N-th tensor power of the Fubini-Study metric, and by k.kSymN
v the N-th symmetric power of the norm k.kv,∗. By [3] Lemma 4.3.6 and Corollary 1.4.3, the following inequality holds for any Archimedean places σ:
Moreover, by [17, Theorem 6.1] (see also §8.2loc. cit.), one has
µbmax(SymN(E∨),(k.kSymN Combining (4.5) and (4.6), we obtain the following result.
Theorem 4.3. — Assume that Conjecture 4.1 is true. For any Hermitian adelic vector bundle E over SpecK, the following inequality holds
(4.7) µbmin(E)6bνminabs(E) +
Remark 4.4. — It is plausible that the method of [11, §4] leads to a finer inequality (4.6) in replacingPn−1
i=1 1
i by 12ln(n).
4.3. Application to the transference problem. — We have observed that a transference inequality for the sum of the last minimum of a Hermitian adelic vector bundle and the first minimum of its dual leads to a comparison result between successive minima and successive slope. In this subsection, we explain how to deduce a general transference inequality from the comparison between minima and slopes.
Proposition 4.5. — Let n be an integer such that n > 1. Let \ ∈ {BV,RT,abs}. For any Hermitian adelic vector bundle E of rank n on SpecK, one has
(4.8) bνi\(E) +νbn+1−i\ (E∨)>−2δ\(n).
Proof. — By definition one has
νbi\(E)>bµi(E)−δ\(n) and bνn+1−i\ (E)>bµn+1−i(E)−δ\(n).
Taking the sum of the two inequality we obtain
νbi\(E) +bνn+1−i\ (E∨)>bµi(E) +µbn+1−i(E∨)−2δ\(n) =−2δ\(n).
The interest of the above proposition is to deduce, from an upper bound of δ\(n), a lower bound for bνi\(E) +bνn+1−i\ (E∨). In particular, we propose the following conjecture.
Conjecture 4.6. — Letnbe an integer such thatn>1andE be a Hermitian adelic vector bundle on SpecK. For anyi∈ {1, . . . , n} one has
νbiabs(E) +bνn+1−iabs (E∨)>−ln(n).
This conjecture is an analogue over Qa of Banaszczyk’s transference theo-rem. It can for example be deduced from Conjecture 4.1 and the following conjectural inequality
lim sup
N→+∞
bµmax(SymN(E∨))
N 6bµmax(E) +1 2ln(n).
Such observation opens a new way to study the transference problem by Arakelov geometry instead of classic geometry of numbers.
Acknowledgement. — This note corresponds to my lecture at the 8th Inter-national Congress of Chinese Mathematicians. I am grateful to the organisers for the invitation. This note has been written during my visit to Beijing In-ternational Center for Mathematical Research. Moreover, the contents of the third section come from my lecture at Kyoto University in the framework of
“Top Global Course Special Lectures”. I would like to thank these institutions for the hospitality. Finally, I thank François Ballaÿ for his careful reading of the text and for his comments.
References
[1] W. Banaszczyk– “New bounds in some transference theorems in the geometry of numbers”,Mathematische Annalen 296(1993), no. 4, p. 625–635.
[2] T. Borek – “Successive minima and slopes of Hermitian vector bundles over number fields”,Journal of Number Theory 113(2005), no. 2, p. 380–388.
[3] J.-B. Bost, H. Gillet&C. Soulé– “Heights of projective varieties and positive Green forms”,Journal of the American Mathematical Society7(1994), no. 4, p. 903–
1027.
[4] J.-B. Bost – “Périodes et isogenies des variétés abéliennes sur les corps de nombres (d’après D. Masser et G. Wüstholz)”, no. 237, 1996, Séminaire Bourbaki, Vol. 1994/95, p. Exp. No. 795, 4, 115–161.
[5] J.-B. Bost– “Algebraic leaves of algebraic foliations over number fields”, Publica-tions Mathématiques. Institut de Hautes Études Scientifiques (2001), no. 93, p. 161–
221.
[6] A. Chambert-Loir – “Théorèmes d’algébricité en géométrie diophantienne (d’après J.-B. Bost, Y. André, D. & G. Chudnovsky)”, no. 282, 2002, Séminaire Bourbaki, Vol. 2000/2001, p. Exp. No. 886, viii, 175–209.
[7] H. Chen– “Convergence des polygones de Harder-Narasimhan”,Mémoires de la Société Mathématique de France. Nouvelle Série (2010), no. 120, p. 116.
[8] H. Chen – “Differentiability of the arithmetic volume function”, Journal of the London Mathematical Society. Second Series 84 (2011), no. 2, p. 365–384.
[9] H. Chen – “Géométrie d’arakelov: théorèmes de limite et comptage des points rationnels”, Habilitation, Université Paris Diderot, December 2011.
[10] H. Chen – “Sur la comparaison entre les minima et les pentes”, Publications mathématiques de Besançon. Algèbre et théorie des nombres 2018(2018), p. 5–23.
[11] H. Chen & A. Moriwaki– “Arakelov geometry over adelic curves”, preprint, 2019.
[12] G. Faltings – “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Inventiones Mathematicae 73(1983), no. 3, p. 349–366.
[13] G. Faltings– “Erratum: “Finiteness theorems for abelian varieties over number fields” ”,Inventiones Mathematicae 75(1984), no. 2, p. 381.
[14] E. Gaudron – “Pentes des fibrés vectoriels adéliques sur un corps global”, Rendiconti del Seminario Matematico della Università di Padova. Mathematical Journal of the University of Padua 119(2008), p. 21–95.
[15] E. Gaudron– “Géométrie des nombres adélique et lemmes de Siegel généralisés”, Manuscripta Mathematica 130(2009), no. 2, p. 159–182.
[16] E. Gaudron– “Minima and slopes of rigid adelic spaces”, inArakelov geometry and Diophantine applications (E. Peyre & G. Rémond, eds.), 2017.
[17] E. Gaudron & G. Rémond– “Minima, pentes et algèbre tensorielle”, Israel Journal of Mathematics 195(2013), no. 2, p. 565–591.
[18] D. R. Grayson – “Reduction theory using semistability”, Commentarii Math-ematici Helvetici 59(1984), no. 4, p. 600–634.
[19] W. Gubler – “Heights of subvarieties over M-fields”, inArithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, p. 190–227.
[20] G. Harder & M. S. Narasimhan – “On the cohomology groups of moduli spaces of vector bundles on curves”,Mathematische Annalen212(1974/75), p. 215–
248.
[21] N. M. Katz – “Nilpotent connections and the monodromy theorem: Applica-tions of a result of Turrittin”,Institut des Hautes Études Scientifiques. Publications Mathématiques (1970), no. 39, p. 175–232.
[22] D. Masser & G. Wüstholz – “Isogeny estimates for abelian varieties, and finiteness theorems”, Annals of Mathematics. Second Series 137 (1993), no. 3, p. 459–472.
[23] J. Neukirch–Algebraic number theory, Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.
[24] C. Soulé – “Hermitian vector bundles on arithmetic varieties”, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, p. 383–419.
[25] U. Stuhler– “Eine Bemerkung zur Reduktionstheorie quadratischer Formen”, Archiv der Mathematik 27(1976), no. 6, p. 604–610.
[26] S. Zhang– “Positive line bundles on arithmetic varieties”,Journal of the Amer-ican Mathematical Society 8(1995), no. 1, p. 187–221.
October 6, 2019
Huayi Chen, Université de Paris, IMJ-PRG, CNRS, F-75013 Paris, France • Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche, IMJ-PRG, F-75005 Paris, France • E-mail : huayi.chen@imj-prg.fr
Url :webusers.imj-prg.fr/~huayi.chen