Maximal slope of tensor product of Hermitian vector bundles
aa r X i v : . [ m a t h . AG ] J a n Maximal slope of tensor product of Hermitian vectorbundles
Chen
Huayi ∗ November 12, 2018
Abstract
We give an upper bound for the maximal slope of the tensor product of several non-zeroHermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski’sFirst Theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian linesubbundle M of the tensor product. In the case where the generic fiber of M is semistablein the sense of geometric invariant theory, the estimation is established by constructing(through the classical invariant theory) a special polynomial which does not vanish onthe generic fibre of M . Otherwise we use an explicte version of a result of Ramanan andRamanathan to reduce the general case to the former one. It is well known that on a projective and smooth curve defined over a field of characteristic0, the tensor product of two semistable vector bundles is still semistable. This result has beenfirstly proved by Narasimhan and Seshadri [NS65] by using analytic method in the complexalgebraic geometry framework. Then this result has been reestablished by Ramanan andRamanathan [RR84] in purely algebraic context, through the geometric invariant theory. Theirmethod is based on a result of Kempf [Kem78], which has also been independently obtainedby Rousseau [Rou78], generalizing the Hilbert-Mumford criterion [MFK94] of semistability inthe sense of geometric invariant theory. By reformulating the results of Kempf and Ramanan-Ramanathan, Totaro [Tot96] (see also [dS] for a review) has given a new proof of a conjecturedue to Fontaine [Fon79], which had been firstly proved by Faltings [Fal89] asserting that thetensor product of two semistable admissible filtered isocristals is still semistable.Let us go back to the case of vector bundles. Consider a smooth projective curve C definedover a field k . For any non-zero vector bundle E on C , the slope of E is defined as the quotientof its degree by its rank and is denoted by µ ( E ). The maximal slope µ max ( E ) of E is themaximal value of slopes of all non-zero subbundles of E . By definition, µ ( E ) ≤ µ max ( E ). Wesay that E is semistable if the equality µ ( E ) = µ max ( E ) holds. If E and F are two non-zerovector bundles on C , then µ ( E ⊗ F ) = µ ( E ) + µ ( F ). The result of Ramanan-Ramanathan[RR84] implies that, if k is of characteristic 0, then the equality holds for maximal slopes, i.e., µ max ( E ⊗ F ) = µ max ( E ) + µ max ( F ). When the characteristic of k is positive, this equalityis not true in general (see [Gie73] for a counter-example). Nevertheless, there always exists aconstant a which only depends on C such that µ max ( E ) + µ max ( F ) ≤ µ max ( E ⊗ F ) ≤ µ max ( E ) + µ max ( F ) + a. (1) ∗ CMLS, Ecole Polytechnique, Palaiseau 91120, France. ([email protected]) K be a number field and O K be its integer ring. We denote by Σ ∞ the setof all embeddings of K into C . A Hermitian vector bundle E = ( E, h ) on Spec O K is bydefinition a projective O K -module of finite type E together with a family of Hermitian metrics h = ( k · k σ ) σ ∈ Σ ∞ , where for any σ ∈ Σ ∞ , k · k σ is a Hermitian norm on E ⊗ O K ,σ C , subject tothe condition that the data ( k · k σ ) σ ∈ Σ ∞ is invariant by the complex conjugation. That is, forany e ∈ E , z ∈ C and σ ∈ Σ ∞ , we have k e ⊗ z k σ = k e ⊗ z k σ .The (normalized) Arakelov degree of a Hermitian vector bundle E of rank r on Spec O K isdefined as d deg n E = 1[ K : Q ] (cid:16) log E/ O K s + · · · + O K s r ) − X σ ∈ Σ ∞ log det( h s i , s j i σ ) (cid:17) , where ( s , · · · , s r ) is an arbitrary element in E r which defines a basis of E K over K . Thisdefinition does not depend on the choice of ( s , · · · , s r ). The function d deg n is invariant by anyfinite extension of K . That is, if K ′ /K is a finite extension and if E ′ = E ⊗ O K O K ′ , then d deg n ( E ′ ) = d deg n ( E ). The slope of a non-zero Hermitian vector bundle E on Spec O K is definedas the quotient b µ ( E ) := d deg n ( E ) / rk( E ). For more details, see [Bos96], [Bos01], [CL02].We say that a non-zero Hermitian vector bundle E is semistable if the maximal slope b µ max ( E ) of E , defined as the maximal value of slopes of its non-zero Hermitian subbundles,equals its slope. If E is a non-zero Hermitian vector bundle on Spec O K , Stuhler [Stu76] andGrayson [Gra84] have proved that there exists a unique Hermitian subbundle E des of E having b µ max ( E ) as its slope and containing all Hermitian subbundle F of E such that b µ ( F ) = b µ max ( E ).Clearly E is semistable if and only if E = E des . If it is not the case, then E des is said to bethe Hermitian subbundle which destabilizes E .In a lecture at Oberwolfach, J.-B. Bost [Bos97] has conjectured that the tensor productof two semistable Hermitian vector bundles on Spec O K is semistable. This conjecture isequivalent to the assertion that for any non-zero Hermitian vector bundles E and F on Spec O K , b µ max ( E ⊗ F ) = b µ max ( E ) + b µ max ( F ) . We always have the inequality b µ max ( E ⊗ F ) ≥ b µ max ( E ) + b µ max ( F ). But the inverse inequalityremains open. Several special cases of this conjecture have been proved. Some estimations oftype (1) have been established with error terms depending on the ranks of the vector bundlesand on the number field K . We resume some known results on this conjecture.1) By definition of the maximal slope, if E is a non-zero Hermitian vector bundle and if L isa Hermitian line bundle, that is, a Hermitian vector bundle of rank one, then b µ max ( E ⊗ L ) = b µ max ( E ) + d deg n ( L ) = b µ max ( E ) + b µ max ( L ) . The geometric counterpart of this equality is also true for positive characteristic case.2) De Shalit and Parzanovski [dSP06] have proved that, if E and F are two semistable Her-mitian vector bundles on Spec Z such that rk E + rk F ≤
5, then E ⊗ F is semistable.3) In [Bos96] (see also [Gra00]), using the comparison of a Hermitian vector bundle to a directsum of Hermitian line bundles, Bost has proved that b µ max ( E ⊗ · · · ⊗ E n ) ≤ n X i =1 (cid:16)b µ max ( E i ) + 3 rk E i log(rk E i ) (cid:17) for any family of non-zero Hermitian vector bundles ( E i ) ni =1 on Spec O K .2) Recently, Bost and K¨unnemann [BK07] have proved that, if K is a number field and if E and F are two non-zero Hermitian vector bundles on Spec O K , then b µ max ( E ⊗ F ) ≤ b µ max ( E ) + b µ max ( F ) + 12 (cid:0) log rk E + log rk F (cid:1) + log | ∆ K | K : Q ] , where ∆ K is the discriminant of K .We state the main result of this article as follows: Theorem 1.1
Let K be a number field and O K be its integer ring. If ( E i ) ni =1 is a family ofnon-zero Hermitian vector bundles on Spec O K , then b µ max ( E ⊗ · · · ⊗ E n ) ≤ n X i =1 (cid:16)b µ max ( E i ) + log(rk E i ) (cid:17) . (2)The idea goes back to an article of Bost [Bos94] inspired by Bogomolov [Ray81], Gieseker[Gie77] and Cornalba-Harris [CH88]. In an article of Gasbarri [Gas00] appears also a similaridea. By Minkowski’s First Theorem, we reduce our problem to finding an upper bound forthe Arakelov degree of an arbitrary Hermitian line subbundle M of E ⊗ · · · ⊗ E n . In thecase where M K is semistable (in the sense of geometric invariant theory) for the action of GL ( E ,K ) × · · · × GL ( E n,K ), the classical invariant theory gives invariant polynomials withcoefficients in Z whose Archimedian norms are “small”. The general case can be reduced tothe former one using an explicit version of a result of Ramanan-Ramanathan [RR84].The structure of this article is as follows. In the second section we fix the notation andpresent some preliminary results. In the third section we recall the first principal theorem inclassical invariant theory and discuss some generalizations in the case of several vector spaces.We then establish in the fourth section an upper bound for the Arakelov degree of a Hermitianline subbundle with semistable hypothesis. The fifth section is contributed to some basicnotions for filtrations in the category of vector spaces. Then in the sixth section, we state anexplicit version of a result of Ramanan-Ramanathan in our context and, following the methodof Totaro, give a proof for it. In the seventh section is presented a criterion of semistability (forHermitian vector bundles) which is an arithmetic analogue of a result of Bogomolov. In theeighth section, we explain how to use the result in previous sections to reduce the majorationof the Arakelov degree of an arbitrary Hermitian line subbundle to the case with semistabilityhypothesis, which has already been discussed in the fourth section. Finally, we give the proofof Theorem 1.1 in the ninth section.The result presented here is part of my doctorial thesis [Che06], supervised by J.-B. Bost.The ideas in this article are largely inspired by his article [Bos94] and his personal notes. Iwould like to thank him deeply for his instruction and his sustained encouragement. Duringmy visit to Institut Joseph Fourier in Grenoble, E. Gaudron pointed out to me that the methodin this article, combined with his recent result [Gau07], leads to an estimation which is similarto (2) for the tensor product of Adelic vector bundles. I am grateful to him for discussions andfor suggestions. I would also like to express my gratitude to the referee for his/her very carefulreading and for his/her numerous useful suggestions to improve the writing of this article.3
Notation and preliminary results
Throughout this article, if K is a field and if V is a vector space of finite rank over K , wedenote by P ( V ) the K -scheme which represents the functor Schemes /K −→ Sets ( p : S → Spec K ) n locally free quotientof rank 1 of p ∗ V o (3)In particular, P ( V )( K ) classifies all hyperplanes in V , or equivalently, all lines in V ∨ . Wedenote by O V (1) the canonical line bundle on P ( V ). In other words, if π : P ( V ) → Spec K isthe structural morphism, then O V (1) is the quotient of π ∗ V defined by the universal object ofthe representable functor (3). For any integer m ≥
1, we use the expression O V ( m ) to denotethe line bundle O V (1) ⊗ m .Let G be an algebraic group over Spec K and X be a projective variety over Spec K .Suppose that G acts on X and that L is an ample G -linearized line bundle on X . We saythat a rational point x of X is semistable for the action of G relatively to L if there exists aninteger D ≥ s ∈ H ( X, L ⊗ D ) invariant by the action of G such that x lies inthe open subset of X defined by the non-vanishing of s . Clearly x is semistable for the actionof G relatively to L if and only if it is semistable for the action of G relatively to any strictlypositive tensor power of L .In particular, if G ( K ) acts linearly on a vector space V of finite rank over K , then the actionof G on V induces naturally an action of G on P ( V ), and O V (1) becomes a G -linearized linebundle. Let R be a vector subspace of rank 1 of V ∨ , which is viewed as a point in P ( V )( K ).Then R is semistable for the action of G relatively to O V (1) if and only if there exists aninteger m ≥ s ∈ H ( P ( V ) , O V ( m )) = S m V which is invariant by theaction of G ( K ) such that the composed homomorphism R / / V ∨ s / / K is non-zero.We present some estimations for maximal slopes in geometric case. Let k be an arbitraryfield and C be a smooth projective curve of genus g defined over k . Let b = min { deg( L ) | L ∈ Pic( C ) , L is ample } and let a = b + g − Lemma 2.1
Let E be a non-zero vector bundle on C . If H ( C, E ) = 0 , then µ max ( E ) ≤ g − .Proof. Since H ( C, E ) = 0, for any non-zero subbundle F of E , we also have H ( C, F ) = 0.Recall that the Riemann-Roch theorem asserts thatrk k H ( C, F ) − rk k H ( C, F ) = deg( F ) + rk( F )(1 − g ) . Then deg( F ) + rk( F )(1 − g ) ≤
0, which implies µ ( F ) ≤ g −
1. Since F is arbitrary, µ max ( E ) ≤ g − ✷ Proposition 2.2
For any non-zero vector bundles E and F on C , we have the inequality µ max ( E ) + µ max ( F ) ≤ µ max ( E ⊗ F ) ≤ µ max ( E ) + µ max ( F ) + a, where a = b + g − only depends on C . roof.
1) Let E be a subbundle of E such that µ ( E ) = µ max ( E ) and let F be a subbundleof F such that µ ( F ) = µ max ( F ). Since E ⊗ F is a subbundle of E ⊗ F , we obtain µ max ( E ) + µ max ( F ) = µ ( E ) + µ ( F ) = µ ( E ⊗ F ) ≤ µ max ( E ⊗ F ) , which is the first inequality.2) We first prove that, if E ′ and E ′′ are two non-zero vector bundles on C such that µ max ( E ′ ) + µ max ( E ′′ ) <
0, then µ max ( E ′ ⊗ E ′′ ) ≤ g −
1. In fact, if µ max ( E ′ ⊗ E ′′ ) > g −
1, thenby Lemma 2.1, H ( C, E ′ ⊗ E ′′ ) = 0. Therefore, there exists a non-zero homomorphism ϕ from E ′∨ to E ′′ . Let G be the image of ϕ , which is non-zero since ϕ is non-zero. The vector bundle G is a subbundle of E ′′ and a quotient bundle of E ′∨ . Hence G ∨ is a subbundle of E ′∨∨ ∼ = E ′ .Therefore, we have µ ( G ) ≤ µ max ( E ′′ ) and µ ( G ∨ ) = − µ ( G ) ≤ µ max ( E ′ ). By taking the sum,we obtain µ max ( E ′ ) + µ max ( E ′′ ) ≥ b , there exists a linebundle M such that − b ≤ µ max ( E ) + µ max ( F ) + deg( M ) = µ max ( E ⊗ M ) + µ max ( F ) <
0. Then,by combining the previously proved result, we obtain µ max ( E ⊗ M ⊗ F ) ≤ g −
1. Therefore, µ max ( E ⊗ F ) ≤ g − − deg( M ) ≤ µ max ( E ) + µ max ( F ) + g + b − . ✷ We now recall some classical results in Arakelov theory, which will be useful afterwards.We begin by introducing the notation.Let E be a Hermitian vector bundle on Spec O K . For any finite place p of K , we denoteby K p the completion of K with respect to p , equipped with the absolute value | · | p which isnormalized as | · | p = O K / p ) − v p ( · ) with v p being the discrete valuation associated to p . Thestructure of O K -module on E induces naturally a norm k · k p on E K p := E ⊗ K K p such that E K p becomes a Banach space over K p .If L is a Hermitian line bundle on Spec O K and if s is an arbitrary non-zero element in L ,then d deg n ( L ) = 1[ K : Q ] (cid:16) log L/ O K s ) − X σ : K → C log k s k σ (cid:17) , which can also be written as d deg n ( L ) = − K : Q ] (cid:16) X p log k s k p + X σ : K → C log k s k σ (cid:17) . (4)Note that this formula is analogous to the degree function of a line bundle on a smoothprojective curve. Similarly to the geometric case, for any Hermitian vector bundle E of rank r on Spec O K , we have d deg n ( E ) = d deg n (Λ r E ) (5)where Λ r E is the r th exterior power of E , that is, the determinant of E , which is a Hermitianline bundle. Furthermore, if 0 / / E ′ / / E / / E ′′ / / O K , the following equality holds: d deg n ( E ) = d deg n ( E ′ ) + d deg n ( E ′′ ) . (6) Lemma 2.3 If E and F are two Hermtian vector bundles of ranks r and r on Spec O K ,respectively. Then d deg n ( E ⊗ F ) = rk( E ) d deg n ( F ) + rk( F ) d deg n ( E ) . (7)5 roof. The determinant Hermitian line bundle Λ r + r ( E ⊗ F ) is isomorphic to (Λ r E ) ⊗ r ⊗ (Λ r F ) ⊗ r . Taking Arakelov degree and using (5) we obtain (7). ✷ We establish below the arithmetic analogue to the first inequality in Proposition 2.2.
Proposition 2.4
Let E and F be two non-zero Hermitian vector bundles on Spec O K . Then b µ max ( E ) + b µ max ( F ) ≤ b µ max ( E ⊗ F ) . Proof.
Let E des and F des be the Hermitian subbundles of E and of F respectively as definedin Section 1. By definition, b µ ( E des ) = b µ max ( E ) and b µ ( F des ) = b µ max ( F ). Since E des ⊗ F des isa Hermitian vector subbundle of E ⊗ F , we obtain b µ max ( E ) + b µ max ( F ) = b µ ( E des ) + b µ ( F des ) = b µ ( E des ⊗ F des ) ≤ b µ max ( E ⊗ F ) , where the second equality results from (7). ✷ Corollary 2.5
Let ( E i ) ≤ i ≤ n be a finite family of non-zero Hermitian vector bundles on Spec O K . Then the following equality holds: b µ max ( E ) + · · · + b µ max ( E n ) ≤ b µ max ( E ⊗ · · · ⊗ E n ) . (8)Let E and F be two Hermitian vector bundles and ϕ : E K → F K be a non-zero K -linearhomomorphism. For any finite place p of K , we denote by h p ( ϕ ) the real number log k ϕ p k ,where ϕ p : E K p → F K p is induced from ϕ by scalar extension. Note that if ϕ is induced byan O K -homomorphism from E to F , then h p ( ϕ ) ≤ p . Similarly, for anyembedding σ : K → C , we define h σ ( ϕ ) = log k ϕ σ k , where ϕ σ : E σ, C → F σ, C is given by thescalar extension σ . Finally, we define the height of ϕ as h ( ϕ ) = 1[ K : Q ] (cid:16) X p h p ( ϕ ) + X σ : K → C h σ ( ϕ ) (cid:17) . Proposition 2.6 ([Bos96])
Let E and F be two Hermitian vector bundles on Spec O K and ϕ : E K → F K be a K -linear homomorphism.1) If ϕ is injective, then b µ ( E ) ≤ b µ max ( F ) + h ( ϕ ) . (9)
2) If ϕ is non-zero, then b µ min ( E ) ≤ b µ max ( E ) + h ( ϕ ) (10) where b µ min ( E ) is the minimal value of slopes of all non-zero Hermitian vector quotientbundles of E . For any non-zero Hermitian vector bundle E on Spec O K , let u d deg n ( E ) be the maximaldegree of line subbundles of E . We recall a result of Bost and K¨unnemann comparing themaximal degree and the maximal slope of E , which is a variant of Minkowski’s First Theorem. Proposition 2.7 ([BK07] (3.27))
Let E be a non-zero Hermitian vector bundle on Spec O K .Then u d deg n ( E ) ≤ b µ max ( E ) ≤ u d deg n ( E ) + 12 log(rk E ) + log | ∆ K | K : Q ] , (11) where ∆ K is the discriminant of K . Reminder on invariant theory
In this section we recall some known results in classical invariant theory. We fix K to be afield of characteristic 0. If V is a vector space over K and if u ∈ N , then the expression V ⊗ ( − u ) denotes the space V ∨⊗ u .Let V be a finite dimensional non-zero vector space over K . For any u ∈ N , we denoteby J u : End K ( V ) ⊗ u → End K ( V ⊗ u ) the K -linear homomorphism (of vector spaces) whichsends the tensor product T ⊗ · · · ⊗ T u of u elements in End K ( V ) to their tensor productas an endomorphism of V ⊗ u . The mapping J u is actually a homomorphism of K -algebras.Furthermore, as a homomorphism of vector spaces, J u can be written as the composition ofthe following natural isomorphisms:End K ( V ) ⊗ u / / ( V ∨ ⊗ V ) ⊗ u / / ( V ∨ ) ⊗ u ⊗ V ⊗ u / / ( V ⊗ u ) ∨ ⊗ V ⊗ u / / End K ( V ⊗ u ) , so is itself an isomorphism. Moreover, there exists an action of the symmetric group S u on V ⊗ u by permuting the factors. This representation of S u defines a homomorphism from thegroup algebra K [ S u ] to End K ( V ⊗ u ). The elements of S u act by conjugation on End K ( V ⊗ u ).If we identify End K ( V ⊗ u ) with End K ( V ) ⊗ u by the isomorphism J u , then the corresponding S u -action is just the permutation of factors in tensor product. Finally the group GL K ( V ) actsdiagonally on V ⊗ u .When u = 0, J reduces to the identical homomorphism Id : K → K , and S reduces tothe group of one element. The “diagonal” action of GL K ( V ) on V ⊗ ∼ = K is trivial.We recall below the “first principal theorem” of classical invariant theory (cf. [Wey97]Chapter III, see also [ABP73] Appendix 1 for a proof). Theorem 3.1
Let V be a finite dimensional non-zero vector space over K . Let u ∈ N and v ∈ Z . If T is a non-zero element in V ∨⊗ u ⊗ V ⊗ v , which is invariant by the action of GL K ( V ) ,then u = v , and T is a linear combination of permutations in S u acting on V (here we identify V ∨⊗ u ⊗ V ⊗ u with End K ( V ⊗ u ) ). We now present a generalization of Theorem 3.1 to the case of several linear spaces. In therest of this section, we fix a family ( V i ) ≤ i ≤ n of finite dimensional non-zero vector space over K . For any mapping α : { , · · · , n } → Z , we shall use the notation V α := V ⊗ α (1)1 ⊗ · · · ⊗ V ⊗ α ( n ) n (12)to simplify the writing. Denote by G the algebraic group GL K ( V ) × K · · · × K GL K ( V n ).Then G ( K ) is the group GL K ( V ) × · · · × GL K ( V n ). For any mapping α : { , · · · , n } → N with natural integer values, we denote by S α the product S α (1) × · · · × S α ( n ) of symmetricgroups. We have a natural isomorphism of K -algebras from End K ( V α ) to End K ( V ) ⊗ α (1) ⊗ K · · · ⊗ K End K ( V n ) ⊗ α ( n ) . The group G ( K ) acts naturally on V α and the group S α acts on V α by permutating tensor factors. By using induction on n , Theorem 3.1 implies the followingcorollary: Corollary 3.2
With the notation above, if α : { , · · · , n } → N and β : { , · · · , n } → Z aretwo mappings and if T is a non-zero element in ( V α ) ∨ ⊗ V β which is invariant by the actionof G ( K ) , then α = β , and T is a linear combination of elements in S α acting on V α . Let A be a finite family of mappings from { , · · · , n } to N and ( b i ) ≤ i ≤ n be a family ofintegers. We denote by W the vector space L α ∈A V α . Note that the group G ( K ) acts naturally7n W . Let L be the G ( K )-module (det V ) ⊗ b ⊗ · · · ⊗ (det V n ) ⊗ b n . For any integer D ≥ α = ( α j ) ≤ j ≤ D ∈ A D , letpr α : W ⊗ D −→ V α ⊗ · · · ⊗ V α D be the canonical projection. For any integer i ∈ { , · · · , n } , let r i be the rank of V i over K .Finally let π : P ( W ∨ ) → Spec K be the canonical morphism. Theorem 3.3
With the notation above, if m is a strictly positive integer and if R is a vectorsubspace of rank of W (considered as a rational point of P ( W ∨ ) ) which is semistable for theaction of G relatively to O W ∨ ( m ) ⊗ π ∗ L , then there exists an integer D ≥ and a family α =( α j ) ≤ j ≤ mD of elements in A such that, by noting A = α + · · · + α mD , we have A ( i ) = Db i r i and hence b i ≥ for any i .Furthermore, there exists an element σ ∈ S A such that the composition of homomorphisms R ⊗ mD ⊗ L ∨⊗ D / / W ⊗ mD ⊗ L ∨⊗ D pr α ⊗ Id / / V A ⊗ L ∨⊗ Dσ ⊗ Id (cid:15) (cid:15) V A ⊗ L ∨⊗ D det V ⊗ Db ⊗···⊗ det Vn ⊗ Dbn ⊗ Id (cid:15) (cid:15) L ⊗ D ⊗ L ∨⊗ D ∼ = K does not vanish, where the first arrow is induced by the canonical inclusion of R ⊗ nD in W ⊗ nD .Proof. Since R is semistable for the action of G relatively to O W ∨ ( m ) ⊗ π ∗ L , there exists aninteger D ≥ s ∈ S mD ( W ∨ ) ⊗ L ⊗ D which is invariant by the action of G ( K )such that the composition of homomorphisms R ⊗ mD ⊗ L ∨⊗ D / / S mD ( W ∨ ) ∨ ⊗ L ∨⊗ D s / / K does not vanish, the first arrow being the canonical inclusion.As K is of characteristic 0, S md ( W ∨ ) is a direct factor as a GL( W )-module of W ∨⊗ mD .Hence S mD ( W ∨ ) ⊗ L ⊗ D is a direct factor as a G ( K )-module of W ∨⊗ mD ⊗ L ⊗ D . So we canchoose s ′ ∈ W ∨⊗ mD ⊗ L ⊗ D invariant by the action of G ( K ) such that the class of s ′ in S mD ( W ∨ ) ⊗ L ⊗ D coincides with s . There then exists α = ( α j ) ≤ j ≤ mD ∈ A D such that thecomposition R ⊗ mD ⊗ L ∨⊗ D / / W ⊗ mD ⊗ L ∨⊗ D pr α ⊗ Id / / V A ⊗ L ∨⊗ D s ′ α / / K is non-zero, where A = α + · · · + α mD and s ′ α is the component of index α of s ′ . Let B : { , · · · , n } → Z be the mapping which sends i to Db i r i . Note that for any i , Λ r i V i = det V i is naturally a direct factor of V ⊗ r i i . We can therefore choose a preimage s ′′ α of s ′ α in ( V A ) ∨ ⊗ V B which is invariant by G ( K ). By Corollary 3.2, A = B and s ′′ α is a linear combination of per-mutations acting on V . Therefore the theorem is proved. ✷ Upper bound for the degree of a Hermitian line sub-bundle with hypothesis of semistability
Let K be a number field and O K be its integer ring. Consider a family ( E i ) ≤ i ≤ n ofnon-zero Hermitian vector bundles on Spec O K . Let A be a non-empty and finite family ofnon-identically zero mappings from { , · · · , n } to N . We define a new Hermitian vector bundleover Spec O K as follows: E := M α ∈A E ⊗ α (1)1 ⊗ · · · ⊗ E ⊗ α ( n ) n . In this section, we shall use the ideas in [Bos94] to obtain an upper bound for the Arakelovdegree of a Hermitian line subbundle M of E under hypothesis of semistability (in the senseof geometric invariant theory) for M K . This upper bound is crucial because, as we shall seelater, the general case can be reduced to this special one through an argument of Ramananand Ramanathan [RR84].For any integer i such that 1 ≤ i ≤ n , let r i be the rank of E i and let V i be the vectorspace E i,K . Let W = E K and π : P ( W ∨ ) → Spec K be the canonical morphism. By definition W = L α ∈A V α , where V α is defined in (12). We denote by G the algebraic group GL K ( V ) ×· · · × GL K ( V n ) which acts naturally on P ( W ∨ ). Let ( b i ) ≤ i ≤ n be a family of strictly positiveintegers such that r i divides b i . Finally let L = (Λ r E ) ⊗ b /r ⊗ · · · (Λ r n E n ) ⊗ b n /r n . Lemma 4.1
Let H be a Hermitian space of dimension d > . Then the norm of the homo-morphism det : H ⊗ d → Λ d H equals √ d ! .Proof. Let ( e i ) ≤ i ≤ d be an orthonormal basis of H and let ( e ∨ i ) ≤ i ≤ d be its dual basis in H ∨ .If we identifies Λ d H with C via the basis e ∧ · · · ∧ e d , then the homomorphism det, viewed asan element in H ∨⊗ d , can be written as X σ ∈ S d sign( σ ) e σ (1) ⊗ · · · ⊗ e σ ( d ) , which is the sum of d ! orthogonal vectors of norm 1 in H ∨⊗ d . So its norm is √ d !. ✷ Theorem 4.2
With the notation above, if m ≥ is an integer and if M is a Hermitian linesubbundle of E such that M K is semistable for the action of G relatively to O W ∨ ( m ) ⊗ π ∗ L K ,then d deg( M ) ≤ m d deg( L ) + 12 m r X i =1 b i log(rk E i ) = n X i =1 b i m (cid:16)b µ ( E i ) + 12 log(rk E i ) (cid:17) . Proof.
By Theorem 3.3, we get, by combining the slope inequality (9) and Lemma 4.1, mD d deg( M ) − D d deg( L ) = mD d deg( M ) − n X i =1 Db i b µ ( E i ) ≤ n X i =1 A ( i ) log( r i !)2 r i = n X i =1 Db i log( r i !)2 r i ≤ D n X i =1 b i log r i , r ! ≤ r r to obtain the last inequality. Finally wedivide the inequality by mD and obtain d deg( M ) ≤ m d deg( L ) + 12 m n X i =1 b i log r i = n X i =1 b i m (cid:16)b µ ( E i ) + log r i (cid:17) . ✷ Let m be a strictly positive integer which is divisible by all r i . We apply Theorem 4.2 tothe special case where A contains a single map α such that α ( i ) = 1 for any i ∈ { , · · · , n } ,in other words, E = E ⊗ · · · ⊗ E n , and where b i = m for any integer i such that 1 ≤ i ≤ n .Then we get the following upper bound: Corollary 4.3 If M is a Hermitian line subbundle of E ⊗· · ·⊗ E n such that M K is semistablefor the action of G relatively to O W ∨ ( m ) ⊗ π ∗ L K , then we have d deg( M ) ≤ n X i =1 (cid:16)b µ ( E i ) + 12 log(rk E i ) (cid:17) . (13) In this section, we introduce some basic notation and results on R -filtrations of vectorspaces, which we shall use in the sequel. We fix a field K . Let V be a non-zero vector space of finite rank r over K . We call R - filtration of V anyfamily F = ( F λ V ) λ ∈ R of subspaces of V such that1) F λ V ⊃ F λ ′ V for all λ ≤ λ ′ ,2) F λ V = 0 for λ sufficiently positive,3) F λ V = V for λ sufficiently negative,4) the function x rk K ( F x V ) on R is left continuous.A filtration F of V is equivalent to the data of a flag V = V ) V ) V ) · · · ) V d = 0 (14)of V together with a strictly increasing sequence of real numbers ( λ i ) ≤ i Let e = ( e , · · · , e r ) be a basis of V and Z be a non-empty subset of R . Themapping Φ e : Fil Z e → Z r defined by Φ e ( F ) = ( λ F ( e ) , · · · , λ F ( e r )) (17) is a bijection. Proposition 5.2 Let v be a non-zero vector in V , F be a subfield of R and e be a basis of V .Then the function F 7→ λ F ( v ) from Fil F e to R can be written as the minimal value of a finitenumber of F -linear forms.Proof. Let v = P ri =1 a i e i be the decomposition of v in the basis e , then for any filtration F ∈ Fil F e , we have λ F ( v ) = min ≤ i ≤ na i =0 λ F ( e i ) . ✷ For any real number ε > 0, we define the dilation of F by ε as the filtration ψ ε F := ( F ελ V ) λ ∈ R (18)of V . Clearly we have E [ ψ ε F ] = ε E [ F ] and λ ψ ε F = ελ F . (19)Let ( V ( i ) ) ≤ i ≤ n be a family of non-zero vector spaces of finite rank over K and V = L ni =1 V ( i ) be their direct sum. For each integer 1 ≤ i ≤ n , let F ( i ) be a filtration of V ( i ) . Weconstruct a filtration F of V such that F λ V = n M i =1 F ( i ) λ V ( i ) . The filtration F is called the direct sum of F ( i ) and is denoted by F (1) ⊕ · · · ⊕ F ( n ) . If foreach 1 ≤ i ≤ n , e ( i ) is a basis of V ( i ) which is compatible with F ( i ) , then the disjoint union e (1) ∐ · · · ∐ e ( n ) , which is a basis of V (1) ⊕ · · · ⊕ V ( n ) , is compatible with F (1) ⊕ · · · ⊕ F ( n ) .Similarly, if W = N ni =1 V ( i ) is the tensor product of V ( i ) , we construct a filtration G of W such that G λ W = X λ + ··· + λ n ≥ λ n O i =1 F ( i ) λ i V ( i ) , tensor product of F ( i ) and denoted by F (1) ⊗ · · · ⊗ F ( n ) . If e ( i ) is a basis of V ( i ) which is compatible with the filtration F ( i ) , then the basis e (1) ⊗ · · · ⊗ e ( n ) := { e ⊗ · · · ⊗ e n | ∀ ≤ i ≤ n, e i ∈ e ( i ) } of V (1) ⊗ · · · ⊗ V ( n ) is compatible with F (1) ⊗ · · · ⊗ F ( n ) . Finally, for any ε > ψ ε ( F (1) ⊗ · · · ⊗ F ( n ) ) = ψ ε F (1) ⊗ · · · ⊗ ψ ε F ( n ) . (20) Let V be a non-zero vector space of finite rank r over K . If F and G are two filtrationsof V , then by Bruhat’s decomposition, there always exists a basis e of V which is compatiblesimultaneously with F and G . We define the scalar product of F and G as hF , Gi := 1 r r X i =1 λ F ( e i ) λ G ( e i ) . (21)This definition does not depend on the choice of e . The number kFk := hF , Fi is called the norm of the filtration F . Notice that kFk = 0 if and only if F is supported by { } . In thiscase, we say that the filtration F is trivial . Proposition 5.3 Let e be a basis of V . Then the function ( x, y ) r h Φ − e ( x ) , Φ − e ( y ) i on R r × R r coincides with the usual Euclidean product on R r , where Φ e : Fil e → R r is thebijection defined in (17) . Let V be a non-zero vector space of finite rank over K and F be a filtration of V corre-sponding to the flag V = V ) V ) V ) · · · ) V d = 0 together with the sequence ( λ j ) ≤ j 1, let G j be a filtration of V j /V j +1 with which e j is compatible. We construct a filtration G on V which is the direct sum via Ψ of ( G j ) ≤ j ≤ d − .Note that the basis e is compatible with the new filtration G . If e i is an element in e , then λ G ( e i ) = λ G τ ( i ) (Ψ( e i )). Therefore we have E [ G ] = 1 r d − X j =0 E [ G j ]rk K ( V j /V j +1 ) , hF , Gi = 1 r d − X j =0 λ j E [ G j ] rk K ( V j /V j +1 ) . (22) We shall establish in this section the explicit version of a result of Ramanan and Ra-manathan [RR84] (Proposition 1.12) for our particular purpose, along the path indicated byTotaro [Tot96] in his proof of Fontaine’s conjecture.Let K be a perfect field. If G is a reductive group over Spec K , we call one-parametersubgroup of G any morphism of K -group schemes from G m ,K to G . Let X be a K -scheme on12hich G acts. If x is a rational point of X and if h is a one-parameter subgroup of G , then weget a K -morphism from G m ,K to X given by the composition G m ,K h / / G ∼ / / G × K Spec K Id × x / / G × K X σ / / X , where σ is the action of the group. If in addition X is proper over Spec K , this morphismextends in the unique way to a K -morphism f h,x from A K to X . We denote by 0 the uniqueelement in A ( K ) \ G m ( K ). The morphism f h,x sends the point 0 to a rational point of X which is invariant by the action of G m ,K . If L is a G -linearized line bundle on X , then theaction of G m ,K on L | f h,x (0) defines a character of G m ,K of the form t t µ ( x,h,L ) , where µ ( x, h, L ) ∈ Z . Furthermore, if we denote by Pic G ( X ) the group of isomorphism classes of all G -linearized linebundles, then µ ( x, h, · ) is a homomorphism of groups from Pic G ( X ) to Z . Remark 6.1 In [MFK94], the authors have defined the µ -invariant with a minus sign.We now recall a well-known result which gives a semistability criterion for rational pointsin a projective variety equipped with an action of a reductive group. Theorem 6.2 (Hilbert-Mumford-Kempf-Rousseau) Let G be a reductive group whichacts on a projective variety X over Spec K , L be an ample G -linearized line bundle on X and x ∈ X ( K ) be a rational point. The point x is semistable for the action of G relatively to L ifand only if µ ( x, h, L ) ≥ for any one-parameter subgroup h of G . This theorem has been originally proved by Mumford (see [MFK94]) for the case where K is algebraically closed. Then it has been independently proved in all generality by Kempf[Kem78] and Rousseau [Rou78], where Kempf’s approach has been revisited by Ramanan andRamanathan [RR84] to prove that the tensor product of two semistable vector bundle on asmooth curve (over a perfect field) is also semistable. The idea of Kempf is to choose a specialone-parameter subgroup h of G destabilizing x , which minimizes a certain function. Theuniqueness of his construction allows us to descend to a smaller field. Later Totaro [Tot96]has introduced a new approach of Kempf’s construction and thus found an elegant proof ofFontaine’s conjecture.In the rest of this section, we recall Totaro’s approach of Hilbert-Mumford criterion in oursetting. We begin by calculating explicitly the number µ ( x, h, L ) using filtrations introducedin the previous section.Let V be a vector space of finite rank over K and ρ : G → GL ( V ) be a representation of G on V . If h : G m ,K → G is a one-parameter subgroup, then the multiplicative group G m ,K actson V via h and ρ . Hence we can decompose V into direct sum of eigenspaces. More precisely,we have the decomposition V = L i ∈ Z V ( i ), where the action of G m ,K on V ( i ) is given by thecomposition G m ,K × K V ( i ) ( t t i ) × Id / / G m ,K × K V ( i ) / / V ( i ) , the second arrow being the scalar multiplication structure on V ( i ). We then define a filtration F ρ,h (supported by Z ) of V such that F ρ,hλ V = X i ≥ λ V ( i ) where λ ∈ R , filtration associated to h relatively to the representation ρ . If there is no ambiguityon the representation, we also write F h instead of F ρ,h to simplify the notation. If G = GL ( V )and if ρ is the canonical representation, then for any filtration F of V supported by Z , thereexists a one-parameter subgroup h of G such that the filtration associated to h equals F .From the scheme-theoretical point of view, the algebraic group G acts via the representation ρ on the projective space P ( V ∨ ).The following result is in [MFK94] Proposition 2.3. Here we work on the dual space V ∨ . Proposition 6.3 Let x be a rational point of P ( V ∨ ) , viewed as a one-dimensional subspace of V and let v x be an arbitrary non-zero vector in x . Then µ ( x, h, O V ∨ (1)) = − λ F ρ,h ( v x ) , where the function λ F ρ,h is defined in (16) .Proof. Let v x = P i ∈ Z v x ( i ) be the canonical decomposition of v x . Let i = λ F ρ,h ( v x ). Bydefinition, it is the maximal index i such that v x ( i ) is non-zero. Furthermore, f h,x (0) is just therational point x which corresponds to the subspace of V generated by v x ( i ). The restrictionof O V ∨ (1) on x identifies with the quotient ( Kv x ( i )) ∨ of V ∨ . Since the action of G m ,K on v x ( i ) via h is the multiplication by t i , its action on ( Kv x ( i )) ∨ is then the multiplication by t − i . Therefore, µ ( x, h, O V ∨ (1)) = − i = − λ F ρ,h ( v x ). ✷ Let ( V i ) ≤ i ≤ n be a finite family of non-zero vector spaces of finite rank over K . For anyinteger 1 ≤ i ≤ n , let r i be the rank of V i . Let G be the algebraic group GL ( V ) × · · ·× GL ( V n ).We suppose that the algebraic group G acts on a vector space V . Let π : P ( V ∨ ) → Spec K bethe canonical morphism. For each integer 1 ≤ i ≤ n , we choose an integer m i which is divisibleby r i . Let M be the G -linearized line bundle on P ( V ∨ ) defined as M := n O i =1 π ∗ (Λ r i V i ) ⊗ m i /r i . It is a trivial line bundle on P ( V ∨ ) with possibly non-trivial G -action. Notice that any one-parameter subgroup of G is of the form h = ( h , · · · , h n ), where h i is a one-parameter subgroupof GL ( V i ). Let F h i be the filtration of V i associated to h i relatively to the canonical represen-tation of GL ( V i ) on V i . The action of G m ,K via h i on Λ r i V i is nothing but the multiplicationby t r i E [ F hi ] . Then we get the following result. Proposition 6.4 With the notation above, for any rational point x of P ( V ∨ ) , we have µ ( x, h, M ) = n X i =1 m i E [ F h i ] . We now introduce the Kempf’s destabilizing flag for the action of a finite product of generallinear groups. Consider a family ( V ( i ) ) ≤ i ≤ n of finite dimensional non-zero vector space over K .Let W be the tensor product V (1) ⊗ K · · · ⊗ K V ( n ) and G be the algebraic group GL ( V (1) ) ×· · · × GL ( V ( n ) ). For any integer i such that 1 ≤ i ≤ n , let r ( i ) be the rank of V ( i ) . Thegroup G acts naturally on W and hence on P ( W ∨ ). We denote by π : P ( W ∨ ) → Spec K thecanonical morphism. Let m be a strictly positive integer which is divisible by all r ( i ) and L bea G -linearized line bundle on P ( W ∨ ) as follows: L := O W ∨ ( m ) ⊗ n O i =1 π ∗ (det V ( i ) ) ⊗ ( m/r ( i ) ) . (23)14or any rational point x of P ( W ∨ ), we define a function Λ x : Fil Q V (1) × · · · × Fil Q V ( n ) → R such that Λ x ( G (1) , · · · , G ( n ) ) = E [ G (1) ] + · · · + E [ G ( n ) ] − λ G (1) ⊗···⊗G ( n ) ( v x )( kG (1) k + · · · + kG ( n ) k ) (24)if at least one filtration among the G ( i ) ’s is non-trivial, and Λ x ( G (1) , · · · , G ( n ) ) = 0 otherwise.We recall that in (24), v x is an arbitrary non-zero element in x . Note that the function Λ x isinvariant by dilation. In other words, for any positive number ε > x ( ψ ε G (1) , · · · , ψ ε G ( n ) ) = Λ x ( G (1) , · · · , G ( n ) ) , where the dilation ψ ε is defined in (18). Proposition 6.5 Let x be a rational point of P ( W ∨ ) . Then the point x is not semistable forthe action of G relatively to L if and only if the function Λ x defined above takes at least onestrictly negative value.Proof. By Propositions 6.3 and 6.4, for any rational point x of P ( W ∨ ), µ ( x, h, L ) = m (cid:16) n X i =1 E [ F h i ] − λ F h ( v x ) (cid:17) . (25)“= ⇒ ”: By the Hilbert-Mumford criterion (Theorem 6.2), there exists a one-parametersubgroup h = ( h , · · · , h n ) of G such that µ ( x, h, L ) < 0. The filtration F h of W associatedwith h coincides with the tensor product filtration F h ⊗ · · · ⊗ F h n , where F h i is the filtrationof V ( i ) associated with h i . Therefore,Λ x ( F h , · · · , F h n ) = µ ( x, h, L ) m ( kF h k + · · · + kF h n k ) < . “ ⇐ =”: Suppose that ( G (1) , · · · , G ( n ) ) is an element in Fil Q V (1) × · · · × Fil Q V ( n ) such thatΛ x ( G (1) , · · · , G ( n ) ) < 0. By equalities (19), (20) and the invariance of Λ x by dilation, we canassume that G (1) , · · · , G ( n ) are all supported by Z . In this case, there exists, for each 1 ≤ i ≤ n ,a one-parameter subgroup h i of GL ( V ( i ) ) such that F h i = G ( i ) . Let h = ( h , · · · , h n ). Bycombining the negativity of Λ x ( F h , · · · , F h n ) with (25), we obtain µ ( x, h, L ) < 0, so x is notsemistable. ✷ Proposition 6.7 below generalizes Proposition 2 of [Tot96]. The proof uses Lemma 6.6,which is equivalent to Lemma 3 of [Tot96], or Lemma 1.1 of [RR84]. See [RR84] for the proofof the lemma. Lemma 6.6 Let n ≥ be an integer and let T be a finite non-empty family of linear forms on R n . Let Λ : R n → R such that Λ( y ) = k y k − max l ∈ T l ( y ) for y = 0 , and that Λ(0) = 0 . Supposethat the function Λ takes at least a strictly negative value. Then1) the function Λ attains its minimum value, furthermore, all points in R n minimizing Λ areproportional;2) if c is the minimal value of Λ and if y ∈ R n is a minimizing point of Λ , then for any y ∈ R n , Λ( y ) ≥ c h y , y ik y k · k y k ; (26)15 ) if in addition all linear forms in T are of rational coefficients, then there exists a point in Z n which minimizes Λ . Proposition 6.7 With the notation of Proposition 6.5, if x is not semistable for the action of G relatively to L , then the function Λ x attains its minimal value. Furthermore, the element in Fil Q V (1) × · · · × Fil Q V ( n ) minimizing Λ x is unique up to dilatation. Finally, if ( F (1) , · · · , F ( n ) ) isan element in Fil Q V (1) × · · · × Fil Q V ( n ) minimizing Λ x and if c is the minimal value of Λ x , thenfor any element ( G (1) , · · · , G ( n ) ) in Fil Q V (1) × · · · × Fil Q V ( n ) , the following inequality holds: n X i =1 E [ G ( i ) ] − λ G (1) ⊗···⊗G ( n ) ( v x ) ≥ c hF (1) , G (1) i + · · · + hF ( n ) , G ( n ) i ( kF (1) k + · · · + kF ( n ) k ) (27) Proof. For each integer 1 ≤ i ≤ n , let e ( i ) = ( e ( i ) j ) ≤ j ≤ r ( i ) be a basis of V ( i ) . Let e =( e ( i ) ) ≤ i ≤ n . Denote by Λ e x the restriction of Λ x on Fil Q e (1) × · · · × Fil Q e ( n ) . The space Fil Q e (1) ×· · · × Fil Q e ( n ) is canonically embedded in Fil e (1) × · · · × Fil e ( n ) , which can be identified with R r (1) × · · · × R r ( n ) through Φ e (1) × · · · Φ e ( n ) (see Proposition 5.3). We extend natually Λ e x to afunction Λ e , † x on Fil e (1) × · · · × Fil e ( n ) , whose numerator part is the maximal value of a finitenumber of linear forms with rational coefficients (see Proposition 5.2) and whose denominatorpart is just the norm of vector in the Euclidean space. Then by Lemma 6.6, the function Λ e , † x attains its minimal value, and there exists an element in Fil Q e (1) × · · · × Fil Q e ( n ) which minimizesΛ e , † x . By definition the same element also minimizes Λ e x . Since the function Λ e x , viewed as afunction on R r (1) + ··· + r ( n ) , only depends on the set n S ⊂ n Y i =1 { , · · · , r ( i ) } (cid:12)(cid:12)(cid:12) v x ∈ X ( j , ··· ,j n ) ∈ S Ke (1) j ⊗ · · · ⊗ e ( n ) j n o . Therefore, there are only a finite number of functions on Euclidean space of dimension r (1) + · · · + r ( n ) of the form Λ e x . Thus we deduce that the function Λ x attains globally its minimalvalue, and the minimizing element of Λ x could be chosen in Fil Q V (1) × · · · × Fil Q V ( n ) .Suppose that there are two elements in Fil Q V (1) × · · · × Fil Q V ( n ) which minimizes Λ x . ByBruhat’s decomposition, we can choose e as above such that both elements lie in Fil Q e (1) ×· · · × Fil Q e ( n ) . Therefore, by Lemma 6.6 they differ only by a dilation. Finally to prove in-equality (27), it suffices to choose e such that ( F (1) , · · · , F ( n ) ) and ( G (1) , · · · , G ( n ) ) are bothin Fil Q e (1) × · · · × Fil Q e ( n ) , and then apply Lemma 6.6 2). ✷ Although the minimizing filtrations ( F (1) , · · · , F ( n ) ) in Proposition 6.7 are a priori sup-ported by Q , it is always possible to choose them to be supported by Z after a dilation.In the rest of the section, let x be a rational point of P ( W ∨ ) which is not semistable forthe action of G relatively to L . We fix an element ( F (1) , · · · , F ( n ) ) in Fil Z V (1) × · · · × Fil Z V ( n ) minimizing Λ x . Define e c := c ( kF (1) k + · · · + kF ( n ) k ) . (28)Note that e c < 0. Moreover, it is a rational number since the following equality holds: e c = Λ x ( F (1) , · · · , F ( n ) )( kF (1) k + · · · + kF ( n ) k ) = E [ F (1) ] + · · · + E [ F ( n ) ] − λ F (1) ⊗···⊗F ( n ) ( v x ) kF (1) k + · · · + kF ( n ) k . 16e suppose that F ( i ) corresponds to the flag D ( i ) : V ( i ) = V ( i )0 ) V ( i )1 ) · · · ) V ( i ) d ( i ) = 0and the strictly increasing sequence of integers λ ( i ) = ( λ ( i ) j ) ≤ j Let e π : P ( f W ∨ ) → Spec K be the canonical morphism and let e L := O f W ∨ ( N ) ⊗ (cid:16) n O i =1 d ( i ) − O j =0 e π ∗ (cid:0) Λ r ( i ) j ( V ( i ) j /V ( i ) j +1 ) (cid:1) ⊗ b ( i ) j (cid:17) . Then the rational point e x of P ( f W ∨ ) is semistable for the action of e G relatively to the G -linearized line bundle e L .Proof. For any integers i and j such that 1 ≤ i ≤ n and 0 ≤ j < d ( i ) , we choose an arbitraryfiltration G ( i ) ,j of V ( i ) j /V ( i ) j +1 supported by Z . We have explained in Subsection 5.5 how toconstruct a new filtration G ( i ) of V ( i ) from G ( i ) ,j . Let G = n O i =1 G ( i ) , e G = M λ (1) j + ··· + λ ( n ) jn = β n O i =1 G ( i ) ,j i . From the construction we know that λ G ( v x ) = λ e G ( e v x ). Using (22), the inequality (27) implies: n X i =1 d ( i ) − X j =0 r ( i ) j r ( i ) E [ G ( i ) ,j ] − n X i =1 d ( i ) − X j =0 e cλ ( i ) j r ( i ) j r ( i ) E [ G ( i ) ,j ] − λ e G ( e v x ) ≥ , (29)17here the constant e c is defined in (28). Hence n X i =1 d ( i ) − X j =0 b ( i ) j r ( i ) j E [ G ( i ) ,j ] − N λ e G ( e v x ) ≥ . (30)Let h be an arbitrary one-parameter subgroup of e G corresponding to filtrations G ( i ) ,j . ByPropositions 6.3 and 6.4, together with the fact that µ ( e x, h, · ) is a homomorphism of groups,we obtain µ ( e x, h, e L ) = µ ( e x, h, O f W ∨ ( N )) + n X i =1 d ( i ) − X j =0 b ( i ) j r ( i ) j E [ G ( i ) ,j ]= − N λ e G ( e v x ) + n X i =1 d ( i ) − X j =0 b ( i ) j r ( i ) j E [ G ( i ) ,j ] ≥ . By Hilbert-Mumford criterion, the point e x is semistable for the action of e G relatively to e L . ✷ Finally we point out the following consequence of the inequality (30). Proposition 6.9 The minimizing filtrations ( F (1) , · · · , F ( n ) ) satisfy E [ F (1) ] = · · · = E [ F ( n ) ] = 0 . In other words, the equality P d ( i ) − j =0 a ( i ) j r ( i ) j = 0 holds, or equivalently, P d ( i ) − j =0 λ ( i ) j r ( i ) j = 0 forany i ∈ { , · · · , n } .Proof. Let ( u i ) ≤ i ≤ n be an arbitrary sequence of integers. For all integers i, j such that1 ≤ i ≤ n and 0 ≤ j < d ( i ) , let G ( i ) ,j be the filtration of V ( i ) j /V ( i +1) j which is supported by { u i } . Note that in this case e G is supported by { u + · · · + u n } . The inequality (30) gives n X i =1 d ( i ) − X j =0 b ( i ) j r ( i ) j u i − N n X i =1 u i = n X i =1 u i d ( i ) − X j =0 a ( i ) j r ( i ) j ≥ . Since ( u i ) ≤ i ≤ n is arbitrary, we obtain P d ( i ) − j =0 a ( i ) j r ( i ) j = 0, and therefore P d ( i ) − j =0 λ ( i ) j r ( i ) j = 0. ✷ We shall give a semistability criterion for Hermitian vector bundles, which is the arithmeticanalogue of a result due to Bogomolov in geometric framework (see [Ray81]).Let E be a non-zero Hermitian vector bundle over Spec O K and let V = E K . We denoteby r its rank. If D : V = V ) V ) · · · ) V d = 0 is a flag of V , it induces a strictly decreasingsequence of saturated sub- O K -modules E = E ) E ) · · · ) E d = 0 of E . For any integer j ≤ j < d , let r j be the rank of E j /E j +1 . If a = ( a j ) ≤ j If the Hermitian vector bundle E is semistable (resp. stable), then for anyinteger d ≥ , any flag D of length d of V , and any strictly increasing sequence a = ( a j ) ≤ j The converse of Proposition 7.1 is also true. Let E be a saturated sub- O K -module of E . Consider the flag D : V ) E ,K ) a = (0 , r ). Then d deg( L a D ) = r rk( E ) (cid:0)b µ ( E ) − b µ ( E ) (cid:1) . Therefore b µ ( E ) ≤ b µ ( E ) (resp. b µ ( E ) < b µ ( E )). Since E is arbitrary, the Hermitian vector bundle E is semistable (resp. stable). In this section, we shall give an upper bound for the Arakelov degree of a Hermitian linesubbundle of a finite tensor product of Hermitian vector bundles. As explained in Section 1, weshall use the results established in Section 6 to reduce our problem to the case with semistabilitycondition (in geometric invariant theory sense), which has already been discussed in Section 4.We point out that, in order to obtain the same estimation as (13) in full generality, we shouldassume that all Hermitian vector bundles E i are semistable, as a price paid for removing thesemistability condition for M K .We denote by K a number field and by O K its integer ring. Let ( E ( i ) ) ≤ i ≤ n be a family of semistable Hermitian vector bundles on Spec O K . For any i ∈ { , · · · , n } , let r ( i ) be the rankof E ( i ) and V ( i ) = E ( i ) K . Let E = E (1) ⊗ · · · ⊗ E ( n ) and W = E K . We denote by π : P ( W ∨ ) → Spec K the natural morphism. The algebraic group G := GL K ( V (1) ) × K · · · × K GL K ( V ( n ) )acts naturally on P ( W ∨ ). Let M be a Hermitian line subbundle of E and m be a strictlypositive integer which is divisible by all r ( i ) ’s.19 roposition 8.1 For any Hermitian line subbundle M of E (1) ⊗ · · · ⊗ E ( n ) , we have d deg( M ) ≤ n X i =1 (cid:16)b µ ( E ( i ) ) + 12 log(rk E ( i ) ) (cid:17) . Proof. We have proved that if M K is semistable for the action of G relatively to O W ∨ ( m ) ⊗ π ∗ (cid:16) N ni =1 (Λ r ( i ) V ( i ) ) ⊗ m/r ( i ) (cid:17) , where m is a strictly positive integer which is divisible by all r ( i ) ,then the following inequality holds: d deg( M ) ≤ n X i =1 (cid:16)b µ ( E i ) + 12 log r ( i ) (cid:17) . If this hypothesis of semistability is not fulfilled, by Proposition 6.8, there exist two strictlypositive integers N and β , and for any i ∈ { , · · · , n } ,1) a flag D ( i ) : V ( i ) = V ( i )0 ) V ( i )1 ) · · · ) V ( i ) d ( i ) = 0of V ( i ) corresponding to the sequence E ( i ) = E ( i )0 ) E ( i )1 ) · · · ) E ( i ) d ( i ) = 0of saturated sub- O K -modules of E ,2) two strictly increasing sequence λ ( i ) = ( λ ( i ) j ) ≤ j 0. Therefore d deg( M ) ≤ n X i =1 b µ ( E ( i ) ) + n X i =1 d ( i ) − X j =0 r ( i ) j b ( i ) j N log r ( i ) . Since P d ( i ) − j =0 r ( i ) j a ( i ) j = 0 for any integer i such that 1 ≤ i ≤ n (see Proposition 6.9), we haveproved the proposition. ✷ Corollary 8.2 The following inequality is verified: b µ max ( E (1) ⊗ · · · ⊗ E ( n ) ) ≤ n X i =1 (cid:16)b µ ( E ( i ) ) + log(rk E ( i ) ) (cid:17) + log | ∆ K | K : Q ] . (32) Proof. Since the Hermitian line bundle M in Proposition 8.1 is arbitrary, we obtainu d deg n ( E (1) ⊗ · · · ⊗ E ( n ) ) ≤ n X i =1 (cid:16)b µ ( E ( i ) ) + 12 log(rk E ( i ) ) (cid:17) . Combining with (11) we obtain (32). ✷ We finally give the proof of Theorem 1.1. Lemma 9.1 Let K be a number field and O K be its integer ring. Let ( E i ) ≤ i ≤ n be a finitefamily of non-zero Hermitian vector bundles (non-necessarily semistable) and E = E ⊗ · · · ⊗ E n . Then the following inequality holds: b µ max ( E ) ≤ n X i =1 (cid:16)b µ max ( E i ) + log(rk E i ) (cid:17) + log | ∆ K | K : Q ] . roof. Let F be a sub- O K -module of E . By taking Harder-Narasimhan flags of E i ’s (cf.[Bos96]), there exists, for any i such that 1 ≤ i ≤ n , a semistable subquotient F i /G i of E i suchthat1) b µ ( F i /G i ) ≤ b µ max ( E i ),2) the inclusion homomorphism from F to E factorises through F ⊗ · · · ⊗ F n ,3) the canonical image of F in ( F /G ) ⊗ · · · ⊗ ( F n /G n ) does not vanish.Combining with the slope inequality (10), Corollary 8.2 implies that b µ min ( F ) ≤ n X i =1 (cid:16)b µ ( F i /G i ) + log(rk( F i /G i )) (cid:17) + log | ∆ K | K : Q ] ≤ n X i =1 (cid:16)b µ max ( E i ) + log(rk E i ) (cid:17) + log | ∆ K | K : Q ] . Since F is arbitrary, the proposition is proved. ✷ Proof of Theorem 1.1 Let N ≥ E ⊗ N as E ⊗ · · · ⊗ E | {z } N copies ⊗ · · · ⊗ E n ⊗ · · · ⊗ E n | {z } N copies , that b µ max ( E ⊗ N ) ≤ n X i =1 N (cid:16)b µ max ( E i ) + log(rk E i ) (cid:17) + log | ∆ K | K : Q ] . On the other hand, by Corollary 2.5, b µ max ( E ⊗ N ) ≥ N b µ max ( E ). 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