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INEQUALITY OF BOGOMOLOV-GIESEKER’S TYPE

arXiv:alg-geom/9305005v1 12 May 1993INEQUALITY OF BOGOMOLOV-GIESEKER’S TYPEON ARITHMETIC SURFACESAtsushi MoriwakiDepartment of MathematicsUniversity of California at Los AngelesLos Angeles, CA 90024, USAe-mail : moriwaki@math.ucla.eduApril, 1993 (Revised Version)Typeset by AMS-TEX1

2Abstract. Let K be an algebraic number field, OK the ring of integers of K, and f :X →Spec(OK) an arithmetic surface.

Let (E, h) be a rank r Hermitian vector bundle onX such that EQ is semistable on the geometric generic fiber XQ of f. In this paper, we willprove an arithmetic analogy of Bogomolov-Gieseker’s inequality:bc2(E, h) −r −12rbc1(E, h)2 ≥0.Table of ContentsIntroduction1. Quick review of arithmetic intersection theory2.

Asymptotic behavior of analytic torsions3. Hermitian modules over arithmetic curves4.

Asymptotic behavior of L2-degree of submodules of H0(Lm)5. Vanishing of a certain Bott-Chern secondary class6.

Donaldson’s Lagrangian and arithmetic second Chern class7. Second fundamental form8.

Proof of the main theorem9. Arithmetic second Chern character of semistable vector bundles10.

Torsion vector bundles

3References

4IntroductionLet M be an n-dimensional compact K¨ahler manifold with a K¨ahler form Φ. For atorsion free sheaf F on M, we define an averaged degree µ(F, Φ) of F with respect to Φbyµ(F, Φ) =ZMc1(F) ∧Φn−1rk F.Let E be a torsion free sheaf on M. E is said to be Φ-stable (resp.

Φ-semisatble) if,for every subsheaf F of E with 0 ⊊F ⊊E, an inequality µ(F, Φ) < µ(E, Φ) (resp.µ(F, Φ) ≤µ(E, Φ)) is satisfied. If a torsion free sheaf E of rank r is Φ-semistable, thenwe haveZMc2(E) −r −12r c1(E)2∧Φn−2 ≥0,which is called Bogomolov-Gieseker’s inequality (cf.

[Bo] and [Gi]). The purpose of thispaper is to establish an arithmetic analogy of the above inequality, that is,Main Theorem.

Let K be an algebraic number field, OK the ring of integers of K, andf : X →Spec(OK) an arithmetic surface. Let (E, h) be a Hermitian vector bundle on X.If EQ is semistable on the geometric generic fiber XQ of f, then we have an inequalitybc2(E, h) −r −12r bc1(E, h)2 ≥0,where r = rk E and bc1(E, h) and bc2(E, h) are arithmetic Chern classes introduced byGillet-Soul´e [GS90b].The theory of stable vector bundles is closely related to the Yang-Mills theory.

LetE be a vector bundle on the compact K¨ahler manifold M. E is said to be Φ-poly-stableif there is a direct sum E = E1 ⊕· · · ⊕Es such that Ei is Φ-stable for all 1 ≤i ≤sand µ(E1) = · · · = µ(Es). Moreover, let h be a Hermitian metric of E. h is calledEinstein-Hermitian (resp.

weakly Einstein-Hermitian) if there is a constant (resp. C∞-function) ϕ such that √−1ΛK(E, h) = ϕ idE, where K(E, h) is the curvature of (E, h).The fundamental theorem concerning Einstein-Hermitian metric isTheorem A.

(cf. [Do83], [Do85], [NS] and [UY])E has an Einstein-Hermitian metricif and only if E is Φ-poly-stable.The above theorem plays a crucial role for the proof of the main theorem.

Here wegive a rough sketch of the proof of the main theorem, which, I think, is a brief summaryof this paper.Step 1. A poly-stable vector bundle is semistable, but a semistable vector bundle is notnecessarily poly-stable.

This means that a semistable bundle does not necessarily havean Einstein-Hermitian metric. In general, a semistable vector bundle F has a filtration:0 = F0 ⊂F1 ⊂· · · ⊂Fl−1 ⊂Fl = F

5such that Fi/Fi−1 is stable for all i and µ(F1/F0) = · · · = µ(Fl/Fl−1), which is called aJordan-H¨older filtration of F. The first step of the proof of the main theorem is a reduc-tion to the case where E is poly-stable on each infinite fiber. Unfortunately, arithmeticChern classes are sensitive for extensions, that is, the Bott-Chern secondary characteris-tic appears in several formula for an exact sequence of Hermitian vector bundles.

So weneed an exact calculation of a certain secondary Bott-Chern characteristic, which will betreated in §7. It follows a good comparison of second Chern classes (cf.

Corollary 7.4).Using this result, we can do our first step.Step 2. By the step 1, we may assume that E is poly-stable on each infinite fiber.

Thusby Theorem A, E has an Einstein-Hermitian metric hEH on each infinite fiber. Here weneed a comparison ofbc2(E, h) −r −12r bc1(E, h)2andbc2(E, hEH) −r −12r bc1(E, hEH)2.The following theorem guarantees that we may assume that h is Einstein-Hermitian.Theorem C. (cf.

Theorem 6.3)Let K be an algebraic number field and OK the ringof integers. We denote by K∞the set of all embeddings of K into C. Let f : X −→Spec(OK) be an arithmetic variety with dim X = d ≥2, and (H, hH) a Hermitian linebundle on X such that, for each σ ∈K∞, c1(Hσ, hHσ) gives a K¨ahler form Φσ on aninfinite fiber Xσ.

Let E be a vector bundle of rank r on X. For a Hermitian metric h ofE, we set∆(E, h) =bc2(E, h) −r −12r bc1(E, h)2· bc1(H, hH)d−2.If E is Φσ-poly-stable on each infinite fiber Xσ, then we have;(1) the set ∆= {∆(E, h) | h is a Hermitian metric of E} has the absolute minimalvalue.

(2) ∆(E, h0) attaches the minimal value of ∆if and only if h0 is weakly Einstein-Hermitian on each infinite fiber.Step 3. Let π : P(E) −→X be the projective bundle of E and O(1) the tautologicalline bundle on P(E).

We set L = O(r) ⊗π∗(det E)−1. The arithmetic analogy of theGrothendieck relation (1.9.1) and Lemma 5.1 implies that(Lr+1) ≤0 =⇒r −12r bc1(E, h)2 ≤bc2(E, h).Thus, it is sufficient to show that (Lr+1) ≤0.Step 4.

Let N be an ample line bundle on X with deg(NK) > 2g(XK) −2.Thearithmetic Riemann-Roch theorem due to Gillet-Soul´e [GS92] shows us thatχL2(Ln ⊗π∗N) + 12τ((Ln ⊗π∗N)∞) = deg( bch(Ln ⊗π∗N) · btdA(TP(E)/OK)).

6Since we havelimn→∞deg( bch(Ln ⊗π∗N) · btdA(TP(E)/OK))nr+1=1(r + 1)! (Lr+1),in order to get (Lr+1) ≤0, it is sufficient to see that(a) τ((Ln ⊗π∗N)∞) ≤O(nr log n).

(b) χL2(Ln ⊗π∗N) ≤O(nr log n).Step 5. (a) is a consequences of the following theorem, which is a generalization of [BV]and Proposition 2.7.8 of [Vo] to the case of semipositive line bundles.Theorem C. (cf.

Theorem 2.1)Let X be a compact K¨ahler manifold of dimension n,(L, h) a Hermitian line bundle on X, and (E, hE) a Hermitian vector bundle on X. Letζq,d be the zeta function of the Laplacian □q,d on A0,q(Ld ⊗E). Let HL be the Hermitianform corresponding to the curvature form K(L, h) of L and k a non-negative integer.

IfHL(x) is positive semi-definite and rk HL(x) ≥k for all x ∈X, then for n −k < q ≤nthere is a constant C such that|ζ′q,d(0)| ≤Cdn log(d)for all d ≥0.Step 6. (b) can be derived from the following theorem.Theorem D. (cf.Theorem 4.1)Let K be an algebraic number field, OK the ringof integers of K and f : X −→Spec(OK) an arithmetic variety over OK. Let (L, h)be a Hermitian line bundle on X, (E, hE) a Hermitian vector bundle on X, and hm aHermitian metric of H0(X, Lm ⊗E) induced by hm ⊗hE.

Then there is a constant Csuch thatdegL2(F, hF )rk F≤Cm log mfor all m > 1 and all submodules F of H0(X, Lm ⊗E), where hF is the Hermitian metricinduced by hm.In §9, we will consider an invariant of a semistable vector bundle arising from thearithmetic second Chern character.Let f : X →Spec(OK) be a regular arithmeticsurface and E a vector bundle on X such that deg(EK) = 0 and EQ is semistable. Wedenote by Herm(E) the set of all Hermitian metrics of E. Here we setch2(E) =suph∈Herm(E)bch2(E, h).As a consequence of Theorem A, we can easily see that ch2(E) is a non-positive realnumber.

Let M XK/K(r, 0) be the moduli scheme of semistable vector bundles on XKwith rank r and degree 0. Our first question is

7Question E. Is there a relation between ch2(E) and a height of the class [EK] inM XK/K(r, 0)?For example, if XK is an elliptic curve, we can give an answer (cf. Corollary 9.7).

For acurve of higher genus, it is still an open problem. To solve the above question, we thinkthat first of all we must construct a certain canonical height of M XK/K(r, 0).

Moreover,we think that the theta divisor on M XK/K(r, 0) gives rise to a canonical height.In §10, we consider the equality condition of the inequality of the main theorem. Fora geometric case, this is closely related to flat vector bundles.

So we need an arithmeticanalogy of flat vector bundles. We give one candidate for arithmetic flatness, that is,torsion vector bundles.

(see Definition 10.6 for the definition of “of torsion type”.) Ourquestion concerning torsion vector bundles is the following.Question F. Let X and E be as above.

If bch2(E, h) = 0, then is E of torsion type?A partial answer is given in Proposition 10.8. Two questions are related to each other.For example, if r = 1, E satisfies some numerical conditions and h is Einstein-Hermitian,then bch2(E, h) = −[K : Q] Height([EK]) (c.f.

Lemma 9.2), where Height is the N´eron-Tate height on Pic0(XK). Thus, if bch2(E, h) = 0, EK is a torsion point of Pic0(XK).Finally we would like to express hearty thanks to Prof. S. Zhang for his valuablesuggestions.

81. Quick review of arithmetic intersection theoryIn this section, we would like to fix several notations of this paper, which gives a quickreview of arithmetic intersection theory.

Details will be found in [GS90a], [GS90b], [Fa2]and [SABK].1.1 Polynomial map associated with a formal power series φ. Let φ be a sym-metric formal power series of n variables over R, A an R-algebra, and Mn(A) the algebraof (n × n)-matrices of A.

We denote by φ(k) the homogeneous component of φ of degreek. Then we can easily construct the unique polynomial mapΦ(k) : Mn(A) −→Asuch that Φ(k) is invariant under conjugation by GLn(A) andΦ(k) (diag(λ1, .

. .

, λn)) = φ(k)(λ1, . .

. , λn).When I is a nilpotent subalgebra, we may defineΦ =XkΦ(k) : Mn(I) −→A.Standard examples of power series are the following:(Chern classes)ci = si(T1, .

. .

, Tn), where si is the i-th elementary symmetricpolynomial. (Total Chern class)c =Xi≥0ci.

(Chern character)ch(T1, . .

. , Tn) =nXi=1exp(Ti).

(Todd class)td(T1, . .

. , Tn) =nYi=1Ti1 −exp(−Ti).1.2 Chern characteristic class attached to φ.

Let M be a complex manifold andAp,p(M) the space of complex C∞-forms of type (p, p). We putA(M) =Mp≥0Ap,p(M)and˜A(M) = A(M)/(Im ∂+ Im ¯∂).Here we set dc =∂−¯∂4π√−1.

Then we have ddc =√−12π ∂¯∂. Let (E, h) be a Hermitianvector bundle on M of rank n, K(E, h) the curvature of (E, h), and φ as above.

Thecharacteristic class of (E, h) attached to φ is defined byφ(E, h) = Φ√−12π K(E, h)∈A(M).

91.3 Bott-Chern secondary characteristic class. Let E : 0 →(S, h′) →(E, h) →(Q, h′′) →0 be an exact sequence of Hermitian vector bundles on M. (Here h′ and h′′are not necessarily the metrics induced by h.) We introduce the Bott-Chern secondarycharacteristic class ˜φ(E) of E attached to φ.

This class is characterized by the followingthree properties:(i) ddc ˜φ(E) = φ(E, h) −φ((S, h′) ⊕(Q, h′′)). (ii) For every homomorphism f : N −→M of complex manifolds,˜φ(f ∗(E)) = f ∗˜φ(E).

(iii) If (E, h) = (S, h′) ⊕(Q, h′′), then ˜φ(E) = 0. (Note that the axiom (i) is different from [BGS] or [GS90b].

)For a later purpose, we will show how to construct it. We denote the homomorphismS −→E by α.

Let P1 be the projective line, O(1) the tautological line bundle on P1and σ a section of H0(P1, O(1)) such that σ(∞) = 0.Let p : M × P1 −→M andq : M × P1 −→P1 be the canonical projections. We set˜E = Coker (p∗(α) ⊕(id ⊗σ) : p∗(S) −→p∗(E) ⊕(p∗(S) ⊗q∗(O(1)))) .Note that ˜EM×{0} ≃E and ˜EM×{∞} ≃S ⊕Q.

Here we give a Hermitian metric ˜h on˜E such that ( ˜E, ˜h)M×{0} and ( ˜E, ˜h)M×{∞} are isometric to (E, h) and (S, h′)⊕(Q, h′′)respectively. Then ˜φ(E) is given as follows:˜φ(E) =ZP1 φ( ˜E, ˜h) log |z|2.1.4 Green current.

Here we assume that the complex manifold M is compact. Wedenote by Dk,k(M) the space of currents on M of type (k, k).

Let Z be a cycle on M ofcodimension p. An element g ∈Dp−1,p−1(M) is called a green current of Z if ddcg + δZis smooth. The smooth form ddcg + δZ is denoted by ω(g).

For example, let D be adivisor on M and h a Hermitian metric of the line bundle OM(D). Let s be a rationalsection of OM(D) such that div(s) = D. Then it is well known that−ddc log(h(s, s)) + δD = c1(OM(D), h).Thus −log(h(s, s)) is a green current of D.1.5 Analytic torsion.

Moreover we assume that M is a K¨ahler manifold with a K¨ahlerform Ω. Let (E, h) be a Hermitian vector bundle on M. Let □q be the Laplacian onA0,q(E) and 0 < λ1 < λ2 < · · · eigenvalues of □q.

We setζq(s) =Xi>0λ−si .

10It is well known that ζq extends meromorphically to the complex plane and is holomorphicat s = 0. We define the analytic torsion τ(E, h) of (E, h) byτ(E, h) =Xq>0(−1)qqζ′q(0).1.6 Arithmetic variety and arithmetic Chow group.

Let K be an algebraic num-ber field and OK the ring of integers of K. We set S = Spec(OK) and the set of C-valuedpoints of K, that is,{σ : K ֒→C | σ is an embedding of the field K to C},is denoted by S∞or K∞. A projective and flat morphism of integral schemes π : X −→Sis called an arithmetic variety over OK if the generic fiber of f is smooth.

For an objectsOb of X, we denote by Obσ the pullback of Ob by an embedding σ ∈S∞. Let Zp(X) bea free abelian group generated by cycles of codimension p. Let bZp(X) be a group of pairs(Z, Pσ gσ) such that Z ∈Zp(X) and gσ is a green current of Zσ on Xσ.

Let dCHp(X)be the quotient group of bZp(X) divided by the subgroup generated by the followingelements:(a) (div(f), Pσ −log |fσ|2), where f is a rational function on some subvariety Y ofcodimension p −1 and log |fσ|2 is the current defined by(log |fσ|2)(γ) =ZYσ(log |fσ|2)γ. (b) (0, Pσ ∂(ασ) + ¯∂(βσ)), where ασ ∈Dp−2,p−1(Xσ), βσ ∈Dp−1,p−2(Xσ).We define three homomorphisms:ω : dCHp(X) −→MσAp,p(Xσ),z : dCHp(X) −→CHp(X),a :Xσ˜Ap−1,p−1(Xσ) −→dCHp(X)byω(Z,Xσgσ) =Xσ(ddcgσ + δZσ) ,z(Z,Xσgσ) = Z,a(Xσgσ) = (0,Xσgσ)respectively.

Moreover, deg : dCHdim X(X) →R is given bydeg(XP ∈XnP P,Xσgσ) =XP ∈XnP log #(OX/mP ) + 12XσZXσgσ,where mP is the maximal ideal of P.

111.7 First Chern class of Hermitian vector bundle. Let E be a vector bundle onthe arithmetic variety X.

We say (E, h) is a Hermitian vector bundle if, for all σ ∈S∞,Eσ is equipped with a Hermitian metric hσ. Let s1, .

. .

, sn be rational sections of E suchthat s1, . .

. , sn give a basis of E at the generic point of X. bc1(E, h) is defined by theclass ofdiv(s1 ∧· · · ∧sn),Xσ∈S∞−log dethσ(s1, s1)· · ·hσ(s1, sn).........hσ(sn, s1)· · ·hσ(sn, sn).1.8 Intersection of arithmetic cycles.

We setdCH(X) =Mp≥0dCHp(X).dCH(X)Q has a natural pairingdCHp(X)Q ⊗dCHq(X)Q −→dCHp+q(X)Qdefined by(Y,XσgYσ) · (Z,XσgZσ) = (Y · Z,XσgYσ ∗gZσ),where gYσ ∗gZσ = gYσδZσ + ω(gYσ)gZσ. In particular,bc1(L, h) · (0,Xσgσ) = (0,Xσc1(Lσ, hσ)gσ).1.9 Arithmetic characteristic classes.

Let φ be a symmetric formal power series ofn variables with real coefficients. Let (E, h) be a Hermitian vector bundle of rank n onthe arithmetic variety X.

We introduce the characteristic classˆφ(E, h) ∈dCH(X)Rattached to φ. This is characterized by the following four axioms (cf.

Theorem 4.1 of[GS90b]) :(I) For every morphism f : Y −→X of arithmetic varieties,f ∗(ˆφ(E, h)) = ˆφ(f ∗(E, h)). (II) If (E, h) is a direct summand of Hermitian line bundles, i.e.

(E, h) = (L1, h1) ⊕· · · ⊕(Ln, hn),

12then we getˆφ(E, h) = φ(bc1(L1, h1), . .

. , bc1(Ln, hn)).

(III) If we setφ(T1 + T, . .

. , Tn + T) =Xi≥0φi(T1, .

. .

, Tn)T i,for a Hermitian line bundle (L, h′),ˆφ((E, h) ⊗(L, h′)) =Xi≥0ˆφi(E, h)bc1(L, h′)i. (IV) ω(ˆφ(E, h)) = Pσ φ(Eσ, hσ).Let E : 0 →(S, h′) →(E, h) →(Q, h′′) →0 be an exact sequence of Hermitian vectorbundles on X.

An important property of ˆφ isˆφ(E, h) −ˆφ((S, h′) ⊕(Q, h′′)) = a(Xσ˜φ(Eσ)).For example, if we putˆct(E, h) =rk EXi=0(−1)iˆci(E, h)tn−i,we haveˆct(E, h) −ˆct(S, h′)ˆct(Q, h′′) = a Xσrk EXi=0(−1)i˜ci(Eσ)tn−i!.We apply this formula to the following special situation. Let f : P(E) −→X be theprojective bundle of E, Q = OY (1) the tautological line bundle of P(E) and S the kernelof f ∗(E) −→Q.

We give the submetric h′ on S and the quotient metric h′′ on Q inducedby f ∗h. Applying the above formula to the exact sequenceE : 0 →(S, h′) →(f ∗E, f ∗h) →(Q, h′′) →0,we getˆct(f ∗E, f ∗h) −ˆct(S, h′)ˆct(Q, h′′) = a Xσrk EXi=0(−1)i˜ci(Eσ)tn−i!,which implies(1.9.1)rk EXi=0(−1)if ∗bci(E, h)bc1(Q, h′′)n−i = a Xσrk EXi=0(−1)i˜ci(Eσ)c1(Qσ, h′′σ)n−i.

!,by evaluating t = bc1(Q, h′′). This is an arithmetical analogy of the Grothendieck relation.Conversely, the relation (1.9.1) defines bci(E, h).

To see this, we will prove the followingfact.

13Claim 1.9.2. (1) A homomorphism ψ : dCH(X)⊕n −→dCH(Y ) byψ(x0, · · · , xn−1) =n−1Xi=0f ∗(xi)bc1(Q, h′′)iis injective, where n = rk E and Y = P(E).

(2) The image of ψ is{x ∈dCH(Y ) | ω(x) ∈Xσn−1Xi=0f ∗σ(A(Xσ))c1(Qσ, h′′σ)i}.Proof. (1) can be easily checked by the formula:f∗(f ∗(γ)bc1(Q, h′′)i) = 0,0 ≤i < n −1,γ,i = n −1.Next we consider (2).

Let x be an element of dCH(Y ) such thatω(x) ∈Xσn−1Xi=0f ∗σ(A(Xσ))c1(Qσ, h′′σ)i.Since ψ is bijective on finite part, we may assume that z(x) = 0.Thus we can setx = (0, Pσ gσ) with gσ ∈A(Yσ). Here we putω(x) =Xσn−1Xi=0f ∗(uσ,i)c1(Qσ, h′′σ)i.Thenddc(gσ) =n−1Xi=0f ∗(uσ,i)c1(Qσ, h′′σ)i.Using integration along fσ, we can find vσ,i with ddcvσ,i = uσ,i.

(For example vσ,n−1 =Zfσgσ. )Thenω x −n−1Xi=0f ∗(0,Xσvσ,i)bc1(Q, h′′)i!= 0.

14Thus we may assume that ω(x) = 0, which implies gσ is harmonic up to Im ∂+ Im ¯∂.Hence by the structure of cohomology ring H∗(Yσ, R), gσ can be written as the formn−1Xi=0gσ,ic1(Qσ, h′′σ)iup to Im ∂+ Im ¯∂. Therefore x =nXi=0f ∗(0,Xσgσ,i)bc1(Q, h′′)i.Let us go back to the construction of bci(E, h).

By this claim, there is a unique sequence{c1, . .

. , cn} of dCH(X) such thatnXi=1(−1)if ∗(ci)bc1(Q, h′′)n−i = a XσnXi=0(−1)i˜ci(Eσ)c1(Qσ, h′′σ)n−i!−bc1(E, h)n.Hence we may set bci(E, h) = ci.1.10 L2-degree of Hermitian module.

Let V be OK-module of finite rank. A pair(V, h) is called a Hermitian module if we give a Hermitian metric hσ on Vσ for eachσ ∈K∞.

For example, OK has the canonical metric canK induced by each embeddingσ : K ֒→C. We define L2-degree degL2(V, h) of (V, h) bydegL2(V, h) = log #VOKx1 + · · · + OKxt−12Xσ∈S∞log dethσ(x1, x1)· · ·hσ(x1, xt).........hσ(xt, x1)· · ·hσ(xt, xt),where x1, .

. .

, xt ∈M and {x1, . .

. , xt} is a basis of M ⊗K.

Using the Hasse prod-uct formula, it is easily checked that degL2(M, h) does not depend on the choice of{x1, . .

. , xt}.1.11 Arithmetic Riemann-Roch Theorem.

Finally we explain the arithmetic Riemann-Roch theorem. Let π : X −→S an arithmetic variety.

We give a K¨ahler form Ωσ foreach infinite fiber Xσ. Let (E, h) be a Hermitian vector bundle on X.

Using this K¨ahlerform, we can give a metric of Hq(Xσ, Eσ) by identifying Hq(Mσ, Eσ) with the spaceof harmonic forms H0,q(Eσ).Hence we can define degL2(Hq(X, E)). Thus we haveL2-Euler characterχL2(E, h) =Xq≥0(−1)q degL2(Hq(X, E)).

15To state arithmetic Riemann-Roch, we need to mention about the arithmetic Todd char-acter. Let ζ(s) = Pn>0 n−s be the standard zeta function and we define a formal powerseries R(T1, .

. .

, Tn) asXm≥1m:odd1m! (2ζ′(−m) + ζ(−m)(1 + 1/2 + 1/3 + · · · + 1/m)) (T m1 + · · · + T mn ).The arithmetic Todd character is defined bybtdA(TX/S, Ω) = btd(TY/S, Ω)(1 −a(XσR(TXσ, Ωσ))).The arithmetic Riemann-Roch due to Gillet-Soul´e [GS92] is as follows:(1.11.1)χL2(E, h) + 12Xσ∈S∞τ(Eσ, hσ) = deg( bch(E, h) · btdA(TY/S, Ω)).

162. Asymptotic behavior of analytic torsionsIn this section, we will consider asymptotic behavior of analytic torsions of powers ofa semi-positive line bundle.

The main theorem of this section is the following theorem.Theorem 2.1. Let X be a compact K¨ahler manifold of dimension n, (L, h) a Hermitianline bundle on X, and (E, hE) a Hermitian vector bundle on X.

Let ζq,d be the zetafunction of the Laplacian □q,d on A0,q(Ld ⊗E). Let HL be the Hermitian form corre-sponding to the curvature form K(L, h) of L and k a non-negative integer.

If HL(x) ispositive semi-definite and rk HL(x) ≥k for all x ∈X, then for n −k < q ≤n there is aconstant C such that|ζ′q,d(0)| ≤Cdn log(d)for all d ≥0.First we prepare the following lemma.Lemma 2.2. With notation as in Theorem 2.1, if HL(x) is positive semi-definite andrk HL(x) ≥k for all x ∈X, then, for n −k < q ≤n, there is a positive constant cL,qsuch that cL,q depends only on (L, h) and q, and that−√−1⟨(Λe(KL) −e(KL)Λ)φ, φ⟩(x) ≥cL,q⟨φ, φ⟩(x)for all φ ∈An,q(E) and x ∈X.

In particular, integrating the above inequality, we have−√−1 ((Λe(KL) −e(KL)Λ)φ, φ) ≥cL,q(φ, φ).Proof. First we claim:Claim 2.2.1.

There is a positive constant c such that, for all x ∈X, we can take anorthogonal basis {w1, . .

. , wn} of TX,x with HL(x)(wi, wj) = hjδijhi ≥c(1 ≤i ≤k).For a point p ∈X, there is an open neighborhood Up of p and a C∞-subvector bundleFp of TUp such that rk Fp = k and HL|Fp is positive definite.

Shrinking Up if necessarily,we may assume that there is a positive constant cp such that HL(y)(v, v) ≥cp for ally ∈Up and all v ∈(Fp)(y) with ||v|| = 1. Since X is compact, we have p1, .

. .

, ps ∈Xwith Sst=1 Upt = X. We set c = min{cp1, .

. .

, cps}. Let x be an arbitrary point of X. Ifx ∈Upt, then we have an orthogonal basis {w1, .

. .

, wn} of TX,x such that w1, . .

. , wk ∈(Fpt)(x) and HL(x)(wi, wj) = hiδij.

Thus hi ≥cpt ≥c for 1 ≤i ≤k. Therefore we getour claim.

17Let {w1, . .

. , wn} be an orthogonal basis of TX,x as in Claim 2.2.1 and {θ1, .

. .

, θn}the dual basis of {w1, . .

. , wn}.

We setφ =Xi1<···

Let us start the proof of Theorem 2.1. The idea of this proof is found in [BV] and[Vo].

Let λq,d be the minimal eigenvalue of the Laplacian □q,d. By Lemma 2.7.7 of [Vo],it is sufficient to see that there are constants c > 0 and c′ ≥0 such thatλq,d ≥cd −c′.Let □′q,d be the Laplacian on An,q(Ld ⊗E ⊗ω−1X ).

As mentioned in the proof of Theorem1 of [BV], □q,d and □′q,d have the same spectrum. We set E′ = E ⊗ω−1X .

Let KE′ andKd be curvature forms of E′ and Ld ⊗E′ respectively. Then we haveKd = dKL + id ⊗KE′.For φ ∈An,q(Ld ⊗E′), by an easy calculation, we get(□′q,dφ, φ) = ||D′φ||2 + ||δ′φ||2 −√−1 ((Λe(Kd) −e(Kd)Λ)φ, φ)≥−d√−1 ((Λe(KL) −e(KL)Λ)φ, φ)−√−1 ((Λ(id ⊗e(KE′)) −(id ⊗e(KE′))Λ)φ, φ) ,where D′ is (1, 0)-part of the Hermitian connection of Ld ⊗E′ and δ′ is the adjointoperator of D′.

By virtue of Lemma 2.2, there is a positive constant c such that−√−1 ((Λe(KL) −e(KL)Λ)φ, φ) ≥c(φ, φ)for all φ ∈Aq,n(Ld ⊗E′). On the other hand, we can easily find c′ ≥0 such that−√−1 ((Λ(id ⊗e(RE′)) −(id ⊗e(RE′))Λ)φ, φ) ≥−c′(φ, φ)for all φ ∈Aq,n(Ld ⊗E′).

Therefore we have an inequality(□′d,qφ, φ) ≥(cd −c′)(φ, φ).We choose φ ∈Aq,n(Ld ⊗E′) such that □′q,dφ = λq,dφ and (φ, φ) = 1. Thus using theabove inequality, we obtainλq,n ≥cd −c′.□

18Corollary 2.4. Let X be a compact K¨ahler manifold of dimension n, (L, h) a Hermitianline bundle on X, and (E, hE) a Hermitian vector bundle on X.

For the Hermitian formHL corresponding to the curvature form of (L, h), we assume that HL(x) is positive semi-definite and rk HL(x) ≥n −1 for all x ∈X. Then there is a positive constant C suchthatτ(Ld ⊗E) ≤Cdn log(d)for all d ≥0.Proof.

Let ζq,d be the zeta function of the Laplacian on A0,q(Ld ⊗E). Thenτ(Ld ⊗E) = −ζ′1,d(0) +Xq≥2(−1)qqζ′q,d(0).By virtue of Theorem 2.1, we have a positive constant C1 such thatXq≥2(−1)qqζ′q,d(0) ≤C1dn log(d).On the other hand, by Proposition 2.7.6 of [Vo], for some positive C2, we getζ′1,d(0) ≥−C2dn log(d).Thus we obtain our corollary.□

193. Hermitian modules over arithmetic curvesIn this section, we will discuss Hermitian modules over arithmetic curves.

Some resultsare found in [St] and [Gr]. For reader’s convenience, we will give however an explicit prooffor each result.Let V be a C-vector space, hV a Hermitian metric on V and W a subvector space ofV .

The metric hV induces a metric hW of W, which is called the submetric of W inducedby hV . Let W ⊥be the orthogonal complement of W. Then the natural homomorphismW ⊥−→V/W is isomorphic.

Thus we have a metric hV/W of V/W. This metric is calledthe quotient metric of V/W induced by hV .Lemma 3.1.

With notation as above, let x1, . .

. , xs be elements of W and xs+1, .

. .

, xnelements of V such that {x1, . .

. , xs} is a basis of W, {x1, .

. .

, xn} is a basis of V and{¯xs+1, . .

. , ¯xn} is a basis of V/W, where ¯xs+1, .

. .

, ¯xn are images of xs+1, . .

. , xn inV/W.

Then we havehV (x1, x1)· · ·hV (x1, xn)......hV (xn, x1)· · ·hV (xn, xn)=hW (x1, x1)· · ·hW (x1, xs)......hW (xs, x1)· · ·hW (xs, xs)×hV/W (¯xs+1, ¯xs+1)· · ·hV/W (¯xs+1, ¯xn)......hV/W (¯xn, ¯xs+1)· · ·hV/W (¯xn, ¯xn)Proof. Let xi = yi + zi be decompositions such that yi ∈W and zi ∈W ⊥.

Then it iseasy to see thathV/W (¯xs+1, ¯xs+1)· · ·hV/W (¯xs+1, ¯xn)......hV/W (¯xn, ¯xs+1)· · ·hV/W (¯xn, ¯xn)=hV (zs+1, zs+1)· · ·hV (zs+1, zn)......hV (zn, zs+1)· · ·hV (zn, zn)and(x1, . .

. , xn) = (x1, .

. .

, xs, zs+1, . .

. , zn)U,whereU =Is∗0In−s.

20HencehV (x1, x1)· · ·hV (x1, xn)......hV (xn, x1)· · ·hV (xn, xn)=UhW (x1, x1)· · ·hW (x1, xs)0· · ·0............hW (xs, x1)· · ·hW (xs, xs)0· · ·00· · ·0hV (zs+1, zs+1)· · ·hV (zs+1, zn)............0· · ·0hV (zn, zs+1)· · ·hV (zn, zn)t ¯U.Thus we have our lemma.□Proposition 3.2. Let K be an algebraic number field and OK the ring of integers ofK.

Let 0 →S →E →Q →0 be an exact sequence of OK-modules and h′, h and h′′Hermitian metrics of S, E and Q respectively. If, for each infinite place σ of K, h′σ isthe submetric of hσ and h′′σ is the quotient metric of hσ, then we havedegL2(E, h) = degL2(S, h′) + degL2(Q, h′′).Proof.

This is an immediate consequence of Lemma 3.1.□Let K be an algebraic number field and OK the ring of integers of K. Let (E, h) bea Hermitian module over OK. We define an averaged L2-degree µL2(E, h) of (E, h) asfollows:µL2(E, h) = degL2(E, h)rk E.Proposition 3.3. With notation as above, we set SK = Spec(OK).

Let h′ be anotherHermitian metrics of E. Assume that, for each σ ∈(SK)σ, there is a constant Cσ suchthathσ(x, x) ≤Cσh′σ(x, x)for all x ∈Eσ. Then we haveµL2(E, h′) ≤µL2(E, h) + 12Xσ∈(SK)∞log CσProof.

Clearly it is sufficient to show the following lemma.□

21Lemma 3.4. Let V be a vector space over the complex number field C and h and h′Hermitian metrics on V such that h(x, x) ≤h′(x, x) for all x ∈V .

Let e1, . .

. , en be abasis of V .

Then we have det(h(ei, ej)) ≤det(h′(ei, ej)). Moreover the equality holds ifand only if h = h′.Proof.

We set H = (h(ei, ej)) and H′ = (h′(ei, ej)). Let U be an unitary matrix suchthat h′(Uei, Uej) = λiδij.

Here we put hij = h(Uei, Uej). Then by virtue of Hadamard’sinequality we havedet H = det tUHU ≤h11 · · · hnn ≤λ1 · · ·λn = det tUH′U = det H′.Furthermore, if the equality holds, then hij = 0 for all i ̸= j and hii = λi for all i by theequality condition of Hadamard’s inequality.

Thus tUHU = tUH′U. Hence h = h′.□Let k ⊆K be an extension of algebraic number fields and Ok and OK the ring ofintegers of k and K respectively.

We set Sk = Spec(Ok) and SK = Spec(OK) and denoteby f the natural morphism SK −→Sk. If for σ ∈(Sk)∞we setf −1∞(σ) = {σ′ ∈(SK)∞| σ′|k = σ},we have a natural isomorphismf∗(V )σ⊗k C∼−→Mσ′∈f −1∞(σ)Vσ′⊗K C,where f∗(V )σ⊗k C is a tensor product by the embedding σ : k ֒→C.

Thus we have ametric h′(σ) on f∗(V )σ⊗k C by the above isomorphism. We denote this Hermitian moduleon Sk by (f∗V, f∗h) and call it the push-forward of (V, h).

The Riemann-Roch theoremin this situation assertsdegL2(f∗V, f∗h) = degL2(V, h) + (rk V ) degL2(f∗OK, f∗canK).Proposition 3.5. Let (E, h) be a Hermitian module on SK.

Then the set{µL2(F, hF ) | F is a sub-sheaf of E and hF is the induced metric by h.}is discrete subset of R and bounded as above.Proof. We setMl(E, h) =degL2(F, hF )F is a sub-sheaf of E with rk(F) = l andhF is the induced metric by h.

22Then we have{µL2(F, hF) | F is a sub-sheaf of E and hF is the induced metric by h.} =M0(E, h) ∪M1(E, h) ∪· · · ∪1l Ml(E, h) ∪· · · ∪1rk E Mrk E(E, h).Thus it is sufficient to see that Ml(E, h) is discrete on R and bounded as above for eachl. Let Etor be the torsion part of E and ¯E = E/Etor.

Let F be a sub-sheaf of E withrk F = l, Ftor the torsion part of F and ¯F = F/Ftor. Then,degL2(F, hF) = log #(Ftor) + degL2( ¯F, h ¯F ),¯F ⊆¯EandFtor ⊆Etor.Thus we haveMl(E, h) ⊆Ml( ¯E, h) + {log(i) | i ∈Z with 1 ≤i ≤#(Etor)}Therefore we may assume E is torsion free.If we denote by f the natural morphism SK −→SQ, by Riemann-Roch theorem, wehavedegL2(f∗F, f∗hF ) = degL2(F, hF ) + l degL2(f∗OK, f∗canK)impliesMl(E, h) ⊆Mln(f∗E, f∗h) −l degL2(f∗OK, f∗canK),where n = [K : Q].

Thus we may assume K = Q. Since Ml(E, h) ⊆M1(Vl E, Vl h), wemay furthermore assume l = 1.We set E = Zx1 ⊕· · · ⊕Zxr andH =h(x1, x1)· · ·h(x1, xr)......h(xr, x1)· · ·h(xr, xr).Since H is a Hermitian matrix, there is an unitary matrix U such thatH = tU diag(λ1, .

. .

, λr) ¯Uand 0 < λ1 ≤· · · ≤λr. Let F be a sub-module of E of rank 1.

We set F = Z(a1x1 +· · · + arxr), ai ∈Z (1 ≤i ≤r) and (b1, . .

. , br) = (a1, .

. .

, ar)tU. ThendegL2(F, hF ) = −12 log h((a1, .

. .

, ar), (a1, . .

. , ar))= −12 log(λ1|b1|2 + · · · + λr|br|2)

23Thus since |a1|2 + · · · + |ar|2 = |b1|2 + · · · + |br|2, we have−12 log(λr(|a1|2 + · · · + |ar|2)) ≤degL2(F, hF ) ≤−12 log(λ1(|a1|2 + · · · + |ar|2))Thus deg(F, hF ) ≤−12 log(λ1), which shows M1(E, h) is bounded as above. Here weassume that M1(E, h) is not discrete in R. Then there is a sequence {Fn} such thatFn are distinct rank 1 sub-sheaves of E and limn→∞degL2(Fn, hFn) exits.

We set Fn =Z(a(n)1 x1 + · · · + a(n)r xr). Since Fn are distinct, we havelimn→∞|a(n)1 |2 + · · · + |a(n)r |2 = ∞.Thus by the above inequality, limn→∞degL2(Fn, hFn) = −∞.

This is a contradiction.□Proposition 3.6. Let k ⊆K be an extension of algebraic number fields and Ok andOK the ring of integers of k and K respectively.

We denote by f the natural morphismSpec(OK) −→Spec(Ok). For a Hermitian module (L, h) on Spec(OK) of rank 1, thereis a constant C such thatµL2(F, hF) ≤Cmfor all m ≥1 and all Ok-submodule F of f∗Lm, where hF is the induced metric by f∗hm.Proof.

First we claim that we may assume L = OK. Let x be a non-zero element of L.We define a homomorphism φm : OK −→Lm by φm(1) = x ⊗· · · ⊗x. Let h1 be theinduced metric of OK by φ1.

Then clearly the induced metric by φm is equal to hm1 . Weset Qm = Coker(φm : OK −→Lm).

We consider the exact sequence:0 →f∗OK →f∗Lm →f∗Qm →0.Let F be a Ok-submodule of f∗Lm. Then,degL2(F) ≤degL2(F ∩f∗OK) + log #(Qm).Thus we have the claim.Let F be a Ok-submodule of f∗OK.

Let hF be the induced metric by f∗hm and h′Fthe induced metric by f∗canK. Then, it is easy to see thatµL2(F, hF) = µL2(F, h′F ) −m2Xσ∈(SK)∞log hσ(1, 1).Hence we have our proposition by Proposition 3.5.□

244. Asymptotic behavior of L2-degree of submodules of H0(Lm)A main purpose of this section is to prove the following theorem.Theorem 4.1.

Let K be an algebraic number field, OK the ring of integers of K andf : X −→Spec(OK) an arithmetic variety over OK. Let (L, h) be a Hermitian linebundle on X, (E, hE) a Hermitian vector bundle on X, and hm a Hermitian metric ofH0(X, Lm ⊗E) induced by hm ⊗hE. Then there is a constant C such thatdegL2(F, hF )rk F≤Cm log mfor all m > 1 and all submodules F of H0(X, Lm ⊗E), where hF is the Hermitian metricinduced by hm.First of all, we prepare the following Lemma.Lemma 4.2.

Let M be a compact K¨ahler manifold of dimension d, (L, h) a Hermitianline bundle on M and Φ a K¨ahler form on M. Let s ∈H0(M, L) be a smooth sectionand V the set of zero points of s. Then there are constants C and C′ such thatZMhm−1(t, t)Φd ≤Cm6dZMhm(s ⊗t, s ⊗t)Φdfor all m ≥1 and t ∈H0(M, Lm−1) and thatZVhm(t, t)Φd−1 ≤C′m2dZMhm(t, t)Φdfor all m ≥1 and t ∈H0(M, Lm).Proof. Here we claim the following sublemma, which can be proved by the same idea(due to M. Gromov) as in the proof of Proposition 3 of [GS88].Sublemma 4.2.1.

For a real number p ≥1, there is a constant C such that||t||sup ≤Cm2d/p||t||Lpfor all m ≥1 and t ∈H0(M, Lm).Proof. Let x be a point of M, Ux an open neighborhood of x,Ux∼−→Vx ⊆Cda local chart, and ux a local basis of L. From now, we identify Ux with Vx.

We setqx(z) = h(ux, ux) andr(a, b) = |a1 −b1| + · · · + |ad −bd|

25for a = (a1, . .

. , ad), b = (b1, .

. .

, bd) ∈Ux. Moreover we setB(a, R) = {(z1, .

. .

, zd) ∈Ux | |zi −ai| ≤R for all i}for a point a = (a1, . .

. , ad) ∈Ux and R > 0.

Shrinking Ux if necessary, we may assumethat there is a constant Kx such that|qx(a) −qx(b)| ≤Kxr(a, b)for all a, b ∈Ux. Thus there are an open neighborhood Wx ⊆Ux of x and a constant rxwith the following properties:(1) B(a, rx) ⊆Ux for all a ∈Wx.

(2) qx(z) ≥qx(a) −Kxr(a, z) > 0 for z ∈B(a, rx) and a ∈Wx.Since M = Sx∈M Wx and M is compact, there are finitely many x1, . .

. , xn such thatM = Sni=1 Wxi.

For simplicity, we set Ui = Uxi, Wi = Wxi, ui = uxi, qi = qxi, Ki = Kxiand ri = rxi. Let t be a section of H0(M, Lm).

Since M is compact, we can take a ∈Mwith ||t||sup =phm(t, t)(a). Let a ∈Wi (1 ≤i ≤n).

Using local basis ui, we writet = f(z)umi . Then,hm(t, t)p/2 = |f(z)|pqi(z)pm/2.We take constants gi, Gi and Ci such that gi ≤qi(z) ≤Gi for all z ∈Ui and Φd ≥Cidx1dy1 · · · dxddyd, where zj = xj + √−1yj for 1 ≤j ≤d.

Let lm be the integral partof pm/2 and l′m = pm/2 −lm. Since |f|p is subharmonic, we have||t||pLp =ZMhm(t, t)p/2Φd≥CiZB(a,ri)|f(z)|pqi(z)pm/2dx1dy1 · · · dxddyd≥Cigl′miZB(a,ri)|f(z)|pqi(z)lmdx1dy1 · · ·dxddyd≥Cigl′miZB(a,ri)|f(z)|p(qi(a) −Kir(a, z))lmdx1dy1 · · · dxddyd≥(2π)dCigl′mi |f(a)|p×Z ri0· · ·Z ri0R1 · · · Rd(qi(a) −Ki(R1 + · · · + Rd))lmdR1 · · · dRd.On the other hand, by an easy calculation, we can seeZ r0x(A −Bx)ldx ≥r2(l + 1)(l + 2)Al,

26where A > 0, B > 0, r > 0, A −Br > 0 and l is a non-negative integer. Hence we have||t||pLp ≥(2π)dCigl′mi r2di(lm + 1)d(lm + 2)d |f(a)|pqi(a)lm≥(2π)dCir2di(lm + 1)d(lm + 2)d giGil′m|f(a)|pqi(a)pm/2≥(2π)dCir2di(pm/2 + 1)d(pm/2 + 2)d giGi||t||psup.Here we take a constant C such that(2π)dCir2di(pm/2 + 1)d(pm/2 + 2)d giGi≥Cm2d ,for all m > 0 and 1 ≤i ≤n.

Then||t||pLp ≥Cm2d ||t||psup.Thus we have our sublemma.□To prove the first part of Lemma 4.2, it is sufficient to see the following claim.Claim 4.2.2. There are constants C1 and C2 such that||t||L1 ≤C1md||s ⊗t||L2for all m ≥1 and t ∈H0(M, Lm−1) and that||t||L2 ≤C2m2d||t||L1for all m ≥1 and t ∈H0(M, Lm−1).Since h(s, s)−1/2 is integrable, we haveZMhm−1(t, t)1/2Φd =ZMh(s, s)−1/2hm(s ⊗t, s ⊗t)1/2Φd≤ZMh(s, s)−1/2Φd||s ⊗t||sup.Thus by Sublemma 4.2.1, we have first inequality of Claim 4.2.2.Clearly there is a constant C3 such that||t||L2 ≤C3||t||sup

27Therefore Sublemma 4.2.1 implies second inequality of Claim 4.2.2.Next we consider the second assertion of Lemma 4.2. Let v be the volume of V .

UsingSublemma 4.2.1, for some constant C4, we haveZVhm(t, t)Φd−1 ≤v|| t|V ||2sup≤v||t||2sup≤vC4m2dZMhm(t, t)Φd.Thus we get Lemma 4.2.□Now let us go back to the proof of Theorem 4.1. To get Theorem 4.1, clearly we mayassume that X is normal.4.3 Reduction 1.

We may assume that E is a line bundle.Let g : Y = P(E) −→X be the projective bundle of E and O(1) the tautological linebundle of E. We can define the quotient metric hO(1) on O(1) by the natural surjectivehomomorphism g∗(E) →O(1). We consider an isomorphism:αm : H0(X, Lm ⊗E) −→H0(Y, g∗(L)m ⊗O(1)).Let F be a submodule of H0(X, Lm ⊗E).

Let hF be a Hermitian metric of F inducedby hm ⊗hE and hαm(F ) a Hermitian metric of αm(F) induced by g∗(h)m ⊗hO(1). Then,by Proposition 3.3, we haveµ(F, hF ) ≤µ(αm(F), hαm(F )).Thus we get Reduction 1.4.4 Reduction 2.

We may assume that H0(X, E) ̸= 0.Let (E′, hE′) be a Hermitian line bundle on X such that H0(X, E′) ̸= 0 and H0(X, E⊗E′) ̸= 0. Let s1 be a non-zero section of H0(X, E′).

We consider an injective homomor-phism:βm : H0(X, Lm ⊗E)⊗s1−→H0(X, Lm ⊗E ⊗E′).We take a constant B1 such that ||(s1)σ||sup ≤B1 for all σ ∈K∞. Then we have||βm(t)σ||L2 ≤B1||tσ||L2for all t ∈H0(X, Lm ⊗E) and all σ ∈K∞.

Thus, by Proposition 3.3, for a submoduleF of H0(X, Lm ⊗E), we obtainµ(F, hF ) ≤µ(βm(F), hβm(F )) + [K : Q] log B1.Therefore, we get Reduction 2.

284.5 Reduction 3. We may assume that E = (OX, canK).Let s2 be a non-zero section of H0(X, E).

We consider an injective homomorphism:γm : H0(X, Lm ⊗E)⊗sm−12−→H0(X, (L ⊗E)m).We take a constant B2 such that ||(s2)σ||sup ≤B2 for all σ ∈K∞. Then, by the sameway as in Reduction 2, for a submodule F of H0(X, Lm ⊗E), we haveµ(F, hF ) ≤µ(γm(F), hγm(F )) + (m −1)[K : Q] log B2.Therefore, we get Reduction 3.4.6 Reduction 4.

We may assume that L is very ample.Let (L′, hL′) be a Hermitian line bundle on X such that H0(X, L′) ̸= 0 and L ⊗L′is very ample.Let s3 be a non-zero section of H0(X, L′).We consider an injectivehomomorphism:δm : H0(X, Lm)⊗sm3−→H0(X, (L ⊗L′)m).We take a constant B3 such that ||(s3)σ||sup ≤B3 for all σ ∈K∞. Then, by the sameway as in Reduction 2, for a submodule F of H0(X, Lm ⊗E), we haveµ(F, hF ) ≤µ(δm(F), hδm(F )) + m[K : Q] log B3.Therefore, we get Reduction 4.4.6.

Let us go to the main part of the proof of Theorem 4.1.Gathering Reduction1–4, we may assume that (E, h) = (OX, canK) and L is very ample. We use inductionon dim f. If dim f = 0, this theorem holds by Proposition 3.6.

Here we remark thefollowing.Remark 4.7.1. For a base extension Spec(OK′) −→Spec(OK) and OK-submoduleF ⊆H0(X, Lm), we getF ⊗OK′ ⊆H0(X ⊗OK′, (L ⊗OK′)m)andµL2(F ⊗OK′, hF ⊗OK′) = [K′ : K]µL2(F, hF).For s ∈H0(X, L), we denote by div(s) the set of zero of s. Since L is very ample, bythe above Remark 4.7.1, considering a base change of f : X −→Spec(OK) if necessarily,we may assume that there is a section s ∈H0(X, L) such that div(s) −→Spec(OK) isgenerically smooth.

Let V ′ = div(s), V the horizontal component of V ′, IV the definingideal of V . We give a Hermitian metric hIV of IV by h−1.

LetXf ′−→Spec(OK′)π−→Spec(OK)

29be the Stein factorization of Xf−→Spec(OK). Then f ′∗(IV ⊗OX(V ′)) is a torsion freesheaf of rank 1 and the natural homomorphismIV ⊗OX(V ′) −→f ′∗(f ′∗(IV ⊗OX(V ′)))is injective because IV ⊗OX(V ′) is isomorphic to OXK′ on the generic fiber XK′ off ′.

Since f ′∗(IV ⊗OX(V ′)) is of rank 1, there is a factorial ideal A of OK such thatf ′∗(IV ⊗OX(V ′)) ⊆π∗(A). In particular, we have IV ⊗OX(V ′) ⊆f ∗(A).

Here we give atrivial Hermitian metric hA to A. Then the inclusion IV ⊗OX(V ′) ⊆f ∗(A) is isometry.Let F be a submodule of H0(X, Lm) and hF the induced metric by hm.

We considerthe exact sequence:0 →H0(X, Lm ⊗IV ) →H0(X, Lm) →H0(V, Lm).We set T = F ∩H0(X, Lm ⊗IV ) and Q = F/F ∩H0(X, Lm ⊗IV ). Let hT be thesubmetric of hF and hQ the quotient metric of hF .

Then by Proposition 3.2, we have(4.7.2)µL2(F, hF) =tt + q µL2(T, hT ) +qt + q µL2(Q, hQ),where t = rk T and q = rk Q. Let h′T be the submetric induced by hm ⊗hIV = hm−1.Since Q is a submodule of H0(V, Lm), we get a metric h′Q induced by (h|V )m. If we setd = dim f, by Proposition 3.3 and Lemma 4.2, there are constants C1 and C2 such that(4.7.3)µL2(T, hT ) ≤µL2(T, h′T ) + log C1 + 6d log m2[K : Q]and(4.7.4)µL2(Q, hQ) ≤µL2(Q, h′Q) + log C2 + 2d log m2[K : Q],for all m ≥1.

By hypothesis of induction, there is a constant C3 such that(4.7.5)µL2(N, hN) ≤C3m log mfor all m > 1 and all submodule N of H0(V, Lm). On the other hand, due to Proposi-tion 3.5, we can take a constant C4 such that(4.7.6)µL2(P, hP) ≤C4for all submodule P of H0(X, L2).

MoreoverµL2(T, h′T ) = µL2(T ⊗A−1, h′T ⊗h−1A ) + µL2(A, hA)and T ⊗A−1 ⊆H0(X, Lm−1). We choose C satisfying(4.7.7)µL2(A, hA) + log C1 + 6d log m2[K : Q] ≤C log mC3m log m + log C2 + 2d log m2[K : Q] ≤Cm log mC4 ≤C(2 log 2)for all m > 0.

Here we claim:

30Claim 4.7.8. µL2(F, hF) ≤Cm log m for all m > 1 and all submodules F of H0(X, Lm).We prove this claim by induction on m. By hypothesis of induction on m, we haveµL2(T ⊗A−1, h′T ⊗h−1A ) ≤C(m −1) log(m −1).Thus by (4.7.3) and (4.7.7), we getµL2(T, hT ) ≤Cm log m.On the other hand, (4.7.4), (4.7.5) and (4.7.7) implyµL2(Q, hQ) ≤Cm log m.Hence by (4.7.2), we obtainµL2(F, hF) ≤tt + q Cm log m +qt + q Cm log m = Cm log m.□

315. Vanishing of a certain Bott-Chern secondary classThis section is devoted to prove the following lemma.Lemma 5.1.

Let C be a compact Riemann surface, E a vector bundle of rank r on Cand h a projectively flat metric of E. Let f : Y = P(E) −→C be the projective bundleof E and OY (1) the tautological line bundle. Let E : 0 →F →f ∗E →OY (1) →0 bethe canonical exact sequence.

We give the canonical Hermitian metrics on F, f ∗E andOY (1) induced by the Hermitian metric h of E. Then we haveZY(c1(OY (1)) −1r f ∗(c1(E))rXi=1(−1)i˜ci(E)c1(OY (1))r−i)= 0.Proof. First we notice that we may assume that det E is divisible by r in Pic(C) consid-ering a finite covering of C. We setΦ(E) =rXi=1(−1)i˜ci(E)c1(OY (1))r−i.Let L be a Hermitian line bundle on C. We will see Φ(E ⊗L) = Φ(E).

Using the formulaof (1.3.3.2) in [GS90b], we haveΦ(E ⊗L) =rXi=1(−1)i˜ci(E ⊗L) (c1(OY (1)) + f ∗(c1(L)))r−i=rXi=1(−1)i (˜ci(E) + (r −i + 1)˜ci−1(E)f ∗(c1(L))) ×c1(OY (1))r−i + (r −i)c1(OY (1))r−i−1f ∗(c1(L))= Φ(E) +rXi=1(−1)i(r −i)˜ci(E)c1(OY (1))r−i−1f ∗(c1(L))+rXi=1(−1)i(r −i + 1)˜ci−1(E)c1(OY (1))r−if ∗(c1(L))= Φ(E).Since det E is divisible by r in Pic(C), we can take a Hermitian line bundle (L, hL)such that (L, hL)⊗r = (det E, det h). We set E′ = E ⊗L−1.

Then c1(E′) = 0. Theseobservations show us that we may assume that c1(E) = 0.Thus, to get our assertion, it is sufficient to see thatZY˜ci(E)c1(OY (1))r−i+1 = 0

32for i = 1, . .

. , r. We denote the homomorphisms F −→f ∗E and f ∗E −→OY (1) byι and τ respectively.Let p : Y × P1 −→Y and q : Y × P1 −→P1 be the naturalprojections.

Let σ ∈H0(P1, OP1(1)) be a non-zero section such that σ(∞) = 0. LethOP1 (1) be a metric of OP1(1) such that hOP1 (1)(σ, σ)0 = 1.

We set˜E = Coker (p∗ι ⊕(idp∗F ⊗σ) : p∗F −→p∗f ∗E ⊕(p∗F ⊗q∗OP1(1))) .Let F ⊥be the orthogonal complement of F in f ∗E. Using the natural bijective homo-morphismp∗F ⊥⊕(p∗F ⊗q∗OP1(1)) −→˜E,we give a metric ˜h on ˜E.

Let x, x′ ∈p∗f ∗E andx = x1 + x2 (x1 ∈p∗F, x2 ∈p∗F ⊥),x′ = x′1 + x′2 (x′1 ∈p∗F, x′2 ∈p∗F ⊥)the orthonormal decompositions. Then we have˜h(x, x′) −hE(x, x′) = (||σ||2 −1)hE(x1, x′1).These observation shows us that ( ˜E, ˜h)Y ×{0} is isometric to (f ∗E, hE).

On the otherhand, clearly ( ˜E, ˜h)Y ×{∞} is isometric to (F, hF ) ⊕(OY (1), hOY (1)).Hence by theconstruction of ˜ci(E) (cf. 1.3)˜ci(E) =ZP1 ci( ˜E, ˜h) log |z|2,where z is an inhomogeneous coordinate of P1.

Thus using Fubini’s theorem, we haveZY˜ci(E)c1(OY (1))r−i+1 =ZYc1(OY (1))r−i+1ZP1 ci( ˜E, ˜h) log |z|2=ZY ×P1 p∗c1(OY (1))r−i+1ci( ˜E, ˜h) log |z|2=ZP1log |z|2ZYp∗c1(OY (1))r−i+1ci( ˜E, ˜h).Hence it is sufficient to see thatZYp∗c1(OY (1))r−i+1ci( ˜E, ˜h) = 0for z ̸= ∞.Since E is flat, we can take a local flat frame s0, . .

. , sr−1 of E suchthat hE(si, sj) = δij.

Let X0, . .

. , Xr−1 be a homogeneous coordinate of Y = P(E)corresponding to s0, .

. .

, sr−1. On X0 ̸= 0, we set zi = Xi/X0 andg = s0 + ¯z1s1 + · · · + ¯zr−1sr−11 + |z1|2 + · · · + |zr−1|2 .

33Then g ∈F ⊥and τ(g) = τ(s0). Sinces0 = (s0 −g) + gandsi = (si −zig) + zig (i = 1, · · · , r −1)are orthogonal decompositions of s0 and si respectively, we have(˜h(si, sj)) = ||σ||2(δij)+1 −||σ||21 + |z1|2 + · · · + |zr−1|21¯z1¯z2· · ·¯zr−1z1|z1|2z1¯z2· · ·z1¯zr−1z2z2¯z1|z2|2...z2¯zr−1.........· · ·...zr−1zr−1¯z1zr−1¯z2· · ·|zr−1|2.Thus we getci( ˜E, ˜h) ∈A0,0[dz, dz1, .

. .

, dzr−1, d¯z, d¯z1, . .

. , d¯zr−1].On the other hand, sincehE(g, g) =11 + |z1|2 + · · · + |zr−1|2 ,we obtainc1(OY (1)) ∈A0,0[dz1, .

. .

, dzr−1, d¯z1, . .

. , d¯zr−1].Therefore since p∗c1(OY (1))r−i+1ci( ˜E, ˜h) is (r + 1, r + 1)-form and lies inA0,0[dz, dz1, .

. .

, dzr−1, d¯z, d¯z1, . .

. , d¯zr−1],we obtainp∗c1(OY (1))r−i+1ci( ˜E, ˜h) = 0on the chart X0 ̸= 0.

Hence we have our lemma.□

346. Donaldson’s Lagrangian and arithmetic second Chern classLet M be an n-dimensional complex manifold, and E a vector bundle of rank r on M.Let h and k be Hermitian metrics of E, and {ht}0≤t≤1 a C∞-deformation of Hermitianmetrics of E such that h0 = h and h1 = k. Let Rt be the curvature of ht.

We setQ1(E, h, k) = log(det(h)/ det(k)),Q2(E, h, k) =√−1Z 10tr(h−1t· (∂tht) · Rt)dt,ech2(E, h, k) = ech2(0 →(E, k) →(E, h) →(0, 0) →0).By Lemma (3.6) in Chapter VI of [Ko], Q2(E, h, k) is uniquely determined by h and kup to ∂(A0,1)+∂(A1,0). Then, by comparing (1, 1)-part of the formula in Corollary 1.30,ii) of [BGS], we have the following lemma.

(Note that ech2(E, h, k) = ech2(0 →(E, h) →(E, k) →(0, 0) →0) in the sense of [BGS]. )Lemma 6.1. ech2(E, h, k) = −12π Q2(E, h, k) modulo ∂(A0,1) + ∂(A1,0).Here we assume that M is compact and K¨ahler.

Let Φ be a K¨ahler form of M. TheDonaldson’s Lagrangian DL(E, h, k; Φ) is defined byDL(E, h, k; Φ) =ZM(Q2(E, h, k) −cnQ1(E, h, k)Φ) ∧Φn−1(n −1)!,wherec =2nπZMc1(E) ∧Φn−1rZMΦn.We fix a Hermitian metric k of E. Then, by Proposition (3.37) in Chapter VI of [Ko],the following are equivalent. (i) DL(E, h0, k; Φ) gives the absolute minimal value of{DL(E, h, k; Φ) | h is a Hermitian metric of E}.

(ii) h0 is Einstein-Hermitian.Lemma 6.2. Let K be an algebraic number field and OK the ring of integers.Letf : X −→Spec(OK) be an arithmetic variety with dim X = d ≥2, and (H, hH) aHermitian line bundle on X such that, for each σ ∈K∞, c1(Hσ, hHσ) gives a K¨ahlerform Φσ on an infinite fiber Xσ.

Let E be a vector bundle on X, and h and h′ Hermitianmetrics of E. If det(h) = det(h′), then we have(bc2(E, h) −bc2(E, h′)) · bc1(H, hH)d−2 = (d −2)!2πXσ∈K∞DL(Eσ, hσ, h′σ; Φσ).

35Proof. Since det(h) = det(h′), we have bc1(E, h) = bc1(E, h′).

Thus we getbc2(E, h) −bc2(E, h′) = −( bch2(E, h) −bch2(E, h′)).Therefore, by Lemma 6.1, we obtainbc2(E, h) −bc2(E, h′) = 12πXσ∈K∞Q2(Eσ, hσ, h′σ).Hence we have our formula because Q1(Eσ, hσ, h′σ) = 0 for all σ ∈K∞.□Theorem 6.3. Let K be an algebraic number field and OK the ring of integers.

Letf : X −→Spec(OK) be an arithmetic variety with dim X = d ≥2, and (H, hH) aHermitian line bundle on X such that, for each σ ∈K∞, c1(Hσ, hHσ) gives a K¨ahlerform Φσ on an infinite fiber Xσ.Let E be a vector bundle of rank r on X.For aHermitian metric h of E, we set∆(E, h) =bc2(E, h) −r −12r bc1(E, h)2· bc1(H, hH)d−2.If E is Φσ-poly-stable on each infinite fiber Xσ, then we have;(1) the set ∆= {∆(E, h) | h is a Hermitian metric of E} has the absolute minimalvalue. (2) ∆(E, h0) attaches the minimal value of ∆if and only if h0 is weakly Einstein-Hermitian on each infinite fiber.Proof.

Since E is poly-stable on each infinite fiber, there is an Einstein-Hermitian metrick of E. We set ρ =rpdet(k)/ det(h) and h′ = ρh. Then it is easy to see that det(h′) =det(h) and ∆(E, h) = ∆(E, h′).

Thus by Lemma 6.2,∆(E, h) −∆(E, k) = ∆(E, h′) −∆(E, k)= (bc2(E, h′) −bc2(E, k)) · bc1(H, hH)d−2= (d −2)!2πXσ∈K∞DL(Eσ, h′σ, kσ; Φσ).Here, since DL is the Donaldson’s Lagrangian, we getXσ∈S∞DL(Eσ, h′σ, kσ; Φσ) ≥0and the equality holds if and only if h′ is Einstein-Hermitian on each infinite fiber. Thuswe have our theorem.□

367. Second fundamental formLet M be a complex manifold and (E, h) a Hermitian vector bundle on M.Let0 →S →E →Q →0 be an exact sequence of vector bundles.

Let h′ and h′′ be Hermit-ian metrics of S and Q induced by h respectively. Let E = S ⊕S⊥be the orthogonaldecomposition of E by h. Let D(E, h), D(S, h′) and D(Q, h′′) be the Hermitian con-nections of (E, h), (S, h′) and (Q, h′′) respectively.

Moreover, let K(E, h), K(S, h′) andK(Q, h′′) be the curvatures of (E, h), (S, h′) and (Q, h′′) respectively. The Hermitianconnection D(E, h) has the following form:D(E, h) =D(S, h′)−A∗AD(Q, h′′),where A ∈A1,0(Hom(S, S⊥)) and A∗is the adjoint of A.

A is called the second funda-mental form of0 →(S, h′) →(E, h) →(Q, h′′) →0.It is well known that the exact sequence 0 →S →E →Q →0 induces the orthogonaldecomposition (E, h) = (S, h′′) ⊕(Q, h′′) if and only if A vanishes identically. We setDt = D(E, h) + (et −1)00A0andKt = (Dt)2.Then, by Proposition 3.28 and Lemma 4.7 of [BC], we havetr(K(E, h)2) −tr(K2t ) =2π√−1ddcZ 0ttrKt ·1000+ tr1000· Ktdt.Therefore, since tr(K(S, h′)) is d-closed, by an easy calculation, we gettr(K(E, h)2) −tr(K2t ) = −4π√−1(1 −et)ddc (tr(A∗∧A)) .Thus we obtaintr(K(E, h)2) −tr(K(S, h′)2 ⊕K(Q, h′′)2) = −4π√−1ddc (tr(A∗∧A))becauselimt→−∞tr(K2t ) = tr(K(S, h′)2 ⊕K(Q, h′′)2).These observations show us the following lemma.

37Lemma 7.1. With notation being as above, we havech2(E, h) −ch2((S, h′) ⊕(Q, h′′)) = ddc√−12πtr(A∗∧A).In particular, by the axiomatic characterization of ech2, we getech2(0 →(S, h′) →(E, h) →(Q, h′′) →0) =√−12πtr(A∗∧A)modulo ∂(A0,1) + ¯∂(A1,0).Remark 7.2.

In the sense of [BGS],ech2(0 →(S, h′) →(E, h) →(Q, h′′) →0) = −√−12πtr(A∗∧A).Here we assume that M is an n-dimensional compact K¨ahler manifold with a K¨ahlerform Φ. Let θ1, .

. .

, θn be a local unitary frame of Ω1M. Then Φ = √−1 P θi ∧¯θi.

Weset A = P Aiθi. Since A∗= P A∗i ¯θi, we get√−1 tr(A∗∧A) ∧Φn−1(n −1)!

=√−1nXi=1,j=1tr(A∗i ∧Aj)(¯θi ∧θj) ∧Φn−1(n −1)!= −nXi=1tr(A∗i ∧Ai)√−1(θi ∧¯θi) ∧Φn−1(n −1)!= −|A|2 Φnn! ,which implies thatZMech2(0 →(S, h′) →(E, h) →(Q, h′′) →0) ∧Φn−1(n −1)!

= −12π ||A||2.Thus we have the following proposition.Proposition 7.3. Let K be an algebraic number field and OK the ring of integers.

Wedenote by K∞the set of all embeddings of K into C. Let f : X −→Spec(OK) be aregular arithmetic variety with dim X = d ≥2, and (H, hH) a Hermitian line bundle onX such that, for each σ ∈K∞, c1(Hσ, hHσ) gives a K¨ahler form Φσ on an infinite fiberXσ. Let 0 →S →E →Q →0 be an exact sequence of torsion free sheaves such thateach torsion free sheaf is locally free on the generic fiber.

Let h be a Hermitian metricof E, and h′ and h′′ Hermitian metrics of S and Q induced by h respectively. Then, wehave the following:bch2(E, h) −bch2((S, h′) ⊕(Q, h′′))· bc1(H, hH)d−2 = −(d −2)!2πXσ∈K∞||Aσ||2,(bc2(E, h) −bc2((S, h′) ⊕(Q, h′′))) · bc1(H, hH)d−2 = (d −2)!2πXσ∈K∞||Aσ||2.

38Corollary 7.4. Let K be an algebraic number field and OK the ring of integers.

Wedenote by K∞the set of all embeddings of K into C. Let f : X −→Spec(OK) be aregular arithmetic variety with dim X = d ≥2, and (H, hH) a Hermitian line bundle onX such that, for each σ ∈K∞, c1(Hσ, hHσ) gives a K¨ahler form Φσ on an infinite fiberXσ. Let0 = E0 ⊂E1 ⊂· · · ⊂El−1 ⊂Elbe a filtration of torsion free sheaves on X such that(i) Ei is locally free on the generic fiber for every 1 ≤i ≤l, and that(ii) Ei/Ei−1 is torsion free and locally free on the generic fiber for every 1 ≤i ≤l.Let hl be a Hermitian metric of El and hi the induced metric of Ei by hl.

Let Qi =Ei/Ei−1 and ki the quotient metric of Qi induced by hi. Then, we havebc2(El, hl) −bc2((Q1, k1) ⊕· · · ⊕(Ql, kl))· bc1(H, hH)d−2 ≥0.Further, the equality of the above inequality holds if and only if (El, hl) is isometric to(Q1, h1) ⊕· · · ⊕(Ql, hl) on each infinite fiber.Proof.

We will prove the following inequality inductively.bc2(El, hl) −bc2((Ei, hi) ⊕(Qi+1, ki+1) ⊕· · · ⊕(Ql, kl))· bc1(H, hH)d−2 ≥0.In the case where i = l −1, the above inequality is an immediate consequence of Propo-sition 7.3. Here we consider an exact sequence:0 →Ei−1 →Ei →Qi →0.By Proposition 7.3, we havebc2(Ei, hi) −bc2((Ei−1, hi−1) ⊕(Qi, ki))· bc1(H, hH)d−2 ≥0.Therefore, we obtainbc2((Ei, hi) ⊕(Qi+1, ki+1) ⊕· · · ⊕(Ql, kl))−bc2((Ei−1, hi−1) ⊕(Qi, ki) ⊕(Qi+1, ki+1) ⊕· · · ⊕(Ql, kl))· bc1(H, hH)d−2 ≥0.Hence, combining hypothesis of induction, we getbc2(El, hl) −bc2((Ei−1, hi−1) ⊕(Qi, ki) ⊕· · · ⊕(Ql, kl))· bc1(H, hH)d−2 ≥0.Thus we have our corollary.□

398. Proof of the main theoremFirst of all, we will prepare the following two lemmas.Lemma 8.1.

Let k an algebraically closed field and K an extension field of k. Let X bea normal projective variety over k, H an ample line bundle on X, and E a vector bundleon X. Then E is H-stable (resp.

H-semistable) if and only if E ⊗k K is H ⊗k K-stable(resp. H ⊗k K-semistable).Proof.

Clearly, if E ⊗k K is H ⊗k K-stable (resp. H ⊗k K-semistable), then E is H-stable(resp.

H-semistable).We assume that E ⊗k K is not H ⊗k K-stable (resp. not H ⊗k K-semistable).

Thenthere is a subsheaf F of E ⊗k K with µ(F) ≥µ(E) (resp. µ(F) > µ(E)).

Here we cantake elements x1, · · · , xn of K such that F is defined over k(x1, . .

. , xn).

We consider anormal projective variety Y over k such that k(Y ) = k(x1, . .

. , xn).

We set Z = X ×k Yand let p : Z →X be the projection. By the assumption, for the generic point η ∈Y ,p∗(E)|X×{¯η} is not H-stable (resp.

not H-semistable). Therefore, by [Ma1], p∗(E)|X×{y}is not H-stable (resp.

not H-semistable) for all closed point y of Y . Thus E is not H-stable (resp.

not H-semistable).□Lemma 8.2. Let k be a Galois extension over Q with a Galois group G = Gal(k/Q).Let X be a smooth projective variety over k, H an ample line bundle on X, and E avector bundle on X.

Then if E ⊗k Q is stable (resp. semistable), then, for every σ ∈G,E ⊗σk Q is also stable (resp.

semistable), where E ⊗σk Q is a tensor product with using anembedding σ : k →Q.Proof. Assume that E ⊗σk Q is not stable (resp.

not semistable). Then there is a subsheafF of E ⊗σk Q such that µ(F) ≥µ(E) (resp.

µ(F) > µ(E)). We may assume that F isdefined over a field k′ such that k ⊂k′ and k′ is a Galois extension over Q.

We takean element σ′ of Gal(k′/Q) with σ′|k = σ. Here we give a right k′-module structure to(E ⊗σk k′) by (e ⊗a) · λ = e ⊗aσ′(λ).

Then if we consider a correspondence e ⊗a ⊗b ⇝e ⊗σ′−1(a)σ′(b), we can easily to see that (E ⊗σk k′) ⊗σ′−1k′k′ ≃E ⊗k k′. Moreover, if wegive a right k′-module structure to (E ⊗σk k′)⊗σ′−1k′k′ by (e⊗a⊗b)·λ = e⊗a⊗bσ′−1(λ),then the above is an isomorphism as k′-modules.

Thus F ⊗σ′−1k′k′ is a subsheaf of E⊗k k′.On the other hand, since the intersection number does not change by an extension of thegrand field, we have µ(F) = µ(F ⊗σ′−1k′k′). This is a contradiction.□Let us start the proof of the main theorem.

We follow steps in Introduction.8.3 Step 1. We try to reduce our theorem to the case where E is poly-stable on eachinfinite fiber.

Since E ⊗K Q is semistable, there is a Jordan-H¨older filtration of E ⊗K Q:0 = E0 ⊂E1 ⊂· · · ⊂El−1 ⊂El = E ⊗K Qsuch that Ei/Ei−1 is stable for all 1 ≤i ≤l and µ(E1/E0) = µ(E2/E1) = · · · =µ(El/El−1). Considering a base change of OK, we may assume that Ei is defined over K

40and K is a Galois extension over Q. Moreover, if we take a suitable birational change ofX, we may assume that X is regular, Ei is defined over X and that Ei/Ei−1 is torsionfree.

We set Qi = Ei/Ei−1. We give a Hermitian metric hi to each Ei induced by h.Here we consider an exact sequence:0 →Ei−1 →Ei →Qi →0.Let ki be the quotient metric of Qi by the above exact sequence, and Q∨∨ithe doubledual of Qi.

Lemma 8.1 and Lemma 8.2 imply Q∨∨1⊕· · · ⊕Q∨∨lis poly-stable on eachinfinite fiber. Thus by hypothesis of reduction, we havebc2((Q∨∨1 , k1) ⊕· · · ⊕(Q∨∨l, kl)) −r −12r bc1((Q∨∨1 , k1) ⊕· · · ⊕(Q∨∨l, kl))2 ≥0.Corollary 7.4 implies thatbc2(E, h) ≥bc2((Q1, k1) ⊕· · · ⊕(Ql, kl)).On the other hand, clearly we havebc2((Q1, k1) ⊕· · · ⊕(Ql, kl)) ≥bc2((Q∨∨1 , k1) ⊕· · · ⊕(Q∨∨l, kl))andbc1(E, h) = bc1((Q∨∨1 , k1) ⊕· · · ⊕(Q∨∨l, kl)).Thus we get Step 1.8.4 Step 2.

By Theorem 6.3, we may assume that h is Einstein-Hermitian on eachinfinite fiber.8.5 Step 3. Let π : Y = P(E) −→X be the projective bundle of E and OY (1) thetautological line bundle.

Let E : 0 →F →π∗E →OY (1) →0 be the canonical exactsequence. We give the canonical Hermitian metrics on F, π∗E and OY (1) induced bythe Hermitian metric of E. We set L = OY (r) ⊗π∗(det E)−1.

Here we considerdegbc1(OY (1)) −1r π∗(bc1(E))r+1.We setΦ =Xσ∈S∞rXi=1(−1)i˜ci(Eσ)c1(OYσ(1))r−i.Then by (1.9.1)bc1(OY (1))r −π∗(bc1(E))bc1(OY (1))r−1 + π∗(bc2(E))bc1(OY (1))r−2 = a(Φ).

41Thus we havebc1(OY (1)) −1r π∗(bc1(E))r+1=r −12r π∗(bc1(E)2)bc1(OY (1))r−1−π∗(bc2(E))bc1(OY (1))r−1+bc1(OY (1)) −1r π∗(bc1(E))a(Φ).Sinceπ∗(π∗(bc1(E)2)bc1(OY (1))r−1) = bc1(E)2andπ∗(π∗(bc2(E))bc1(OY (1))r−1) = bc2(E)by projection formula, we haveπ∗bc1(OY (1)) −1r π∗(bc1(E))r+1= r −12r (bc1(E)2) −(bc2(E))+π∗bc1(OY (1)) −1r π∗(bc1(E))a(Φ).On the other hand, sincedegbc1(OY (1)) −1r π∗(bc1(E))a(Φ)is equal toXσ∈S∞ZYσ(c1(OYσ(1)) −1r π∗(c1(Eσ))rXi=1(−1)i˜ci(Eσ)c1(OYσ(1))r−i),Lemma 5.1 impliesdegbc1(OY (1)) −1r π∗(bc1(E))a(Φ)= 0.Therefore, we getdegbc1(OY (1)) −1r π∗(bc1(E))r+1= r −12r (bc1(E)2) −(bc2(E)).On the the other hand,(Lr+1) = rr+1 degbc1(OY (1)) −1r π∗(bc1(E))r+1.Thus we have(Lr+1) ≤0 =⇒r −12r bc1(E, h)2 ≤bc2(E, h).Hence it is sufficient to show that (Lr+1) ≤0.

428.6 Step 4. Let (N, h) be a Hermitian line bundle on X such that N is ample, deg(NK) >2g(XK) −2 and that L ⊗π∗N is ample.

By the arithmetic Grothendieck-Riemann-Rochtheorem (1.11.1), we haveχL2(Ln ⊗π∗N) + 12Xσ∈S∞τ((Ln ⊗π∗N)σ) = deg( bch(Ln ⊗π∗N) · btdA(TY/S)).Clearlylimn→∞deg( bch(Ln ⊗π∗N) · btdA(TY/S))nr+1=1(r + 1)! (Lr+1).Thus it is sufficient to show that(a)Xσ∈S∞τ((Ln ⊗π∗N)σ) ≤O(nr log n).

(b) χL2(Ln ⊗π∗N) ≤O(nr log n).8.7 Step 5. If we denote by Hσ the Hermitian form of Lσ for all σ ∈S∞, Hσ(y) ispositive semi-definite and rk Hσ(y) ≥r −1 for all y ∈Yσ as in p.89-p.90 of [Ko].

Thusby Corollary 2.4 we have (a).8.8 Step 6. Since deg(N) > 2g(X)−2 and Symrn(E) is semistable on the generic fiber,H1(Y, Ln ⊗π∗N) is torsion module.

Thus it is sufficient to see thatdegL2(H0(Ln ⊗π∗N)) ≤O(nr log n).Since L is nef and (Lr) = 0 on the generic fiber, we haverk H0(Y, Ln ⊗π∗N) ≤O(nr−1).Hence Theorem 4.1 implies thatdegL2(H0(Y, Ln ⊗π∗N)) ≤O(nr log n).Thus we get (b), which completes the proof of the main theorem.□Corollary 8.9. Let K be an algebraic number field, OK the ring of integers of K, andf : X −→Spec(OK) an arithmetic surface.Let E be a torsion free sheaf and h aHermitian metric on X.

If X is regular and EQ is semistable on the geometric genericfiber XQ of f, then we have an inequalitybc2(E, h) −r −12r bc1(E, h)2 ≥0,where r = rk E.Proof. Let E∨∨be the double dual of E. Since X is a regular scheme of dimension 2,E∨∨is locally free and T = E∨∨/E is a finite module.

Moreover, bc1(E, h) = bc1(E∨∨, h)and bc2(E, h) = bc2(E∨∨, h) + log #(T). Thus we have our corollary.□

439. Arithmetic second Chern character of semistable vector bundlesLet K be an algebraic number field and OK the ring of integers.Let f : X −→Spec(OK) be a regular arithmetic surface and E a torsion free sheaf on X.

We denoteby Herm(E) the set of all Hermitian metrics of E. We setch2(E) =suph∈Herm(E)bch2(E, h)(∈(−∞, ∞]).First of all, we will see several properties of ch2(E) when E is semistable on the geometricgeneric fiber.Proposition 9.1. With being notation as above, we assume that EQ is semistable anddeg(EQ) = 0.

Then, we have the following. (1) bch2(E, h) ≤0 for every h ∈Herm(E).

Moreover, the equality holds if and only ifbc1(E, h)2 = bc2(E, h) = 0. (2) ch2(E) ≤0.

(3) For h ∈Herm(E), ch2(E) = bch2(E, h) if and only if h is Einstein-Hermitian. (4) If rk E = 1, then ch2(E) ≤−[K : Q] Height([E|K]), where Height is the N´eron-Tate height function on Pic0(Y ).

Moreover, the equality holds if and only if E islocally free and deg(E|C) = 0 for all vertical curves C on X. (5) If there is a filtration of E: 0 = E0 ⊂E1 ⊂· · · ⊂El−1 ⊂El = E such thatEi/Ei−1 is torsion free and deg((Ei/Ei−1)|K) = 0 for every i, then we havech2(E) =lXi=1ch2(Ei/Ei−1).

(6) We assume that E is locally free. Let K′ be a finite extension field of K, X′ adesingularization of X ×OK OK′, and g : X′ −→X the induced morphism.

Then,we havech2(g∗(E)) = [K′ : K]ch2(E).Proof. (1) First, we will see that bc1(E, h)2 ≤0.

We set(E′, h′) = (det(E), det(h)) ⊕(OX, canK).Then, since E′ is semistable, by Corollary 8.9, we getbc1(E, h)2 = bc1(E′, h′)2 ≤4bc2(E′, h′) = 0.Hence, by virtue of Corollary 8.9,bch2(E, h) = 12bc1(E, h)2 −bc2(E, h)≤12rbc1(E, h)2 −bc2(E, h) −r −12r bc1(E, h)2≤0.

44(2) This is a direct consequence of (1). (3) We fix a Hermitian metric k of E. Then, since deg(LK) = 0, by Lemma 6.1,bch2(E, h) −bch2(E, k) = −12πXσ∈K∞DL(Eσ, hσ, kσ; Φσ).Thus, bch2(E, h) = ch2(E) if and only if DL(Eσ, hσ, kσ; Φσ) gives the absolute minimalvalue for every σ ∈K∞.

Thus, we obtain (3). (4) Let h be an Einstein-Hermitian metric of E. By (3), we have ch2(E) = bch2(E, h).Let E∨∨be the double dual of E. Then, bch2(E, h) ≤bch2(E∨∨) and the equality holdsif and only if E = E∨∨.

So we may assume that E is locally free. Here, we need thefollowing lemma.Lemma 9.2.

Let (L, h) a Hermitian line bundle on X. If h is Einstein-Hermitian anddeg(L|C) = 0 for all vertical curves C on X, thenbch2(L, h) = −[K : Q] Height([LK]).Proof.

For example, see [Fa1] and [Hr].□Let {q1, . .

. , ql} be the set of all critical values of f. Let Supp(Xqt) = C(t)1 +· · ·+C(t)etbe the irreducible decomposition of the fiber Xqt.

Then, since deg(E|K) = 0, we canfind rational numbers {a(t)i } such that(E · C(t)j ) + (etXi=1a(t)i C(t)i· C(t)j ) = 0for all t and j. Let n be a positive integer such that n · a(t)i∈Z for all t and i.

We setL = En ⊗OX(lXt=1etXi=1na(t)i C(t)i ).Then, due to Lemma 9.2, we havebch2(L, hn) = −[K : Q] Height([LK]) = −n2[K : Q] Height([EK]).On the other hand, since deg(L|C) = 0 for all vertical curves C on X, we havebch2(L, hn) + n22 (lXt=1etXi=1a(t)i C(t)i )2 = n2 bch2(E, h).

45Let Plt=1Peti=1 a(t)i C(t)i= D+ −D−be the decomposition of Plt=1Peti=1 a(t)i C(t)isuchthat D+ and D−are effective and that Supp(D+) and Supp(D−) have no commoncomponent. Then,(lXt=1etXi=1a(t)i C(t)i )2 = (D2+) + (D2−) −2(D+ · D−) ≤0.Moreover, the equality holds if and only if(D+ · C) = (D−· C) = 0for all vertical curves C on X.

Thus we have (4). (5) Let h be a Hermitian metric of E. Let hi be the sub-metric of Ei induced by hand ki the quotient metric of Ei/Ei−1 induced by hi.

Then, by Corollary 7.4, we havebch2(E, h) ≤bch2(E1/E0, k1) + · · · + bch2(El/El−1, kl)≤ch2(E1/E0) + · · · + ch2(El/El−1)Therefore, we havech2(E) ≤lXi=1ch2(Ei/Ei−1).In order to consider the converse inequality of the above, we need the following lemma.Lemma 9.3. Let 0 →S →E →Q →0 an exact sequence of torsion free sheaves on X.Let h′ and h′′ be Hermitian metrics of S and Q respectively.

If deg(SK) = deg(EK) =deg(QK) = 0, then there is a family {ht}t∈R of Hermitian metrics of E withlimt→∞bch2(E, ht) = bch2((S, h′) ⊕(Q, h′′)).Proof. For σ ∈K∞, let Eσ = Sσ ⊕Pσ be a decomposition of Eσ as C∞-vector bundles.Then, since Pσ is isomorphic to Qσ, using the above decomposition, we define a Hermitianmetric ht of E by et · h′ ⊕h′′.

Let At be the second fundamental form of0 →(S, et · h′) →(E, ht) →(Q, h′′) →0.If we denote A0 by A, then it is easy to see that At = A and (At)∗= e−tA∗. Thus, byProposition 7.3, we havebch2(E, ht) −bch2((S, et · h′) ⊕(Q, h′′)) = −e−t2πXσ∈K∞||Aσ||2.

46On the other hand, since deg(SK) = 0, by an easy calculation, we obtain bch2(S, et · h′) =bch2(S, h′). Thus, we get our lemma.□Let us start to prove the inequalitych2(E) ≥lXi=1ch2(Ei/Ei−1).by induction on l. Let h′ and h′′ be arbitrary Hermitian metrics of El−1 and Ql respec-tively.

Then, by Lemma 9.3, there is a family {ht}t∈R of Hermitian metric of E suchthatlimt→∞bch2(E, ht) = bch2(El−1, h′) + bch2(Ql, h′′).Therefore, we havech2(E) ≥bch2(El−1, h′) + bch2(Ql, h′′)Since h′ and h′′ are arbitrary, it follows thatch2(E) ≥ch2(El−1) + ch2(Ql)Thus, by hypothesis of induction, we have (5). (6) Let k be a fixed Hermitian metric of E. For each σ ∈K∞, we setDL(Eσ) = infhσ DL(Eσ, hσ, kσ; Φσ),where hσ runs over all Hermitian metrics of Eσ.

Let {hn} be a sequence of Hermitianmetrics of E withlimn→∞bch2(E, hn) = ch2(E).Sincebch2(E, hn) −bch2(E, k) = −12πXσ∈K∞DL(Eσ, (hn)σ, kσ; Φσ),for each σ ∈K∞, we haveDL(Eσ) = limn→∞DL(Eσ, (hn)σ, kσ; Φσ).Therefore, we obtainDL(g∗(E)σ′) = limn→∞DL(E σ′|K, (hn) σ′|K, k σ′|K; Φ σ′|K)for all σ′ ∈K′∞because DL(g∗(E)σ′) = DL(E σ′|K). It follows thatch2(g∗(E)) = limn→∞bch2(g∗(E, hn))= limn→∞[K′ : K] bch2(E, hn)= [K′ : K]ch2(E)Thus we have (6).□

47Corollary 9.4. Let f : X −→Spec(OK) be a regular arithmetic surface and E a torsionfree sheaf on X such EQ is semistable and deg(EK) = 0.

If there is a filtration of E:0 = E0 ⊂E1 ⊂· · · ⊂Er−1 ⊂Er = E such that Ei/Ei−1 is torsion free of rank 1 anddeg((Ei/Ei−1)|K) = 0 for every i, then we havech2(E) ≤−[K : Q]rXi=1Height((Ei/Ei−1)K).Moreover, the equality holds if and only if Ei/Ei−1 is locally free and deg((Ei/Ei−1)|C) =0 for all vertical curves C on X.Proof. By (5) of Proposition 9.1, we havech2(E) =rXi=1ch2(Ei/Ei−1).Moreover, due to (4) of Proposition 9.1, for every i,ch2(Ei/Ei−1) ≤−[K : Q] Height((Ei/Ei−1)K),and the equality holds if and only if Ei/Ei−1 is locally free and deg((Ei/Ei−1)|C) = 0for all vertical curves C on X.

Thus we have our corollary.□Definition 9.5. Let Y be a smooth algebraic curves over an algebraic number field K.Let M Y/K(r, d) be the moduli scheme of semistable vector bundles on Y with rank rand degree d (cf.

[Ma2]).We would like to consider M Y/K(r, 0) when Y is an elliptic curve. For this purpose,we need the following lemma.Lemma 9.6.

Let C be a smooth projective curve of genus 1 over an algebraically closedfield and E a rank r semistable vector bundle on X with deg(E) = 0. Then, there is afiltration of E:0 = E0 ⊂E1 ⊂· · · ⊂Er−1 ⊂Er = Esuch that Ei/Ei−1 is a locally free of rank 1 and deg(Ei/Ei−1) = 0.Proof.

We prove this proposition by induction on rk E. Let Q be a minimal quotient linebundle of E and S the kernel of E →Q. Then, by [MS],deg(Q) −deg(S)r −1 = deg(Q) ·rr −1 ≤1.Since deg(Q) is non-negative integer, we have deg(Q) = 0.

Thus S is semistable anddeg(S) = 0. Therefore, S has a desired filtration by hypothesis of induction.

Hence, weobtain our lemma.□

48The above lemma shows us thatM Y/K(r, 0)(Q) =r timesz}|{Pic0(Y )(Q) × · · · × Pic0(Y )(Q) /Sr,where Sr is the rth symmetric group. Thus, M Y/K(r, 0)(Q) has the canonical heightfunction by using N´eron-Tate height, that is,Height(e) := Height(l1) + · · · + Height(lr)for an element e = (l1, · · · , lr) of M Y/K(r, 0)(Q).

In terms of this height function Heightof M Y/K(r, 0), we haveCorollary 9.7. Let f : X →Spec(OK) be a regular arithmetic surface with the genusof the generic fiber being one, and E a torsion free sheaf of rank r on X.

Then, we havech2(E) ≤−[K : Q] Height([EK]),where [EK] is the class of EK in M XK/K(r, 0). Moreover, the equality holds if and onlyif there is a finite field extension K′ of K with the following properties:(a) Let X′ be a desingularization of X ×OK OK′ and g : X′ →X the induced mor-phism.

(b) There is a filtration of g∗(E) : 0 = E′0 ⊂E′1 ⊂· · · ⊂E′r−1 ⊂E′r = g∗(E) suchthat E′i/E′i−1 is locally free of rank 1 and deg((E′i/E′i−1)K) = 0 for every i. (c) deg((E′i/E′i−1)C) = 0 for all vertical curves of X′.Proof.

By (6) of Proposition 9.1 and Lemma 9.6, we may assume that EK has a filtration0 = F0 ⊂F1 ⊂· · · ⊂Fr−1 ⊂Fr = EK such that Fi/Fi−1 is locally free of rank 1 anddeg(Fi/Fi−1) = 0 for every i. Therefore, there is a filtration of E : 0 = E0 ⊂E1 ⊂· · · ⊂Er−1 ⊂Er = E such that Ei/Ei−1 is torsion free and (Ei)K = Fi.

Thus byCorollary 9.4, we have our corollary.□Generalizing the above corollary, we would like to pose the following question.Question 9.8. Let f : X →Spec(OK) be a regular arithmetic surface and E asemistable vector bundle on X.

We have two questions. (1) Is there a canonical height function on M XK/K(r, 0)?

(2) If it exists, is there a relation between ch2(E) and the canonical height function?

4910. Torsion vector bundlesIn this section, we consider conditions for the equality of bch2(E, h) ≤0.

First of all,let us consider the case where E is a line bundle.Proposition 10.1. Let K be an algebraic number field and OK the ring of integers.

Letf : X −→Spec(OK) be a regular arithmetic surface and (L, h) a Hermitian line bundleon X with deg(LQ) = 0. Then, bc1(L, h)2 = 0 if and only ifi) h is Einstein-Hermitian,ii) deg(L|C) = 0 for all vertical curves C on X, andiii) LK is a torsion point of Pic0(XK).Proof.

First, we assume the conditions i), ii) and iii). By Lemma 9.2, we getbc1(L, h)2 = −2[K : Q] Height([LK]) = 0.Next we assume that bc1(L, h)2 = 0.

By (3) of Proposition 9.1, h is Einstein-Hermitian.Moreover, by virtue of (4) of Proposition 9.1, we have deg(L|C) = 0 for all vertical curvesC of f. Hence, by Lemma 9.2, we getbc1(L, h)2 = −2[K : Q] Height([LK]),which implies Height([LK]) = 0. Therefore, LK is a torsion of Pic0(XK).□Definition 10.2.

Let M be a complex manifold and E a rank r flat vector bundle onM. The vector bundle E defines a representationρ : π1(M) →GLr(C)of the fundamental group π1(M) of M. E is said to be of torsion type if the image of ρis a finite group.Lemma 10.3.

Let X be a smooth algebraic variety over C and E a flat vector bundle onX. Then, E is of torsion type if and only if there is a dominant morphism of algebraicvarieties f : Y →X over C such that the composition of homomorphisms:π1(Y ) →π1(X) →GLr(C)is trivial.Proof.

First we assume that E is of torsion type. Then, since Ker(ρ) has a finite indexin π1(X), there is a finite etale covering f : Z →X of algebraic varieties such thatπ1(Z) = Ker(ρ).

Thus we have the first assertion.Next we assume that there is a dominant morphism of algebraic varieties f : Y →Xover C such that the composition of homomorphisms π1(Y ) →π1(X) →GLr(C) istrivial. By Proposition 2.9.1 of [Kol], the image of π1(Y ) →π1(X) has a finite index inπ1(X).

Thus the image of ρ is a finite group.□

50Proposition 10.4. Let X be a smooth projective curve over an algebraically closed fieldk with k ⊂C.

Let E be a rank r vector bundle on X such that E is semistable anddeg(E) = 0. Then, the following are equivalent.

(a) There is a finite etale covering f : Z →X of X over k with f ∗(E) ≃O⊕rZ . (b) There is a surjective morphism g : Y →X of smooth projective varieties over kwith g∗(E) ≃O⊕rY .

(c) There is a finite etale covering f ′ : Z′ →XC of XC over C such that f ′∗(EC) ≃O⊕rZ′ . (d) There is a surjective morphism g′ : Y ′ →XC of smooth projective varieties overC with g′∗(EC) ≃O⊕rY ′ .

(e) EC is flat and of torsion type.Proof. First of all, we prepare the following lemma.Lemma 10.5.

Let X be a smooth projective curve over an algebraically closed field ofcharacteristic zero and E a rank r vector bundle on X. If E is semistable, deg(E) = 0and π∗(E) ≃O⊕rYfor some finite covering π : Y →X, then E is poly-stable.Proof.

We prove this lemma by induction on rk E. Clearly we may assume that E is notstable. Moreover, we may assume that π : Y →X is a Galois covering.

Since E is notstable, there is an exact sequence:0 →F →E →Q →0such that F and Q are locally free, deg(F) = deg(Q) = 0 and that F and Q aresemistable. Then, it is easy to see that the exact sequence:0 →π∗(F) →π∗(E) →π∗(Q) →0splits, π∗(F) ≃O⊕sYand that π∗(Q) ≃O⊕tY , where s = rk F and t = rk Q.

Hence F andQ are poly-stable by hypothesis of induction. Therefore, it is sufficient to see that thenatural homomorphismExt1X(Q, F) −→Ext1Y (π∗(Q), π∗(E))is injective because the injectivity of the above homomorphism implies that the exactsequence0 →F →E →Q →0splits.

Since π is the Galois covering, we have a trace map π∗(OY ) →OX, which showsus that an exact sequence:0 →OX →π∗(OY ) →π∗(OY )/OX →0

51splits.Therefore, H1(X, F ⊗Q∨) −→H1(X, F ⊗Q∨⊗π∗(OY )) is injective.ThusExt1X(Q, F) −→Ext1Y (π∗(Q), π∗(E)) is injective.□Let us start the proof of Proposition 10.4. (a) =⇒(b), (b) =⇒(d) and (e) =⇒(c)are trivial.

So it is sufficient to show (d) =⇒(e) and (c) =⇒(a). (d) =⇒(e) : By Lemma 10.5, EC is poly-stable.

Thus EC is a flat vector bundle. SoEC comes from a representationρ : π1(XC) →GLr(C)of the fundamental group π1(XC) of XC.

Since g′∗(EC) is also flat, it has an EinsteinHermitian metric h.On the other hand, since g′∗(EC) ≃O⊕rY ′ , by [Ko, Chap.V,Proposition 8.2], (g′∗(EC), h) is isometric to (OY ′, h1) ⊕· · · ⊕(OY ′, hr) as Einstein-Hermitian vector bundles, which shows us that the composition of homomorphisms:π1(Y ′) →π1(XC) →GLr(C)is trivial. Thus, by Lemma 10.3, EC is of torsion type.

(c) =⇒(a) : Clearly we can find a field F such that k ⊂F ⊂C, F is finitelygenerated over k and that f ′ : Z′ →XC and f ′∗(EC) ≃O⊕rZ′ are defined over F. Thusthere are algebraic varieties T and eZ over k and projective morphisms h : eZ →T and˜f : eZ →X × T over k with the following properties. (1) The function field of T is F.(2) If p : X × T →T and q : X × T →X are the natural projections, then we havep · ˜f = h.(3) ˜f ×T Spec(C) : eZ ×T Spec(C) →(X × T) ×T Spec(C) is nothing more thatf ′ : Z′ →XC.eZ˜f−−−−→X × ThyypTTAt the generic point η of T, we have ˜fη is etale and ( ˜f ∗(q∗(E)))η ≃O⊕reZη .

Hence there isa non-empty open set U of T such that ˜ft is etale and ( ˜f ∗(q∗(E)))t ≃O⊕reZt for all t ∈U.Therefore, if we choose a closed point t0 of U, we have (a).□Definition 10.6. Let K be an algebraic number field and OK the ring of integers.

Letf : X −→Spec(OK) be a regular arithmetic surface and E a vector bundle on X. Eis said to be of torsion type if EQ satisfies one of equivalent conditions (a) – (d) inProposition 10.4.

52Question 10.7. Let K be an algebraic number field and OK the ring of integers.

Letf : X −→Spec(OK) be a regular arithmetic surface and (E, h) a Hermitian vectorbundle on X such that deg(EK) = 0 and EQ is semistable. Then, if bch2(E, h) = 0, is Eof torsion type?For this question, we have the following partial answer.Proposition 10.8.

Let K be an algebraic number field and OK the ring of integers.Let f : X −→Spec(OK) be a regular arithmetic surface and (E, h) a rank r Hermitianvector bundle on X such that deg(EK) = 0 and EQ is semistable. Assume that there isa filtration of EQ:0 = E0 ⊂E1 ⊂E2 ⊂· · · ⊂Er−1 ⊂Er = EQsuch that Ei/Ei−1 is a locally free sheaf of rank 1 and deg(Ei/Ei−1) = 0 for every1 ≤i ≤r.

Then, if bch2(E, h) = 0, E is of torsion type.Proof. We may assume that the filtration is defined over K. Hence we can construct afiltration of E:0 = F0 ⊂F1 ⊂F2 ⊂· · · ⊂Fr−1 ⊂Fr = Esuch that Fi|Q = Ei, Fi is locally free and that Fi/Fi−1 is torsion free.

Let hFi be theinduced sub-metric of h and hi the quotient metric of Fi/Fi−1 induced by hFi. Let Libe the double dual of Fi/Fi−1.

Then, by Corollary 7.4, we have0 = bch2(E, h) ≤bch2((F1/F0, h1) ⊕· · · ⊕(Fr/Fr−1, hr))≤bch2((L1, h1) ⊕· · · ⊕(Lr, hr)) ≤0.Therefore, we getbch2((L1, h1) ⊕· · · ⊕(Lr, hr)) = 12(bc1(L1, h1)2 + · · · + bc1(Lr, hr)2) = 0and (E, h) is isometric to (L1, h1)⊕· · ·⊕(Lr, hr) on each infinite fiber. Since bc1(Li, hi)2 ≤0 for each i, we obtain bc1(Li, hi)2 = 0.

Therefore, by Proposition 10.1, (Li)C is of torsiontype. Therefore EC is torsion type because EC = (L1)C ⊕· · · ⊕(Lr)C.□

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133 (1991), 509–548.Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-01,JapanCurrent address: Department of Mathematics, University of California, Los Angeles, 405 HilgardAvenue, Los Angeles, California 90024, USA

54E-mail address: moriwaki@math.ucla.edu

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