A. Moriwaki proved the following arithmetic analogue of the Bogomolov
unstability theorem. If a torsion-free hermitian coherent sheaf on an
arithmetic surface has negative discriminant then it admits an arithmetically
destabilising subsheaf. In the geometric situation it is known that such a
subsheaf can be found subject to an additional numerical constraint and here we
prove the arithmetic analogue. We then apply this result to slightly simplify a
part of C. Soul\'e's proof of a vanishing theorem on arithmetic surfaces.Comment: final version, to appear in Math. Res. Let

Naumann, Niko

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A quantitative sharpening of Moriwaki’s arithmetic Bogomolov inequality

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Universita¨t RegensburgMathematikA quantitative sharpening of Moriwaki’sarithmetic Bogomolov inequalityNiko NaumannPreprint Nr. 10/2005A quantitative sharpening of Moriwaki’s arithmeticBogomolov inequalityN. NaumannAbstractA. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability the-orem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negativediscriminant then it admits an arithmetically destabilising subsheaf. In the geometric sit-uation it is known that such a subsheaf can be found subject to an additional numericalconstraint and here we prove the arithmetic analogue. We then apply this result to slightlysimplify a part of C. Soule´’s proof of a vanishing theorem on arithmetic surfaces.1. Introduction and statement of resultLet K be a number field with ring of integers OK and X/Spec (OK) an arithmetic surface, i.e. aregular, integral, purely two-dimensional scheme, proper and flat over Spec (OK) and with smoothand geometrically connected generic fibre. Attached to a hermitian coherent sheaf on X are theusual characteristic classes with values in the arithmetic Chow-groups ĈHi(X) (cf. [GS1], 2.5), andin particular the discriminant of E∆(E) := (1− r)cˆ1(E)2 + 2rcˆ2(E) ∈ ĈH2(X)where r := rk(E). The arithmetic degree mapd̂eg : ĈH2(X)R −→ Ris an isomorphism [GS2] and we will use the same symbol to to denote an element in ĈH2(X)Rand its arithmetic degree in R, see [GS2], 1.1 for the definition of arithmetic Chow-groups with realcoefficients ĈH∗(X)R. Following [Mo2] we define the positive cone of X to beCˆ++(X) := {x ∈ ĈH1(X)R |x2 > 0 and degK(x) > 0} .Given a torsion-free hermitian coherent sheaf E of rank r > 1 on X and a subsheaf E′ ⊆ E weendow E′ with the metric induced from E and consider the difference of slopesξE′,E:=cˆ1(E′)rk(E′)− cˆ1(E)r∈ ĈH1(X)R.Recall that a subsheaf E′ ⊆ E is saturated if the quotient E/E′ is torsion-free. Our main result isthe following.2000 Mathematics Subject Classification 14G40Keywords: Bogomolov inequality, successive minimaN. NaumannTheorem 1. Let E be a torsion-free hermitian coherent sheaf of rank r > 2 on the arithmeticsurface X, satisfying∆(E) < 0 .Then there is a non-zero saturated subsheaf E′ ⊆ E such that ξE′,E∈ Cˆ++(X) and(1) ξ2E′,E> −∆r2(r − 1) .Remark 2. The existence of an E′ ⊆ E with ξE′,E∈ Cˆ++(X) is the main result of [Mo2] and meansthat E′ ⊆ E is arithmetically destabilising with respect to any polarisation of X, c.f. loc. cit. formore details on this. The new contribution here is the inequality (1) which is the exact arithmeticanalogue of a known geometric result, c.f. for example [HL], Theorem 7.3.4.Remark 3. A special case of Theorem 1 appears in disguised form in the proof of [So], Theorem2: Given a sufficiently positive hermitian line bundle L on the arithmetic surface X and somenon-torsion element e ∈ H1(X,L−1) ' Ext1(L,OX), C. Soule´ establishes a lower bound for||e||2 := sup σ:K↪→C ||σ(e)||2L2by considering the extension determined by eE : 0 −→ OX −→ E −→ L −→ 0and suitably metrised as to have cˆ1(E) = L and 2cˆ2(E) =∑σ ||σ(e)||2L2 , hence ∆(E) = −L2 +2∑σ ||σ(e)||2L2 (where we write L = cˆ1(L) following the notation of loc. cit.).If EQ is semi-stable the arithmetic Bogomolov inequality concludes the proof. Otherwise, the mainpoint is to show the existence of of an arithmetic divisor D satisfyingdegK(D) 6 degK(L)/2 and(2)2(L−D)D 6 [K : Q] · ||e||2,(3)c.f. (28) and (32) of loc. cit. where these inequalities are established by some direct argument. Wewish to point out that the existence of some D satisfying (2) and (3) is a special case of Theorem1. In fact, let E′ ⊆ E be as in Theorem 1 and define D := L− cˆ1(E′). We then computeξE′,E=L2−Dand ξE′,E∈ Cˆ++(X) implies (2). Furthermore, the inequality (1) in the present case readsξ2E′,E=L24+D2 − L D > −∆4=L24− 12∑σ||σ(e)||2L2 , i.e.2(L−D)D 6∑σ||σ(e)||2L2 ,hence the trivial estimate [K : Q] · ||e||2 >∑σ ||σ(e)||2L2 gives (3).2A quantitative sharpening of Moriwaki’s arithmetic Bogomolov inequalityAcknowledgementsI would like to thank K. Ku¨nnemann for useful conversations about a preliminary draft of thepresent note.2. Proof of Theorem 1We collect some lemmas first. We call a short exact sequenceE : 0 −→ E′ −→ E −→ E′′ −→ 0of hermitian coherent sheaves on X isometric if the metrics on E′ and E′′ are induced from the oneon E. This implies that cˆ1(E) = cˆ1(E′) + cˆ1(E′′) (i.e. c˜1(E) = 0). We also havecˆ2(E) = cˆ2(E′ ⊕ E′′)− a(c˜2(E)) in ĈH2(X) ,wherea : A˜1,1(XR) −→ ĈH2(X)is the usual map [SABK], chapter III.Lemma 4. IfE : 0 −→ E′ −→ E −→ E′′ −→ 0is an isometric short exact sequence of hermitian coherent sheaves on X with ranks r′, r, r′′ > 1 anddiscriminants ∆′,∆,∆′′, then∆′r′+∆′′r′′− ∆r=rr′r′′ξ2E′,E+ 2a(c˜2(E)) in ĈH2(X)R .Proof. We omit the computation using the formulas for cˆi(E) recalled above which shows that theleft hand side of the stated equality equalscˆ1(E)2(r − 1r+1− r′r′)+ cˆ1(E′′)2(r − 1r+1− r′′r′′)++cˆ1(E′)cˆ1(E′′)(2(r − 1)r− 2)+ 2a(c˜(E)).Similarly one writes ξ2E′,Eas a rational linear combination of cˆ1(E)2, cˆ1(E′′)2 and cˆ1(E′)cˆ1(E′′) andcomparing the results, the stated formula drops out.Lemma 5. For E as in Lemma 4 and G′′ ⊆ E′′ a saturated subsheaf of rank s > 1 carrying theinduced metric, putG := ker(E −→ E′′ −→ E′′/G′′) ⊆ Ewith the induced metric. ThenξG,E =r′(r′′ − s)(r′ + s)r′′ξE′,E+sr′ + sξG′′,E′′ in ĈH1(X)R .Observe that the coefficients in the last expression are non-negative rational numbers.3N. NaumannProof. We have a commutative diagram with exact rows and columns0 0E/GOO' // E′′/G′′OOE : 0 // E′ // EOO// E′′ //OO00 // H'OO// G //OOG′′OO// 00OO0.OOHere, we have endowed E/G,E′′/G′′ and H with the metrics induced from E,E′′ and G, henceall rows and columns are isometric by definition. A minor point to note is that with this choiceof metrics the two indicated isomorphisms are isometric, indeed this only means that taking sub-(resp. quotient-)metrics is transitive. One hasξE′,E=r′′cˆ1(E′)− r′cˆ1(E′′)r′rand analogously for any isometric exact sequence in place of E . Using this and the diagram onewrites both sides of the stated equality as a Q-linear combination of cˆ1(E′), cˆ1(G′′) and cˆ1(E′′/G′′)to obtain the same result, namelyr′′ − s(r′ + s)rcˆ1(E′) +r′′ − s(r′ + s)rcˆ1(G′′)− 1rcˆ1(E′′/G′′).Finally, we will need the following observation about the intersection theory on X where, for x ∈Cˆ++(X), we write |x| := (x2)1/2 ∈ R+.Lemma 6. The subset Cˆ++(X) ⊆ ĈH1(X)R is an open cone, i.e. x, y ∈ Cˆ++(X) and λ ∈ R+ impliesthat x+ y, λx ∈ Cˆ++(X). For x, y ∈ Cˆ++ we have |x+ y| > |x|+ |y|.Proof. This is [Mo2], (1.1.2.2) except for the final assertion which is obvious if x ∈ Ry and we canthus assume that V := Rx + Ry ⊆ ĈH1(X)R is two-dimensional. We claim that the restrictionof the intersection-pairing makes V a real quadratic space of type (1,−1). As we have x ∈ V andx2 > 0 we only have to exhibit some v ∈ V with v2 < 0. To achieve this let h ∈ ĈH1(X)R be thefirst arithmetic Chern class of some sufficiently positive hermitian line bundle on X such that thearithmetic Hodge index theorem holds for the Lefschetz operator defined by h, c.f. [GS2], Theorem2.1, ii). Then a := xh (resp. b := yh) are non-zero real numbers for otherwise we would have x2 < 0(resp. y2 < 0). Thus v := xa − yb ∈ V satisfies v 6= 0 and vh = 0 , hence v2 < 0.4A quantitative sharpening of Moriwaki’s arithmetic Bogomolov inequalityFix a basis e, f ∈ V with e2 = 1, f2 = −1 and writex = αe+ βf andy = γe+ δf.To show that |x+y| > |x|+ |y| we can assume, changing both the signs of x and y if necessary, thatα > 0. We then claim that γ > 0. For otherwise there would be λ1, λ2 ∈ R+ such that v := λ1x+λ2ywould have e- coordinate equal to zero, hence v2 6 0 contradicting the fact that either −v or v liesin Cˆ++(X) (depending on whether or not we changed the signs of x and y above).From x2 = α2 − β2, y2 = γ2 − δ2 > 0 we obtain α = |α| > |β| and γ = |γ| > |δ| and thenαγ > |βδ| > βδ, i.e.(4) xy = αγ − βδ > 0.To conclude, we use the following chain of equivalent statements|x+ y| > |x|+ |y| ⇔(x+ y)2 − (|x|+ |y|)2 > 0⇔2xy − 2|x||y| > 0⇔xy > |x||y| (4)⇔(xy)2 > |x|2|y|2 ⇔(αγ − βδ)2 > (α2 − β2)(γ2 − δ2)⇔α2γ2 + β2δ2 − 2αβγδ > α2γ2 − α2δ2 − β2γ2 + β2δ2 ⇔2αβγδ 6 α2δ2 + β2γ2 ⇔0 6 (αδ − βγ)2.Proof of Theorem 1. We first remark that for a torsion-free hermitian coherent sheaf F of rankone on X we always have ∆(F ) > 0. In fact,F ' L⊗ IZfor some line-bundle L and IZ the ideal sheaf of some closed subscheme Z ⊆ X of codimension 2.This becomes an isometry for the trivial metric on IZ and a suitable metric on L (since IZ is trivialon the generic fibre of X). Then∆(F ) = 2cˆ2(L ⊗ IZ) = 2cˆ2(IZ) = 2 length(Z) > 0 .By the main result of [Mo2], there is 0 6= E′ ⊆ E saturated such that ξE′,E∈ Cˆ++(X). We canassume that, as E′ varies through these subsheaves, the real numbers ξ2E′,Eremain bounded forotherwise there is nothing to prove. So we can choose 0 6= E′ ⊆ E saturated with ξE′,E∈ Cˆ++(X)and ξ2E′,Emaximal subject to these conditions. Put E′′ := E/E′ and consider the isometric exact5N. NaumannsequenceE : 0 −→ E′ −→ E −→ E′′ −→ 0with discriminants ∆′,∆,∆′′ and ranks r′, r, r′′. We claim that ∆′ > 0. This is clear in case r = 2from the remark made at the beginning of the proof. In case r > 3 we assume that ∆′ < 0 and welet G ⊆ E′ be a saturated subsheaf with ξG,E′ ∈ Cˆ++. Then G ⊆ E is saturated and using lemma6 we get|ξG,E | = |ξG,E′ + ξE′,E | > |ξG,E′ |+ |ξE′,E | > |ξE′,E |contradicting the maximality of |ξE′,E|. So we have indeed ∆′ > 0. Assume now, contrary to ourassertion, that(5)∆r< −r(r − 1)ξ2E′,E.Then from Lemma 4, ∆′ > 0, (5) and c˜2(E) 6 0 ([Mo1], 7.2) we get∆′′r′′6 ∆r+rr′r′′ξ2E′,E<(−r(r − 1) + rr′r′′)ξ2E′,E= −r2 r′′ − 1r′′ξ2E′,E6 0 ,hence ∆′′ < 0. By induction, there is 0 6= G′′ ⊆ E′′ saturated with ξG′′,E′′ ∈ Cˆ++(X) and(6) ξ2G′′,E′′ > −∆′′r′′2(r′′ − 1) >r2r′′2ξ2E′,E.Clearly G := ker(E → E′′/G′′) ⊆ E is saturated and from Lemma 5, the positivity of the coefficientsappearing there and lemma 6 we get|ξG,E | >r′(r′′ − s)(r′ + s)r′′|ξE′,E|+ sr′ + s|ξG′′,E′′ |(6)>r′(r′′ − s)(r′ + s)r′′|ξE′,E|+ sr′ + srr′′|ξE′,E|=(r′(r′′ − s) + rsr′′(r′ + s))|ξE′,E| = |ξE′,E| .This again contradicts the maximality of |ξE′,E| and concludes the proof. 2ReferencesGS1 H. Gillet, C. Soule´, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), no. 3, 473–543.GS2 H. Gillet, C. Soule´, Arithmetic analogs of the standard conjectures, Motives (Seattle, WA, 1991),129–140, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.HL D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31,Friedr. Vieweg and Sohn, Braunschweig, 1997.Mo1 A. Moriwaki, Inequality of Bogomolov-Gieseker type on arithmetic surfaces, Duke Math. J. 74 (1994),no. 3, 713–761.Mo2 A. Moriwaki, Bogomolov unstability on arithmetic surfaces, Math. Res. Lett. 1 (1994), no. 5, 601–611.So C. Soule´, A vanishing theorem on arithmetic surfaces, Invent. Math. 116 (1994), no. 1-3, 577–599.SABK C. Soule´, Lectures on Arakelov geometry, wth the collaboration of D. Abramovich, J.-F. Burnol and J.Kramer,Cambridge Studies in Advanced Mathematics, 33, Cambridge University Press, Cambridge,1992.6A quantitative sharpening of Moriwaki’s arithmetic Bogomolov inequalityN. Naumann niko.naumann@mathematik.uni-regensburg.deNWF I- Mathematik, Universita¨t Regensburg, Universita¨tsstrasse 31, 93053 Regensburg7

A quantitative sharpening of Moriwaki's arithmetic Bogomolov inequality

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