In this note, we study the differentiability of the arithmetic volumes along
arithmetic R-divisors, and give some equality conditions for the
Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big
arithmetic R-divisors.Comment: 35 page

Ikoma, Hideaki

English

arXiv

Kyoto University Research Information Repository

Title On the concavity of the arithmetic volumesAuthor(s)Ikoma, HideakiCitation代数幾何学シンポジウム記録 (2014), 2014: 88-98Issue Date2014URL http://hdl.handle.net/2433/215017RightType Departmental Bulletin PaperTextversionpublisherKyoto University＆ ＆vol(P ) v̂ol(P )deg(P i ·QdimX−i) d̂eg(P i ·QdimX−i)P Q P ≡num Q P ∼R Qs(P,Q) = s(P,Q) = s(P,Q)sup{t : P ⊃ tQ+ c,∃c} sup{t : P − tQ }X P,QR R Xv̂ol(P +Q)1dimX ! v̂ol(P ) 1dimX + v̂ol(Q) 1dimX .A = diag(a1, . . . , an) B = diag(b1, . . . , bn)A,BV (A(k) ·B(n−k)) = 1(nk) ∑I⊂{1,...,n},♯I=k∏i∈Iai ·∏j /∈Ibj.88代数幾何学シンポジウム記録2014年度   pp.88-98∀kV (A(k) ·B(n−k)) !⎛⎜⎜⎝ ∏I⊂{1,...,n},♯I=k∏i∈Iai ·∏j /∈Ibj⎞⎟⎟⎠(nk)−1= det(A)kn det(B)n−kndet(A+B) =n∑k=0(nk)V (A(k) ·B(n−k))!n∑k=0(nk)det(A)kn det(B)n−kn =(det(A)1n + det(B)1n)n.∀k a1/b1 = · · · = an/bnC = diag(c1, . . . , cn)V((A+B)(n−1) · C) 1n−1 ! V (A(n−1) · C) 1n−1 + V (B(n−1) · C) 1n−1 .s = s(A,B) = min{ai/bi}0 "(V (A(n−1) ·B) 1n−1 − s det(B) 1n−1)n" V (A(n−1)·B) nn−1−det(A)·det(B) 1n−1 .s = sup{t ∈ R : det(A− tB) > 0}det(A) = n∫ st=0V((A− tB)(n−1) ·B) dt" n∫ st=0(V (A(n−1) ·B) 1n−1 − t det(B) 1n−1)n−1dts(A,B) = s(B,A)−1 = (det(A)/ det(B))1n89RXSpec(Z)d := dimX−1 X Rat(X)R R D =(D, g) R R D = a1D1 + · · · + alDl Dg : (X \ ⋃ Supp(Di))(C) → R g∀p ∈ X(C)up(x) := g(x) +l∑i=1ai log |fi(x)|2C0 p fi Dip ∞ RR X D̂iv(X)L = (L, | · |) Xs L d̂iv(s) := (div(s),− log |s|2)R C0φ ∈ Rat(X)× ⊗Z R φe11 · · ·φerrφi ∈ Rat(X)× ei ∈ R φ R(̂φ) := e1((φ1),− log |φ1|2) + · · ·+ er((φr),− log |φr|2).R D XH0(D) :={φ ∈ Rat(X)× : (φ) +D ! 0} ∪ {0}Ĥ0(D) :={φ ∈ H0(D) : ∥φ∥gsup " 1},∥ · ∥gsup H0(D)⊗Z R∥φ∥gsup := ess.supx∈X(C)|φ(x)| exp(g(x)2).R D D ! 0 g ! 0 D1 ∈ Ĥ0(D)Dv̂ol(D) = lim supm→∞log ♯Ĥ0(mD)mdimX/ dimX!.90D → v̂ol(D)dimXD v̂ol(D) > 0 RB̂ig(X)D v̂ol(A) > 0 v̂ol(D + A) > 0D = (a1D1+ · · ·+ alDl, g) R XDix ∈ X(Q)x K(x) {x} Cx∗ x /∈ Supp(Di) ∀i xhD(x) :=1[K(x) : Q]⎛⎝ l∑i=1ai log ♯OCx(Di)/OCx +12∑σ:K(x)→Cg(xσ)⎞⎠ .φ ∈ Rat(X)×⊗Z R D+ (̂φ)∗D D up ∀phD(x) ! 0 ∀x ∈ X(Q) RX N̂ef(X)D D ( ) −( ) ∞ RR X Înt(X)PdZ = Proj(Z[X0, . . . , Xd])H := {X0 = 0}gFS := log(1 + |X1/X0|2 + · · · + |Xd/X0|2).H = (H, gFS)λ > 0 (H, gFS + λ)x := (x0 : · · · : xd) ∈ PdZ(Q)hnaive(x) :=1[K(x) : Q]∑v∈MK(x)log(maxi{|xi|v}),α ∈ K(x)×hnaive(x) = hH(x) + O(1)hD +O(1) D91D0, . . . , Dd−1d̂eg :d︷ ︸︸ ︷Înt(X)× · · ·× Înt(X)×D̂iv(X)→ R,(D0, . . . , Dd−1;Dd) /→ d̂eg(D0 · · ·Dd)R N d̂eg(N ·d+1) = v̂ol(N)D0, . . . , Dd−1 Dd d̂eg(D0 · · ·Dd) !0C∞ ∗Dϕ : X ′ → X R H X ′d̂eg(H·d · ϕ∗D) ! 0DR H1, . . . , Hd R H1,Q, . . . , Hd,Qdeg(DQ ·H2,Q · · ·Hd,Q) = 0 d̂eg(D·2 ·H2 · · ·Hd) " 0d̂eg(D ·H1 · · ·Hd) = 0 d̂eg(D·2 ·H2 · · ·Hd) " 0t = deg(DQ ·H2,Q · · ·Hd,Q)/ deg(H1,Q ·H2,Q · · ·Hd,Q)D − tH1, H2, . . . , HdD Spec(H 0(OX))D,E,H1, . . . , Hd R X92i 1 " i " dd̂eg(D·i · E·(d−i+1))2 ! d̂eg(D·(i−1) · E·(d−i+2)) · d̂eg(D·(i+1) · E·(d−i)).k 1 " k " d+ 1 i 0 " i " kd̂eg(D·i · E·(k−i) ·Hk · · ·Hd)k! d̂eg(D·k ·Hk · · ·Hd)i · d̂eg(E·k ·Hk · · ·Hd)k−i.k 1 " k " d+ 1d̂eg((D + E)·k ·Hk · · ·Hd) 1k! d̂eg(D·k ·Hk · · ·Hd) 1k + d̂eg(E·k ·Hk · · ·Hd) 1k .D (ϕ : X ′ → X,M)ϕ X ′ X ′QM R X ′ ϕ∗D −MD Θ̂(D) DΘ̂(D) ̸= ∅0 " n " d D0, . . . , DnDn+1, . . . , Dd(D0, . . . , Dn;Dn+1, . . . , Dd)⟨D0 · · ·Dn⟩Dn+1 · · ·Dd := sup(ϕ,M i)∈bΘ(Di) d̂eg(M0 · · ·Mn · ϕ∗Dn+1 · · ·ϕ∗Dd).B̂ig(X)×(n+1) × (N̂ef(X) ∩ B̂ig(X))×(d−n) → R,(D0, . . . , Dn;Dn+1, . . . , Dd) /→ ⟨D0 · · ·Dn⟩Dn+1 · · ·Dd,Dn+1, . . . , DdB̂ig(X)×(n+1) × Înt(X)×(d−n) → R,(D0, . . . , Dn;Dn+1, . . . , Dd) /→ ⟨D0 · · ·Dn⟩Dn+1 · · ·Dd.93n = d− 1B̂ig(X)×d × D̂iv(X)→ R,(D0, . . . , Dd−1;Dd) /→ ⟨D0 · · ·Dd−1⟩Dd.Dv̂ol(D) = ⟨D·(d+1)⟩D,E R i 1 " i "d− 1⟨D·i · E·(d−i+1)⟩ ! v̂ol(D) id+1 · v̂ol(E) d−i+1d+1⟨D·d⟩E ! ⟨D·d · E⟩ ! v̂ol(D) dd+1 · v̂ol(E) 1d+1 .v̂ol(D + E) !d+1∑i=0(d+ 1i)⟨D·i · Ed−i+1⟩ !(v̂ol(D)1d+1 + v̂ol(E)1d+1)d+1.D,E Rv̂ol(D − E) ! v̂ol(D)− (dimX)d̂eg(D·d · E).D /→ v̂ol(D)R D Elimt→0v̂ol(D + tE)− v̂ol(D)t= (dimX)⟨D·d⟩E.D Xh+D(X) :=v̂ol(D)(dimX) vol(DQ).(xn) XX (xn)lim infn→∞hD(xn) ! h+D(X).94hD(xn) h+D(X) D D + t(0, 2f)DRf : X(C)→ R(xn)hD(xn) h+D(X)limn→∞1[K(xn) : Q]∑σ:K(xn)→Cf(xσn) =⟨D·d⟩(0, 2f)vol(DQ).D P0 "((⟨D·d⟩P ) 1d − sv̂ol(P ) 1d)d+1" (⟨D·d⟩P )1+ 1d − v̂ol(D) · v̂ol(P ) 1d ,s = s(D,P ) = sup{t ∈ R : D − tP }D,E Rv̂ol(D + E)1d+1 = v̂ol(D)1d+1 + v̂ol(E)1d+1i 1 " i " d d̂eg(D·i · E·(d−i+1)) = v̂ol(D) id+1 · v̂ol(E) d−i+1d+1d̂eg(D·d · E) = v̂ol(D) dd+1 · v̂ol(E) 1d+1∃φ ∈ Rat(X)×Dv̂ol(D)1d+1− Ev̂ol(E)1d+1= (̂φ).⇒ ⇒ ⇒⇒s = s(D,E) =(v̂ol(D)v̂ol(E)) 1d+1s(E,D) = s−1.D−sE sE−D95XQ ω∫X(C) ω = 1D Xb(mD) := Image(〈Ĥ0(mD)〉Z⊗Z OX(−mD)→ OX).µm : Xm → X Xm Xm,Qb(mD)OXmF (mD) := b(mD)OXm M(mD) := µ∗m(mD)− F (mD).L2 e1, . . . , erm〈Ĥ0(mD)〉CBerg(mD)(x) := |e1(x)|2 + · · ·+ |erm(x)|2, x ∈ X(C),OXm(F (mD))|1F (mD)|(x) =√Berg(mD)(µm(x)), x ∈ Xm(C).F (mD) := (F (mD),−µ∗m log Berg(mD)) M(mD) :=µ∗m(mD)− F (mD)XQ Dk 1 " k " d+1 Dk, . . . , DnR Dn+1, . . . , Dd R⟨D·k ·Dk · · ·Dn⟩Dn+1 · · ·Dd = limm→∞⟨Dk · · ·Dn⟩M(mD)·k ·Dn+1 · · ·Ddmk.limm→∞d̂eg(M(mD)·d · F (mD))md+1= 0.R DD = P +N P R NR v̂ol(P ) = v̂ol(D)96dimX = 2DdimX ! 3XP2Z = Proj(Z[X0, X1, X2]) zi := Xi/X0H := {X0 = 0}g := max{−2, log |X1/X0|2 + 2, log |X2/X0|2 + 2} ,Hρ : P2(C)→ R!0Supp(ρ) # {|z1| < exp(−2)}× {|z2| < exp(−2)} .H = (H, g + ρ) hH(1 : 0 : 0) < 0 g + ρP2Z (1 : 0 : 0) {X0 ̸= 0}ϕ : Proj(Z[z1, z2][Y1, Y2]/(z2Y1 − z1Y2))→ {X0 ̸= 0}.ϕ∗H Ewij := Yj/YiP =(ϕ∗H − 12E,max{log |ziwi1|, log |ziwi2|, log |ziwi1|2 + 2, log |ziwi2|2 + 2}),N =(12E,max {0,−2−max {log |ziwi1|, log |ziwi2|}} + ϕ∗ρ)! 0,v̂ol(H) = v̂ol(P ) = 5/4P,Q R v̂ol(P ) = v̂ol(Q)P ! Q P = Q2v̂ol(P )1d+1 = v̂ol(P )1d+1 + v̂ol(Q)1d+1 " v̂ol(P +Q) 1d+1 " v̂ol(2P ) 1d+1 .∃φ ∈ Rat(X)× ⊗Z R P −Q = (̂φ) ! 0(̂φ) ! 0 ⇔ (̂φ) = 0 (⇔ φ ∈ H0(O∗X)⊗Z R).R9798

On the concavity of the arithmetic volumes

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