A UNIVERSAL METRIC FOR THE CANONICAL BUNDLE OF A HOLOMORPHIC FAMILY OF PROJECTIVE ALGEBRAIC MANIFOLDS

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1 A UNIERSAL METRIC FOR THE CANONICAL BUNDLE OF A HOLOMORPHIC FAMILY OF PROJECTIE ALGEBRAIC MANIFOLDS DROR AROLIN Dedicated to M Saah Baouendi on the occasion of his 60th birthday 1 Introduction In his ceebrated wor [S-98, S-0], Siu proved that the purigenera of any agebraic manifod are invariant in famiies More precisey, et π : D be a hoomorphic submersion (ie, dπ is nowhere zero) from a compex manifod to the unit dis D, and assume that every fiber t := π 1 (t) is a compact proective manifod Then for every m N, the function P m : D N defined by P m (t) := h 0 ( t, mk t ) is constant Siu s approach to the probem begins with the observation that the function P m is upper semicontinuous Thus in order to prove that P m is continuous (hence constant) it suffices to show that given a goba hoomorphic section s of mk 0, there is a famiy of goba hoomorphic sections s t of t, for a t in a neighborhood of 0, that varies hoomorphicay with t and satisfies s 0 = s To prove such an extension theorem, Siu estabishes a generaization of the Ohsawa-Taegoshi Extension Theorem to the setting of compex submanifods of a Kaher manifod having codimension 1 and cut out by a singe, bounded hoomorphic function This theorem, which we wi discuss beow, requires the existence of a singuar Hermitian metric on the ambient manifod having non-negative curvature current, with respect to which the section to be extended is L Thus in the presence of the extension theorem, the approach reduces to construction of such a metric The case where the fibers t of our hoomorphic famiy are of genera type was treated in [S-98] In this setting, Siu produced a singe singuar Hermitian metric e κ for K so that every m-canonica section is L with respect to e (m 1)κ However, in the case where the fibers t of our hoomorphic famiy are assumed ony to be agebraic, and not necessariy of genera type, Siu s proof in [S-0] does not construct a singe metric as in the case of genera type Instead, Siu constructs for every section s of mk 0 a singuar Hermitian metric for mk of non-negative curvature so that s is L with respect to this metric Definition Let be a hoomorphic famiy of compex manifods and 0 the centa fiber of A universa canonica metric for the pair (, 0 ) is a singuar Hermitian metric e κ for the canonica bunde K of such that for every goba hoomorphic section s H 0 ( 0, mk 0 ), 0 s e (m 1)κ < + The goa of this paper is to prove that for any hoomorphic famiy of compact compex agebraic manifods with centra fiber 0, the pair (, 0 ) has a universa canonica metric having non-negative curvature current To this end, our main theorem is the foowing resut Theorem 1 Let be a compex manifod admitting a positive ine bunde A, and a smooth compact compex submanifod of codimension 1 Assume there is a subvariety not containing such that is a Stein manifod Let T H 0 (, ) be a hoomorphic section of 000 Mathematics Subect Cassification 3L10 14F10 Partiay supported by an NSF grant 1

2 the ine bunde associated to, thought of as a divisor Let E be a hoomorphic ine bunde and denote by K the canonica bunde of Assume we are given singuar metrics e ϕ E for E and e ϕ for the ine bunde associated to Suppose in addition that the above data satisfy the foowing assumptions (R) The metrics e ϕ E and e ϕ restrict to singuar metrics on (B) sup T e ϕ < + (G) The ine bundes p(k + + E) + A, 0 p m 1, are gobay generated, in the sense that a finite number of sections of H 0 (, p(k + + E) + A) generate the sheaf O (p(k + + E) + A) (P) 1 ϕ E 0 and there exists a constant µ such that µ 1 ϕ E 1 ϕ (T) The singuar metric e (ϕ +ϕ E ) has trivia mutipier idea: I (, e (ϕ +ϕ E ) ) = O Then there is a metric e κ for K + + E with the foowing properties: (C) 1 κ 0 (L) For every m > 0 and every section s H 0 (, m(k + E )), s e ((m 1)κ+ϕ E+ϕ ) is ocay integrabe (I) For every integer m > 0 and every section s H 0 (, m(k + E)), s e (m 1)κ+ϕ E < + Remars (i) For the ambient manifod, we have in mind the foowing two exampes: either is compact compex proective (in which case the variety coud be taen to be a hyperpane section of some embedding of ) or ese is a famiy of compact compex agebraic manifods In the former case, it is we-nown that the hypothesis (G) hods for any sufficienty ampe A, whie in the atter case, one might have to shrin a itte to obtain (G) Of course, there are many other exampes of such (ii) Note that in condition (L), the oca functions s e ((m 1)κ+ϕ E+ϕ ) depend on the oca triviaizations of the ine bundes in question However, the oca integrabiity condition is independent of these choices Together with a variant of the Ohsawa-Taegoshi Theorem (Theorem 4 beow), Theorem 1 impies a generaization of Siu s extension theorem to the case where the norma bunde of the submanifod is not necessariy trivia The first extension theorem of this type was estabished by Taayama [Ta-05, Theorem 41]under some additiona hypotheses The genera case was done in [-06], where Theorem 4 was aso estabished The argument here is reated to that of [-06], but the focus is on construction of the metric rather than on the extension theorem As a resut of Theorem 1, we have the foowing coroary, which is our stated goa Coroary For every hoomorphic famiy of smooth proective varieties with centra fiber 0, the pair (, 0 ) has, perhaps after sighty shrining the famiy, a universa canonica metric having non-negative curvature current Proof Let be a famiy of compact proective manifods π : D, and = 0 the centra fiber Tae T = π, E = O and ϕ E 0 Since 0 is cut out by a singe hoomorphic function, the ine bunde associated to 0 is trivia Tae ϕ 0 Then the hypotheses of Theorem 1 are satisfied, perhaps after shrining the famiy, and we obtain a metric e κ for K such that 1 κ 0 and s e (m 1)κm is integrabe for every integer m > 0 and every section s H 0 ( 0, mk 0 )

3 Remar Note that in the setting of famiies, the constant µ is not needed, and the hypotheses (L) and (I) are the same Remar In his paper [Ts-0], Tsui has caimed the existence of a metric with the properties stated in Coroary As in our approach, Tsui s proof maes use of an infinite process It seems that convergence of this process was not checed; in fact, it is demonstrated in [S-0] that Tsui s process, as we as any reasonabe modification of it, diverges Proposition 3 For each integer m > 0, fix a basis s (m) 1,, s (m) N m of H 0 (, m(k + E )) Choose constants ε m such that the metric ( Nm ) 1/m κ 0 := og ε m m=1 is convergent Suppose e ϕ E is ocay integrabe Then for each m > 0 and every s H 0 (, m(k + E )), s e ((m 1)κ 0+ϕ E ) < + ( Proof Fix s H 0 Nm ) 1/m (, m(k + E )), and et κ 0,m = og =1 s(m) Note that e κ 0 e κ 0,m, and thus we have s e (m 1)κ 0+ϕ E s e (m 1)κ 0,m+ϕ E = ( s/m ( s = N m s/m e γ E ϕ E e γ E ) (m 1)/m e γ E ϕ E e γ E s e γ E ϕ E e mγ E ω (n 1)(m 1) ) 1/m ( e γ E ϕ E ω n 1 ) (m 1)/m, where ω is a fixed Käher form for and e γ is a smooth metric for E The ast inequaity is a consequence of Höder s Inequaity Since e ϕ E is ocay integrabe, we are done A cacuation simiar to the proof of Proposition 3 shows that s e ((m 1)κ 0+ϕ +ϕ E ) is ocay integrabe on Thus in view of Proposition 3, Theorem 1 foows if we construct a metric e κ with non-negative curvature current such that e κ = e κ 0 This is precisey what we do We empoy a technica simpification, due to Paun [P-05], of Siu s origina idea of extending metrics using an Ohsawa-Taegoshi-type extension theorem for sections Contents 1 Introduction 1 The Ohsawa-Taegoshi Extension theorem 4 3 Inductive construction of certain sections by extension 4 4 Construction of the metric 7 41 A metric associated to m(k + + E) 7 4 The metric for K + + E; Proof of Theorem 1 9 References 10 3

4 The Ohsawa-Taegoshi Extension theorem Let Y be a Käher manifod of compex dimension n Assume there exists an anaytic hypersurface Y such that Y is Stein Exampes of such manifods are Stein manifods (where is empty) and proective agebraic manifods (where one can tae to be the intersection of Y with a proective hyperpane in some proective space in which Y is embedded) Fix a smooth hypersurface Y such that In [-06] we proved the foowing generaization of the Ohsawa-Taogoshi Extension Theorem Theorem 4 Suppose given a hoomorphic ine bunde H Y with a singuar Hermitian metric e ψ, and a singuar Hermitian metric e ϕ for the ine bunde associated to the divisor, such that the foowing properties hod (i) The restrictions e ψ and e ϕ are singuar metrics (ii) There is a goba hoomorphic section T H 0 (Y, ) such that = {T = 0} and sup T e ϕ = 1 Y (iii) 1 ψ 0 and there is an integer µ > 0 such that µ 1 ψ 1 ϕ Then for every s H 0 (, K + H) such that s e ψ < + and s dt I (e (ϕ+ψ) ), there exists a section S H 0 (Y, K Y + + H) such that S = s dt and S e (ϕ+ψ) 40πµ Y s e ψ 3 Inductive construction of certain sections by extension Fix a hoomorphic ine bunde A such that the property (G) in Theorem 1 hods Let us fix bases { σ (m,0,p) ; 1 M p } of H 0 (, p(k + + E) + A) We et σ (m,0,p) We aso fix smooth metrics respectivey Finay, et us fix bases H 0 (, p(k + E ) + A ) be such that σ (m,0,p) = σ (m,0,p) (dt ) p e γ and e γ E for, and E s (m) 1,, s (m) N m for H 0 (, m(k + E )), m = 1,,, orthonorma with respect to the singuar metric (ω (n 1) e γ E) m 1 e ϕ E for (m 1)K + me (Since e ϕ E is ocay integrabe, every hoomorphic section is integrabe with respect to this metric) Proposition 5 For each m = 1,, there exist a constant C m < + and sections σ (m,,p), H 0 (, (m + p)(k + + E) + A) where p = 1,,, m 1, 1 M p, 1 N m and = 1,,, with the foowing properties (a) σ (m,,p), = (s (m) ) σ (m,0,p) (dt ) (m+p) 4

5 (b) If 1, (c) For 1 p m 1, M0 =1 σ(m,,0), e (γ +γ E ) =1 σ (m, 1,m 1) C m, Mp =1 σ(m,,p), e (γ +γ E ) Mp 1 =1 σ (m,,p 1) C m, Proof (Doube induction on and p) Fix a constant Ĉm such that the and and for a 0 p m, and ( = 0) We set σ (m,0,p), sup sup sup M0 =1 σ(m,0,0) M0 =1 σ(m,0,0) sup := σ (m,0,p) ω n(m 1) e (m 1)(γ +γ E ) =1 σ (m,0,m 1) Ĉm ω (n 1)(m 1) e (m 1)γ E =1 σ (m,0,m 1) Ĉm, Np+1 =1 σ(m,0,p+1) ω n e (γ +γ E ) Mp =1 σ(m,0,p) Ĉm, Np+1 =1 σ(m,0,p+1) ω (n 1) e γ E Mp =1 σ(m,0,p) Ĉm and simpy observe that Mp =1 σ(m,0,p), e (γ +γ E ) Mp 1 =1 σ (m,0,p 1), Ĉm ( 1) Assume the resut has been proved for 1 ((p = 0)): Consider the sections (s (m) ) σ (m,0,0), and define the semi-positivey curved metric M m 1 ψ,,0 := og σ (m, 1,m 1), for the ine bunde (m 1)(K + + E) + A Observe that ocay on, (s (m) =1 dt m ) σ (m,0,0) e (ϕ +ψ,,0 +ϕ E ) = ω n dt m σ(m,0,0) e (ϕ +ϕ E ) =1 σ (m,0,m 1) e (ϕ +ϕ E ) Moreover, we have 1 (ψ,,0 + ϕ E ) 0 and µ 1 (ψ,,0 + ϕ E ) 1 ϕ 5

6 Finay, = (s (m) ) σ (m,0,0) e (ψ,,0+ϕ E ) We may thus appy Theorem 4 to obtain sections such that and σ(m,0,0) e (m 1)γ Ee ((m 1)γ E+ϕ E ) =1 σ (m,0,m 1) < + σ (m,,0), H 0 (, m(k + + E) + A), 1 M 0, 1 N m, σ (m,,0), Summing over, we obtain = (s (m) ) σ (m,0,0), (dt ) m, 1 M 0, 1 N m, σ (m,,0), e (ψ,,0+ϕ +ϕ E ) 40πµ M0 =1 σ(m,,0), e (γ +γ E ) =1 σ (m, 1,m 1) sup e ϕ +ϕ E γ γ E, M0 =1 σ(m,,0) 40π sup e ϕ +ϕ E γ γ E, e (ϕ +ϕ E ) =1 σ (m, 1,m 1), M0 σ(0) e (ϕ E+ϕ B ) Nm 1 =1 σ (m 1) e ϕ E =1 σ(m,0,0) =1 σ (m,0,m 1) e κ 40πĈm sup e ϕ +ϕ E γ γ E ω (n 1)(m 1) e ((m 1)γ E+ϕ E ) = 40πĈm sup e ϕ +ϕ E γ γ E ((1 p m 1)): Assume that we have obtained the sections σ (m,,p 1),, 1 M p 1, 1 N m Consider the non-negativey curved singuar metric for (m + p 1)(K + + E) + A We have (s (m) M p 1 ψ,,p := og σ (m,,p 1), =1 ) σ (m,0,p) e (ϕ +ψ,,p +ϕ E ) = σ(m,0,p) e (ϕ +ϕ E ) Mp 1 =1 σ (m,0,p 1) e (ϕ+ϕe), which is ocay integrabe on by the hypothesis (T) Next, (s (m) ) σ (m,0,p) e (ψ σ (m,0,p),,p+ϕ E ) e ϕ E = Mp 1 =1 σ (m,0,p 1) C e γ σ(m,0,p) e (ϕ +ϕ E ) Mp 1 < +, 6 =1 σ (m,0,p 1)

7 where C := sup e ϕ γ Moreover, 1 (ψ,,p + ϕ E ) 0 and 1 (ψ,,p + ϕ E ) 1 ϕ By Theorem 4 there exist sections σ (m,,p), H 0 (, (m + p)(k + + E) + A), 1 M 0 such that σ (m,,p), = (s (m) ) σ (m,0,p), (dt ) m+p, 1 M p, and σ (m,,p), e (ψ,,p+ϕ +ϕ E ) σ (m,0,p) e ϕ E 40πµ Mp 1 =1 σ (m,0,p 1) Summing over, we obtain Mp =1 σ(m,,p), e (γ +γ E ) Mp 1 =1 σ (m,,p 1) 40πµ sup e ϕ +ϕ E γ γ E Ĉ m e ϕ E ω n 1, Letting ( ) C m := 40πµĈm max ω n, sup e ϕ +ϕ E +ϕ B γ γ E, sup e ϕ +ϕ E γ γ E e ϕ E ω n 1 competes the proof 4 Construction of the metric 41 A metric associated to m(k + + E) Fix a smooth metric e ψ for A Consider the functions Mp λ (m),n := og σ (m,,p), ω n(m+p) e (m(γ +γ E )+ψ), where N = m + p Set =1 λ (m) N := og Nm =1 e λ(m),n Lemma 6 For any non-empty open subset and any smooth function f : R +, ( ) 1 (λ (m) fωn N Nm C m sup λ(m) N 1 )fωn og f fωn Proof Observe that by Proposition 5, there exists a constant C m such that for any open subset, (e λ(m),n λ(m),n 1)fω n C m sup f, and thus N m (e λ(m) N λ(m) N 1 )fω n = (e λ(m),n λ(m),n 1)fω n N m C m sup f =1 An appication of (the concave version of) Jensen s inequaity to the concave function og then gives ( ) 1 (λ (m) fωn N Nm C m sup λ(m) N 1 )fωn og f fωn 7

8 The proof is compete Consider the function = 1 λ(m) m Note that is ocay the sum of a purisubharmonic function and a smooth function By appying Lemma 6 and using the teescoping property, we see that for any open set and any smooth function f : R +, (1) 1 fωn fω n m og Proposition 7 There exists a constant C (m) o such that (x) C o (m), x ( ) Nm C m sup f fωn Proof Let us cover by coordinate charts 1,, N such that for each there is a bihoomorphic map F from to the ba B(0, ) of radius centered at the origin in C n, and such that if U = F 1 (B(0, 1)), then U 1,, U N is aso an open cover Let W = \ F 1 (B(0, 3/)) Now, on each, is the sum of a purisubharmonic function and a smooth function Say = h + g on, where h is purisubharmonic and g is smooth Then for constant A we have sup sup g + sup h U U U sup g + A h F d U W sup g A g F d + A U W W F d Let C (m) and define the smooth function f by := sup g A g F d U W f ω n = F d Then by (1) appied with = W and f = f, we have ( sup C (m) Nm C m sup W f + ma og U W f ω n Letting C (m) o competes the proof := max 1 N { C (m) + ma og ) W f ω n ( ) } Nm C m sup W f W f ω n f ω n W Since the upper reguarization of the im sup of a uniformy bounded sequence of purisubharmonic functions is purisubharmonic (see, eg, [H-90, Theorem 16]), we essentiay have the foowing coroary 8

9 Coroary 8 The function (x) := im sup y x im sup (y) is ocay the sum of a purisubharmonic function and a smooth function Proof One need ony observe that the function Λ is obtained from a singuar metric on the ine bunde m(k + + E) (this singuar metric e κ(m) fixed smooth metric of the dua ine bunde Consider the singuar Hermitian metric e κ(m) This singuar metric is given by the formua ( e κ(m) (x) = exp where The curvature of e κ(m) is thus 1 κ (m) = wi be described shorty) by mutipying by a for m(k + + E) defined by e κ(m) = e Λ(m) ω nm e m(γ +γ E ) e κ(m) im sup y x im sup = e Λ(m) ω nm e m(γ +γ E ) 1 og 1 1 ψ N m N 0 =1 =1 ) κ (m) (y), σ (m,,0), 1 1 ψ We caim next that the curvature of e κ is non-negative To see this, it suffices to wor ocay Then we have that the functions are purisubharmonic But im sup y x im sup κ (m) + 1 κ (m) + 1 ψ ψ = im sup y x It foows that κ (m) is purisubharmonic, as desired im sup κ (m) = κ (m) 4 The metric for K + + E; Proof of Theorem 1 Let ε m be constants, chosen so ε m 0 sufficienty rapidy that the sum e κ := ε m e 1 m κ(m) = exp( 1 m κ(m) + og ε m ) m=1 m=1 converges everywhere on (to a metric for (K + + E)) It is possibe to find such constants since, by Proposition 7, each κ (m) is ocay uniformy bounded from above (The ower bound e κ(m) 0 is trivia) Moreover, by eementary properties of purisubharmonic functions, κ is purisubharmonic Indeed, for any r N, the function r ψ r := og exp( 1 m κ(m) + og ε m ) m=1 is purisubharmonic, and ψ r κ It foows that κ = sup r ψ r is purisubharmonic (Again, see [H-90, Theorem 16])Thus e κ is a singuar Hermitian metric for K + + E with non-negative curvature current 9

10 Observe that, after identifying K with (K + ) by dividing by dt, ( Nm ) κ (m) = og + 1 M 0 og σ (m,0,0) Thus we obtain e κ(m) = ( Nm =1 s(m) =1 ) 1 It foows that =1 e κ 1 = ( m=1 ε Nm ) /m m =1 s(m) In view of the short discussion foowing the proof of Proposition 3, the metric e κ satisfies the concusions of Theorem 1 The proof of Theorem 1 is thus compete Acnowedgment I am indebted to Lawrence Ein and Mihnea Popa It is to a discussion with them that the present paper owes its existence References [H-90] Hörmander, L, An introduction to compex anaysis in severa variabes Third edition North-Hoand Mathematica Library, 7 North-Hoand Pubishing Co, Amsterdam, 1990 [P-05] Paun, M, Siu s invariance of purigenera: a one-tower proof Preprint 005 [S-98] Siu, Y-T, Invariance of purigenera Invent Math 134 (1998), no 3, [S-0] Siu, Y-T, Extension of twisted puricanonica sections with purisubharmonic weight and invariance of semipositivey twisted purigenera for manifods not necessariy of genera type Compex geometry Coection of papers dedicated to Hans Grauert Springer-erag, Berin, 00 (3 77) [Ta-05] Taayama, S, Puricanonica systems on agebraic varieties of genera type, preprint 005 [Ts-0] Tsui, H, Deformation invariance of purigenera Nagoya Math J 166 (00), [-06] aroin, D, A Taayama-type Extension Theorem Preprint 006 Department of Mathematics Stony Broo University Stony Broo, NY