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Kawamata, Yujiro Citation 代数幾何学シンポジウム記 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Kyoto University Research Information Repository Title On the extension problem of pluricanonical forms Author(s) Kawamata, Yujiro Citation 代数幾何学シンポジウム記録 (1998), 1998: 83-97 Issue Date 1998 URL http://hdl.handle.net/2433/214686 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University 代数幾何学シンポジウム記録 1998年度 pp.83-97 ON THE EXTENSION PROBLEM OF PLURICANONICAL FORMS YUJIRO KAWAMATA Dedicated to Professor Friedrich Hirzebruch on his seventt'eth birthday ABSTRACT. We review some recent development on the extension problem of pluri- canonical forms from a divisor to the ambient space by Siu, Kawamata ancl Nakayama with simplified proofs, 1. INTRODUCTION The purpose of this paper is to review some recent development on the extension problem of pluricanonical forms from a divisor to the ambient space. The main tools of the proofs are the multiplier ideal sheaves and the vanishing theorems for them. Let X be a compact complex manifold. The Tn-genus P. (X) of X for a positive integer m is defined by P.(X) = dim HO(X,mKx). The growth order of the plurigenera for large m is cailed the Kodaira dimension K(X): we have P.(X) fiv mrc(X) for any suMciently large and divisible m. We have the fo}}owing possibilities: rc(X) = -oo,O, 1,•••, or dim X. In particular, if rc(X) = dim X, then X is said to be of Jc eneral type. It is important to note that these invariant,s are independent of the birational models of X. The plurigenera are fundamental discrete invariants for the classification of alge- braic varieties, But they are by definition not topological invariants. However, in order that such classification is reasonable, the following conjecture due to Iitaka should be true: Conjecture 1.1. Let S be an algebraic variety, and let f:,V -År S be a srnooth algebraic morphism. Then the plurigenus P.(Xt) is constant on t E S for any positive integer m. A morphism between complex varieties which is birationally equivalent to a projective morphism wi}l be called an algebraic morphism in this paper. The alge- braicity assumption in the conjecture is slightly weaker than the projectivity. This conjecture is confirmed by Iitaka [Il, I2] in the case in which dim Xt = 2 by using the classification theory of surfaces. Nakayama [Nl] proved that the conjecture follows if the minima} model exists for the fami!y and the abundance 1991 Mathematics Subj'ect Classification. 14B07, 14E30, 14F17. Key uiords and phrases. plurigenus, canonical singularity, cleformation, multiplier ideal sheaf, vanishing theorem. The final version of this paper will appear in Proceeclings Hirzebruch 70. Typeset by AMb--rlEX 83 2 YUJIRO KAWAMATA conjecture conjecture holds for the generic fiber. Thus the conjecture is true if dim Xi = 3 by [K4] [K4] and [KM]. On the other hand, Nakamura [Nm] provided a counterexample for the gener- alization alization of the conjecture in the c笛 e where the morphism f is not algebraic. In his 邑xample, the central fiber Xo is a quotient of a 与dimensional simply connected solvable solvable Liξgroup by a discrete subgroup. We note that X0 is a non-K 込hiξr man- ifold ifold which has non-clos 付 holomorphic 1-forms. So w ♀ only consider algebraic morphisms in this paper. It is interesting tむをxtξnd our results to the casξin which the the fibers are in Fujiki ’s class C. The following theorem of Siu was the starting point of the recent progress on this this conjecture which we shall review. Theorem 1.2. [Si} Let S be αcomplex variety ，αM let f :X →S be a smooth projective projective morphism. Assume th αt the gene パc fiber Xη off is a variety of general type. type. Then the plur 伊 nus Pm (Xi) is co ηst α川 on t E S for any pos 的 ve integer m. We have also a slightly stronger version: Theorem 1.2'. [I<5} Let S be an 均 εbrai ε凶 riety, let X be a complex 回 riety, and let f :X →S be αprop εT fiat alge るraic morphism. Assume that the fibers Xi = 1-1(t) have only canonical singularities for any t E S and that the gε neric fiber fiber Xη is S 叩 riety of general type. Then the plurigenus Pm(Xt) is constant on tεS for any positive integer m. According According to Nakayama [N2], we define the m 附 ric αl I<od α ira dimension ν（X) as as follows (this isκσ （X) in [N2]; there is anotl 悶 version κν （X) of numerical Ko daira daira dirr － and let k be a nonn 配gativ 配 integer. 羽le de 白neν （X ）；ど k if there exist a divisor ． H on X and a positiv 控 numl コer c such that dim H0(X, mli. ’x + H ）三en/' for an SU 缶ciently large and divisible m 司 If there is no such k, then we put v(X) = -oo. It is is easy もG se 告 thatκ （X ）三ν（X ）壬dim X. By the Kodaira lemma ，κ（X) =dim X if ‘ if and only if v(X ）ロ dim X. ThξG おndance conjecture states that the equal- ity ity 11:( X) ：：：：ジ （X) always holds. Nakayama co1 命 med this conjectur 号 in the c総 v(X) = 0 ([N2]). By considering ml¥x 十 H instead of mKx, Nakayama obtained the following: Theorem 1.3. [N3} Let S be an alg 訪問ic variety, let X be 臼 complex veriety, and let f x →S be a proper Fαtα lgebraic morphism. Assume that the fibers Xi= f→ （t) hα ve onl νcα nonic αl singularities for αnνt E S. Then the numerical f{ odaira dimension ν（X t) is constant on t ε S. In pα rticular, if one fiber X0 is of general general type, then so αre αll the fibers. For the finξr classification of algebraic varieties, it is useful to consider not only thξdisc 出邑 invariants Pm(X) but also 也君 infinite sum of v在ctor spacξS R(X) = Efj H0(X, mKx) m>O which has a natural graded ring structure over C = H0(X, Ox). This continuous invariant invariant R(X), called tl 日 Cαnonicαl n吋 of X, is also inde1 河川ent of the birational 84 PLURICANONICAL FORMS 3 models of X. It is con ctured that R(X) is always 自nitely generated as a graded C-algebra. C-algebra. If this is the case, then Proj R(X) is called a canonic αl model of X. A canonical si 吋 ular 均（ resp. terminal singularity) is defined 出 a singularity which may appear on a canonical model of a variety of general type whose canonical ring ring is finitely generated (resp. on a minimal model of an algebraic variety). The formal formal definition by Reid is 笛 follows: a normal variety X is said to have only canonical canonical singularities (resp. terminal singularities) if the canonical divisor I<x is Q-Cartier Q-Cartier and, for a of resolution singularities μ ・Y →X which has exceptional divisors divisors Fj, if we writeµ• I<x =Ky ＋乞Jα jFj, then 町三 0 (resp. 円＜ 0) for all j. j. For For example, the canonical singularities in dimension 2 have been studied exten- sively. sively. They are called in many names such as du Val singularities, rational double points, points, simple singularities, or A-D-E singularities. The terminal singularity in di- mension 2 is smooth, and the terminal singularities in dimension 3 are classified by Mori and others (cf. [R2)). Let Let us consider the subset of a Hilbert scheme with a given Hilbert polynomial which consists of points corresponding to the canonical models of var 目ieties of general type. type. This set should be open from the view point of the moduli problem of varieties (cf. (cf. [V2]). The following is a local version of Theor 叩 1 1.2' and says that this is the case case (this result was previously known up to dimer Theorem 1.4. ｛λ可 Let f: X →B be a fiat morphism from a germ of an algebraic variety variety to a germ of a smooth curve. Assume that the central fiber X0 = f-1 (P) has has only canonical singularities. Then so has the total space Xαs well as αny fiber Xi off. Moreover ，ザ μ： V →X is a b・irational morphism from a normal variety 山 ：th the strict transform X of X0, then J( v + X 三 µ•（ Kx + Xo)- The following theorem answers a similar question for the deformations of minimal models （山 is was previously known up to dimension 3): Theorem 1.5. {N3} Let f:X →B be a fiat morphism from a germ of an α lgebrazc variety variety to a germ of α smooth curve. Assume that the central fiber X0 = f-1 (P) has has only terminal singularities. Then so hαs the total space Xα s well as any fiber X1 off. Moreover, ifμ: V →X is a birational morphism from αnormal uα riety 山 ；th the strict transform X of X0, then the support of I<v + X - μ"(I<x + Xo) contains contains all the exceptional divisors ofμ. The following theorem, which is stronger than Theorem 1.2', says that only the abundance conjecture for the generic fiber implies the deformation invariance of the plurigenera: plurigenera: Theorem 1.6. {N3} Let S be an αlgebn α ic uαη ety, let X be αcomplex variety, and let f :X →S be αproper Fαtα lgebraic morphism.
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