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A Geometric Approach to Problems in Birational Geometry

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A Geometric Approach to Problems in Birational Geometry A geometric approach to problems in birational geometry Chen-Yu Chi1 and Shing-Tung Yau Department of Mathematics, Harvard University, Cambridge, MA 02138; Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved September 15, 2008 (received for review July 18, 2008) A classical set of birational invariants of a variety are its spaces We call this kind of theorem a Torelli-type theorem because of pluricanonical forms and some of their canonically defined sub- the classical Torelli theorem says that the periods of inte- spaces. Each of these vector spaces admits a typical metric struc- grals determine an algebraic curve. This remarkable theorem ture which is also birationally invariant. These vector spaces so was generalized to higher-dimensional algebraic varieties. The metrized will be referred to as the pseudonormed spaces of the most notable one was the work of Piatetsky-Shapiro and Sha- original varieties. A fundamental question is the following: Given farevich (3) for algebraic K3 surfaces, which was generalized two mildly singular projective varieties with some of the first vari- to Kähler K3s by Burns and Rapoport (4), where they proved ety’s pseudonormed spaces being isometric to the corresponding the injectivity of period maps. The surjectivity of period maps ones of the second variety’s, can one construct a birational map for K3 surfaces was done using Ricci flat metrics by Siu (5) between them that induces these isometries? In this work, a posi- and Todorov (6) following the work of Kulikov (7) and of tive answer to this question is given for varieties of general type. Perrson and Pinkham (8). This phenomenon of subjectivity is This can be thought of as a theorem of Torelli type for birational known to be rather generic, and in many cases the period map equivalence. can be proved to have degree one for hypersurfaces [see, e.g., Donagi (9)]. pseudonorms | pluricanonical systems | varieties of general type | 2. Existence. Characterize geometrically and algebraically those Torelli-type theorems | classification normed vector spaces that can be realized as the pluricanonical spaces of some algebraic varieties as above. Hopefully, there n this article, we initiate a program to study problems in bira- may be some effective way to construct the birational models I tional geometry. This approach will be more geometric than of these varieties. other more algebraic approaches. Most of the arguments can, 3. Computation. In the case of the classical Torelli theorems, the however, be phrased in a purely algebraic way. It is quite likely periods can be effectively computed by methods dating back to that some of them can be applied to deal with the geometry over Picard, Leray, Dwork, and others. We hope to calculate these different ground fields. normed spaces effectively too. Some differential geometric methods will be brought in. 1. Introduction 4. Relations with questions of GIT and other invariants. Making Given a projective variety M, we shall study the geometric infor- use of the pluricanonical series, we are able to form new invari- 0 mation provided by the pluricanonical space H (M, mKM ). Note ant (pseudo-)metrics on the algebraic manifolds. There should that the minimal model program, as led by Mori, Kawamata, be some relationship between these metrics and other well- Kollár, and others, has achieved great success. Although the ear- known canonical metrics such as Kähler–Einstein metrics. We lier workers had solved the problem for threefolds completely, the hope to build a link between our approach with other metrical spectacular finite generation question was recently solved by sev- approaches to algebraic geometry. eral people, using different approaches: the analytic approach of | | Siu (1) and the algebraic approach of Birkar, Cascini, Hacon, and In this article, we shall prove that when mKM has no base point McKernan (2). In our approach, instead of using the full canoni- and defines a birational map, the normed space is indeed powerful cal ring, we shall focus our study on the pluricanonical space for enough to determine the birational type of the algebraic varieties. a fixed m. We can achieve this when m is large enough (depending on the Ideally, we would like to determine the birational type of our dimension of M only). Indeed, we prove a Torelli theorem that is algebraic variety based on the information on this space only. Any described in 1 under rather general assumptions. We should say birational transformations of algebraic manifolds will induce a that in case the manifold is 1-dimensional and m = 2, the problem linear map between the corresponding pluricanonical spaces (for was treated by Royden in his study of the biholomorphic transfor- each fixed m). The plurigenera are, of course, invariant under the mations of Teichmüller space (10). We think that it is possible to birational transformations. But more importantly, there are other generalize Royden’s work to higher-dimensional manifolds. finer invariants that are preserved by these transformations. The We shall study 2 by using a more differential geometric most important ones are the natural norm-like functions (called approach. We here outline what kind of metric we can obtain. At ∈ 0 ∈ 0 norms in this introduction) induced by integrating over M the mth every point η0 H (M, mKM ) and η1, η2 H (M, mKM ), viewed 0 root of the product of a m-pluricanonical form and its conjugate. as 2 tangent vectors at η0 in H (M, mKM ), we define a hermitian metric The norm defines an interesting geometry that was not explored extensively before. η η h(η , η ) = 1 2 η In our work, we shall initiate a program to study this geometry. 1 2 η η 0 m The first major questions we address are the following: M 0 0 1. Torelli-type theorem. Given 2 algebraic varieties M and M, suppose there is a linear map that defines an isometry (with Author contributions: S.-T.Y. and C.-Y.C. designed research, performed research, analyzed respect to the norm mentioned above) between the 2 normed data, and wrote the paper. 0 0 The authors declare no conflict of interest. vector spaces H (M, mKM ) and H (M , mKM ). We claim that with a few exceptional cases of M and M, the linear isometry This article is a PNAS Direct Submission. is induced by a birational map between M and M . This can be 1To whom correspondence should be addressed. E-mail: [email protected]. considered as a Torelli-type theorem in birational geometry. © 2008 by The National Academy of Sciences of the USA 18696–18701 PNAS December 2, 2008 vol. 105 no. 48 www.pnas.org / cgi / doi / 10.1073 / pnas.0809030105 Downloaded by guest on September 27, 2021 η1 η2 a1 an b where and are viewed as meromorphic functions on M and We abbreviate (x1, y1, ..., xn, yn), (z1, ..., zn), z ...z , |z1| 1 ... η0 η0 1 n | |bn A | |B η0m is a real nonnegative continuous (n, n)-form as defined in zn , and dx1dy1 ...dxndyn as (X, Y ), Z, Z , Z , and dX dY , 0 2.1. H (M, mKM ) is then given with the structure of a hermitian respectively. Let manifold. This hermitian structure is closely related to the norm 0 A 2 2B we mentioned above. Actually the norm function on H (M, mKM ) (t) = χ(X, Y )|Z + tφ(Z)| m |Z| dXdY . is the Kähler potential of this hermitian metric in a suitable sense 0 (C.-Y.C. and S.-T.Y., unpublished work). Theorem 2.2.1. ⎧ 0 µ 2. Pseudonorms on H (M, mKM ) and their asymptotic ⎪ O |t| ln 1 if 2l + 2 ≥ 1; ⎪ |t| m properties ⎨ µ−1 + 2 | |2l m 1 2.1. The Pseudonorm m. Let M be a complex manifold of dimen- (t) − (0) = c(A, B, φ) t ln |t| 0 ⎪ 2 ∈ ⎪ µ−1 if 2l + < 1, sion n. To every η H (M, mKM ) we can associate a real ⎪ + 2 m ⎩ + | |2l m 1 nonnegative continuous (n, n)-form on M, denoted as ηm,as o t ln |t| follows: U ={ j = j + j n } Let (U,(wU uU ivU )j=1) be an open cover of where c(A, B, φ) is a real number depending on φ. In the last case we 1 n ⊗m ≥ | = ∧ ··· ∧ have c(A, B, φ) 0, and M of coordinate charts. If η U ηU (dwU dwU ) with ηU ∈ OM (U), we can define on U a real nonnegative continuous c(A, B, φ) = 0 ⇐⇒ φ(0, ...,0,zµ+1, ..., zn) ≡ 0. (n, n) form Remark 2.2.2. More specific information when 2l + 2 ≥ 1 can be 2 1 1 n n m | =| | m ∧ ···∧ ∧ 2 η U m ηU duU dvU duU dvU + + given. We actually can show that when k < 2l m < k 1 where (j) { | } ∈U k ∈ N, (0) exists for j ≤ k. In addition, we get in this case an and can verify that η U m U does give a globally defined form, denoted as η . It is routine to see that this definition does not k j m asymptotic expansion for (t)− (0) tj. We will not need these depend on the choice of U. j=0 j! If M is compact, we define in this article so we omit them here (C.-Y.C., unpublished work). In 2.4, we will apply this result to obtain the main global result. ηm = ηm 0 Let η0, η ∈ H (M, mKM ). We hope to describe the asymptotic M behavior of η0 + tη as t → 0. and will abbreviate it as η if m is clear in the context.
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