Yum-Tong Siu

Harvard University, Cambridge, Massachusetts, United States

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Publications (25)14.57 Total impact

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    Yum-Tong Siu
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    ABSTRACT: We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.
    09/2012;
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    Yum-Tong Siu
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    ABSTRACT: The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions.
    Science China Mathematics 04/2011; 54(8). · 0.50 Impact Factor
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    Yum-Tong Siu
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    ABSTRACT: We sketch a proof of the abundance conjecture that the Kodaira dimension of a compact complex algebraic manifold equals its numerical Kodaira dimension. The proof consists of the following three parts: (i) the case of numerical Kodaira dimension zero, (ii) the general case under the assumption of the coincidence of the numerically trivial foliation and fibration for the canonical bundle, and (iii) the verification of the coincidence of the numerically trivial foliation and fibration for the canonical bundle. Besides the use of standard techniques such as the L2 estimates of d-bar, the first part uses Simpson's method of replacing the flat line bundle in a nontrivial flatly twisted canonical section by a torsion flat line bundle. Simpson's method relies on the technique of Gelfond-Schneider for the solution of the seventh problem of Hilbert. The second part uses the semi-positivity of the direct image of a relative pluricanonical bundle. The third part uses the technique of the First Main Theorem of Nevanlinna theory and its use is related to the technique of Gelfond-Schneider in the first part.
    12/2009;
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    Yum-Tong Siu
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    ABSTRACT: This note is written for the Festschrift in honor of Professor Christer Kiselman. Multiplier ideal sheaves identify the location and the extent of the failure of crucial estimates. In this note we will discuss and explain the historic evolution of the notion of multiplier ideal sheaves, especially the interpretation from the viewpoint of destabilizing subsheaves in the context of terminating or bounding an infinite process. We will also discuss the approach of constructing rational curves in Fano manifolds by using dynamic multiplier ideal sheaves and singularity-magnifying complex Monge-Ampere equations. This approach is still under development with details in the process of being worked out. We will indicate where details still need to be worked out.
    03/2009;
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    Yum-Tong Siu
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    ABSTRACT: This article is written for the Proceedings of the Conference on Current Developments in Mathematics in Harvard University, November 16-17, 2007. It is an exposition of the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type. It lists and discusses the main techniques and explains how they are put together in the proof. Of the various main techniques some special attention is given to (i) the technique of discrepancy subspaces and (ii) the technique of subspaces of minimum additional vanishing.
    12/2008;
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    Yum-Tong Siu
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    ABSTRACT: In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425).
    Science in China Series A Mathematics 04/2008; · 0.70 Impact Factor
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    Yum-Tong Siu
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    ABSTRACT: This note discusses the problem of the effective termination of Kohn's algorithm for subelliptic multipliers for bounded smooth weakly pseudoconvex domains of finite type. We give a complete proof for the case of special domains of finite type and indicate briefly how this method is to be extended to the case of general bounded smooth weakly pseudoconvex domains of finite type.
    07/2007;
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    Yum-Tong Siu
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    ABSTRACT: This set of notes provides some additional explanatory material on the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type.
    05/2007;
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    Yum-Tong Siu
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    ABSTRACT: On August 5, 2005 in the American Mathematical Society Summer Institute on Algebraic Geometry in Seattle and later in several conferences I gave lectures on my analytic proof of the finite generation of the canonical ring for the case of general type. After my lectures many people asked me for a copy of the slides which I used for my lectures. Since my slides were quite sketchy because of the time limitation for the lectures, I promised to post later on a preprint server my detailed notes from which my slides were extracted. Here are my detailed notes giving the techniques and the proof.
    11/2006;
  • Yum-Tong Siu
    11/2006: pages 158-174;
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    Yum-Tong Siu
    11/2006: pages 169-192;
  • Yum-Tong Siu
    11/2006: pages 285-311;
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    Yum-Tong Siu
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    ABSTRACT: This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the deformational invariance of plurigenera for general compact algebraic manifolds. Finally its application to the conjecture of the finite generation of the canonical ring is explored and the use of complex algebraic geometry in complex Neumann estimates is discussed.
    Science in China Series A Mathematics 05/2005; · 0.70 Impact Factor
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    Yum-Tong Siu
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    ABSTRACT: This article discusses the recent transcendental techniques used in the proofs of the following three conjectures. (1)~The plurigenera of a compact projective algebraic manifold are invariant under holomorphic deformation. (2)~There exists no smooth Leviflat hypersurface in the complex projective plane. (3)~A generic hypersurface of sufficiently high degree in the complex projective space is hyperbolic in the sense that there is no nonconstant holomorphic map from the complex Euclidean line to it.
    12/2002;
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    Yum-Tong Siu, Sai-Kee Yeung
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    ABSTRACT: This is an addendum to our earlier paper on the defect of an ample divisor of an abelian variety. It modifies an argument of the original paper to handle one difficulty there. At the same time the modification improves the result in the original paper by replacing the counting function by one truncated at a multiplicity given by an explicit function of the dimension of the abelian variety and the Chern number of the divisor.
    09/2002;
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    Yum-Tong Siu
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    ABSTRACT: Matsusaka's Big Theorem gives the very ampleness of mL for an ample line bundle L over an n-dimensional compact complex manifold X when m is no less than a number m 0 depending on L n and L nāˆ’1 K X . The de-pendence of m 0 on L n and L nāˆ’1 K X is not effective. An earlier result of the author gives an effective m 0 containing the factor L nāˆ’1 ((n + 2)L + K X) q with q of order 4 n . Demailly improved the order of q to 3 n by reducing the twisting required for the existence of nontrivial global holomorphic sections for anticanonical sheaves of subvarieties which occur in the verification of the numerical effectiveness of pL āˆ’ K X for some effective p. Twisted sections of anticanonical sheaves are needed to offset the addition of the canonical sheaf in vanishing theorems. We introduce here a technique to get a new bound with q of order 2 n . The technique avoids the use of sections of twisted anticanonical sheaves of subvarieties by transferring the use of vanishing the-orems on subvarieties to X and is more in line with techniques for Fujita conjecture type results.
    Houston Journal of Mathematics University of Houston Volume. 01/2002; 28.
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    Yum-Tong Siu
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    ABSTRACT: In this paper we prove the following theorem. Main Theorem. Let n >= 3 and m >= 3n/2 +7. Then there exists no C^m Levi-flat real hypersurface M in P_n. The condition that M is Levi-flat means that when M is locally defined by the vanishing of a C^m real-valued function f, at every point of M the restriction of d d-bar f to the complex tangent space of M is identically zero. The case of the nonexistence of C^\infty Levi-flat real hypersurface in P_2 is motivated by problems in dynamical systems in P_2.
    05/2000;
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    Yum-tong Siu
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    ABSTRACT: We explain the motivations and main ideas regarding the new techniques in hyperbolicity problems recently introduced by the author and Sai-Kee Yeung and by Michael McQuillan. Streamlined proofs and alternative approaches are given for previously known results. We say that a complex manifold is hyperbolic if there is no nonconstant holomorphic map from C to it. This paper discusses the new techniques in hyperbolicity problems introduced in recent years in a series of joint papers which I wrote with Sai-Kee Yeung [Siu and Yeung 1996b; 1996a; 1997] and in a series of papers by Michael McQuillan [McQuillan 1996; 1997]. The goal is to explain the motivations and the main ideas of these techniques. In the process we examine known results using new approaches, providing streamlined proofs for them. The paper consists of three parts: an Introduction, Chapter 1, and Chapter 2. The Introduction provides the necessary background, states the main problems, and discusses the motivations and the m...
    05/2000;
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    Yum-Tong Siu
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    ABSTRACT: The following conjecture on the deformation invariance of plurigenera is proved. For a smooth projective holomorphic family of compact complex manifolds over the open unit 1-disk such that all the fibers are of general type, every plurigenus of the fiber is independent of the fiber. The proof uses Nadel's multiplier ideal sheaves, Skoda's result on the generation of ideals with L2 estimates with respect to a plurisubharmonic weight, and the extension theorem of Ohsawa-Takegoshi-Manivel for holomorphic top-degree forms which are L2 with respect to a plurisubharmonic weight. Comment: 13 pages, LaTeX file
    Inventiones mathematicae 12/1997; · 2.26 Impact Factor
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    Yum-Tong Siu, Sai-Kee Yeung
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    ABSTRACT: The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let $f:{\bf C}\rightarrow A$ be a holomorphic map. Then the defect for the map $f$ and the divisor $D$ is zero. Corollary to Main Theorem. The complement of an ample divisor $D$ in an abelian variety $A$ is hyperbolic in the sense that there is no nonconstant holomorphic map from $\bf C$ to $A-D$.
    11/1996;

Publication Stats

519 Citations
14.57 Total Impact Points

Institutions

  • 1993–2011
    • Harvard University
      • Department of Mathematics
      Cambridge, Massachusetts, United States