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Jun-muk Hwang - One of the best experts on this subject based on the ideXlab platform.

  • Base manifolds for fibrations of projective irreducible symplectic manifolds
    Inventiones mathematicae, 2008
    Co-Authors: Jun-muk Hwang
    Abstract:

    Given a projective irreducible symplectic manifold M of dimension 2 n , a projective manifold X and a surjective Holomorphic Map f : M → X with connected fibers of positive dimension, we prove that X is biHolomorphic to the projective space of dimension n . The proof is obtained by exploiting two geometric structures at general points of X : the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X .

  • BIRATIONALITY OF THE TANGENT Map FOR MINIMAL RATIONAL CURVES
    Asian Journal of Mathematics, 2004
    Co-Authors: Jun-muk Hwang, Ngaiming Mok
    Abstract:

    For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a Holomorphic Map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biHolomorphic Map.postprin

  • Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles
    Journal of Algebraic Geometry, 2003
    Co-Authors: Jun-muk Hwang, Ngaiming Mok
    Abstract:

    Let X be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and f : X′ → X be a generically finite surjective Holomorphic Map from a projective manifold X′ onto X. When the domain manifold X′ is fixed and the target manifold X is a priori allowed to deform we prove that the Holomorphic Map f : X′ → X is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563– 575) for the analogous situation where the target manifold X is a Fano manifold of Picard number 1 on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold X′ of Picard number 1, we prove a finiteness result for generically finite surjective Holomorphic Maps of X′ onto Fano manifolds (necessarily of Picard number 1) admitting rational curves with trivial normal bundles. As a consequence, any 3-dimensional Fano manifold of Picard number 1 can only dominate a finite number of isomorphism classes of projective

Donovan Humphries - One of the best experts on this subject based on the ideXlab platform.

  • S. Diverio - Kobayashi hyperbolicity of complex projective manifolds and foliations (Part 4)
    2019
    Co-Authors: Simone Diverio, Fanny Bastien, Donovan Humphries
    Abstract:

    The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and Holomorphic foliations. A compact complex space is Kobayashi hyperbolic if and only if every Holomorphic Map from the complex plane to it is constant. Projective (or more generally compact Kähler) Kobayashi hyperbolic manifolds share many features with projective manifolds of general type, and it is nowadays a classical and important conjecture (due to S. Lang) that a complex projective manifold should be hyperbolic if and only if it is of general type together with all of its subvarieties. One essential tool in this business (introduced by Green-Griffiths, and later refined by Demailly) are the so-called (invariant) jet differentials: they are algebraic differential equations which every Holomorphic image of the complex plane must satisfy, provided they are with values in an anti-ample divisor. The abundance of such jet differentials provide then a strong constraint to the existence of non constant Holomorphic Map from the complex plane. In this series of lectures we shall first of all introduce the basic notions and facts about Kobayashi hyperbolicity, explain what jet differentials are, and how to use them. Next, we shall describe a series of counterexamples built using Holomorphic foliations in an essential way, which explain why jet differentials are not enough to obtain results on hyperbolicity of projective manifolds in full generality (even if lots of spectacular results have been obtained in the last decades in special cases). Last, if time permits, we shall overview (in a toy case) McQuillan’s celebrated proof of the the fact the a projective surface of general type with positive second Segre number is "almost" hyperbolic: again this is a combination of jet differentials and Holomorphic foliations.

  • S. Diverio - Kobayashi hyperbolicity of complex projective manifolds and foliations (Part 3)
    2019
    Co-Authors: Simone Diverio, Fanny Bastien, Donovan Humphries
    Abstract:

    The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and Holomorphic foliations. A compact complex space is Kobayashi hyperbolic if and only if every Holomorphic Map from the complex plane to it is constant. Projective (or more generally compact Kähler) Kobayashi hyperbolic manifolds share many features with projective manifolds of general type, and it is nowadays a classical and important conjecture (due to S. Lang) that a complex projective manifold should be hyperbolic if and only if it is of general type together with all of its subvarieties. One essential tool in this business (introduced by Green-Griffiths, and later refined by Demailly) are the so-called (invariant) jet differentials: they are algebraic differential equations which every Holomorphic image of the complex plane must satisfy, provided they are with values in an anti-ample divisor. The abundance of such jet differentials provide then a strong constraint to the existence of non constant Holomorphic Map from the complex plane. In this series of lectures we shall first of all introduce the basic notions and facts about Kobayashi hyperbolicity, explain what jet differentials are, and how to use them. Next, we shall describe a series of counterexamples built using Holomorphic foliations in an essential way, which explain why jet differentials are not enough to obtain results on hyperbolicity of projective manifolds in full generality (even if lots of spectacular results have been obtained in the last decades in special cases). Last, if time permits, we shall overview (in a toy case) McQuillan’s celebrated proof of the the fact the a projective surface of general type with positive second Segre number is "almost" hyperbolic: again this is a combination of jet differentials and Holomorphic foliations.

  • S. Diverio - Kobayashi hyperbolicity of complex projective manifolds and foliations (Part 2)
    2019
    Co-Authors: Simone Diverio, Fanny Bastien, Donovan Humphries
    Abstract:

    The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and Holomorphic foliations. A compact complex space is Kobayashi hyperbolic if and only if every Holomorphic Map from the complex plane to it is constant. Projective (or more generally compact Kähler) Kobayashi hyperbolic manifolds share many features with projective manifolds of general type, and it is nowadays a classical and important conjecture (due to S. Lang) that a complex projective manifold should be hyperbolic if and only if it is of general type together with all of its subvarieties. One essential tool in this business (introduced by Green-Griffiths, and later refined by Demailly) are the so-called (invariant) jet differentials: they are algebraic differential equations which every Holomorphic image of the complex plane must satisfy, provided they are with values in an anti-ample divisor. The abundance of such jet differentials provide then a strong constraint to the existence of non constant Holomorphic Map from the complex plane. In this series of lectures we shall first of all introduce the basic notions and facts about Kobayashi hyperbolicity, explain what jet differentials are, and how to use them. Next, we shall describe a series of counterexamples built using Holomorphic foliations in an essential way, which explain why jet differentials are not enough to obtain results on hyperbolicity of projective manifolds in full generality (even if lots of spectacular results have been obtained in the last decades in special cases). Last, if time permits, we shall overview (in a toy case) McQuillan’s celebrated proof of the the fact the a projective surface of general type with positive second Segre number is "almost" hyperbolic: again this is a combination of jet differentials and Holomorphic foliations.

  • S. Diverio - Kobayashi hyperbolicity of complex projective manifolds and foliations (part 1)
    2019
    Co-Authors: Simone Diverio, Fanny Bastien, Donovan Humphries
    Abstract:

    The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and Holomorphic foliations. A compact complex space is Kobayashi hyperbolic if and only if every Holomorphic Map from the complex plane to it is constant. Projective (or more generally compact Kähler) Kobayashi hyperbolic manifolds share many features with projective manifolds of general type, and it is nowadays a classical and important conjecture (due to S. Lang) that a complex projective manifold should be hyperbolic if and only if it is of general type together with all of its subvarieties. One essential tool in this business (introduced by Green-Griffiths, and later refined by Demailly) are the so-called (invariant) jet differentials: they are algebraic differential equations which every Holomorphic image of the complex plane must satisfy, provided they are with values in an anti-ample divisor. The abundance of such jet differentials provide then a strong constraint to the existence of non constant Holomorphic Map from the complex plane. In this series of lectures we shall first of all introduce the basic notions and facts about Kobayashi hyperbolicity, explain what jet differentials are, and how to use them. Next, we shall describe a series of counterexamples built using Holomorphic foliations in an essential way, which explain why jet differentials are not enough to obtain results on hyperbolicity of projective manifolds in full generality (even if lots of spectacular results have been obtained in the last decades in special cases). Last, if time permits, we shall overview (in a toy case) McQuillan’s celebrated proof of the the fact the a projective surface of general type with positive second Segre number is "almost" hyperbolic: again this is a combination of jet differentials and Holomorphic foliations.

Kaushal Verma - One of the best experts on this subject based on the ideXlab platform.

  • Condition R and proper Holomorphic Maps between equidimensional product domains
    Advances in Mathematics, 2013
    Co-Authors: Debraj Chakrabarti, Kaushal Verma
    Abstract:

    We consider proper Holomorphic Mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such Mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper Holomorphic Map splits as a product of proper Mappings between the factor domains. (C) 2013 Elsevier Inc. All rights reserved.

  • Condition R and proper Holomorphic Maps between equidimensional product domains
    arXiv: Complex Variables, 2012
    Co-Authors: Debraj Chakrabarti, Kaushal Verma
    Abstract:

    We consider proper Holomorphic Mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such Mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper Holomorphic Map splits as product of proper Mappings between the factor domains.

Indranil Biswas - One of the best experts on this subject based on the ideXlab platform.

  • Holomorphic BUNDLES TRIVIALIZABLE BY PROPER SURJECTIVE Holomorphic Map
    2020
    Co-Authors: Indranil Biswas, Sorin Dumitrescu
    Abstract:

    Given a compact complex manifold M , we investigate the Holomorphic vector bundles E on M such that ϕ * E is trivial for some surjective Holomorphic Map ϕ, to M , from some compact complex manifold. We prove that these are exactly those Holomorphic vector bundles that admit a flat Holomorphic connection with finite monodromy homomorphism. A similar result is proved for Holomorphic principal G-bundles, where G is a connected reductive complex affine algebraic group.

  • a criterion for a degree one Holomorphic Map to be a biholomorphism
    arXiv: Complex Variables, 2016
    Co-Authors: Gautam Bharali, Indranil Biswas, Georg Schumacher
    Abstract:

    Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective Holomorphic Map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied.

  • RIGIDITY OF Holomorphic MapS BETWEEN FIBER SPACES
    International Journal of Mathematics, 2014
    Co-Authors: Gautam Bharali, Indranil Biswas
    Abstract:

    In the study of Holomorphic Maps, the term ``rigidity'' refers to certain types of results that give us very specific information about a general class of Holomorphic Maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one Holomorphic Map f :Y -> X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for Holomorphic self-Maps of the total space of a Holomorphic fiber space. Lastly, we consider products X = X-1 x X-2 and Y = Y-1 x Y-2 of compact connected complex manifolds. When X-1 is a Riemann surface of genus >= 2, we show that any non-constant Holomorphic Map F:Y -> X is of a special form.

  • Rigidity of Holomorphic Maps between fiber spaces
    arXiv: Complex Variables, 2013
    Co-Authors: Gautam Bharali, Indranil Biswas
    Abstract:

    In the study of Holomorphic Maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of Holomorphic Maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds $X$ and $Y$, a degree-one Holomorphic Map $f: Y\to X$ is a biholomorphism. Given that the real manifolds underlying $X$ and $Y$ are diffeomorphic, we provide a condition under which $f$ is a biholomorphism. Using this result, we deduce a rigidity result for Holomorphic self-Maps of the total space of a Holomorphic fiber space. Lastly, we consider products $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ of compact connected complex manifolds. When $X_1$ is a Riemann surface of genus $\geq 2$, we show that any non-constant Holomorphic Map $F:Y\to X$ is of a special form.

Israel Or Weinstein - One of the best experts on this subject based on the ideXlab platform.