Displaying similar documents to “Structure of a neighbourhood of a complex compact submanifold in a complex manifold”

The generic dimension of the first derived system

Robert P. Buemi (1978)

Annales de l'institut Fourier

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Any r -dimensional subbundle of the cotangent bundle on an n -dimensional manifold M partitions M into subsets M 0 , ... , M m ( m being the minimum of r and C ( n - r , 2 ) , the combinations of n - r things taken 2 at a time). M i is the set on which the first derived systems of the subbundle has codimension i . In this paper we prove the following: Theorem. Let s 2 and let Q be a generic C s r -dimensional subbundle of the cotangent bundle of an n -dimensional manifold M . The codimension...

Liftings of 1-forms to ( J r T * ) *

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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Let J r T * M be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let ( J r T * M ) * be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on ( J r T * M ) * is given.

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

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We classify all m , n -natural operators : J ² J ² V A transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections ( Γ ) : V A Y J ² V A Y on the vertical Weil bundle V A Y M corresponding to a Weil algebra A.

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold X have a pseudo-effective (instead of generically nef) determinant. A first consequence is that X is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of X is not covered by compact positive-dimensional analytic...

Almost invariant submanifolds for compact group actions

Alan Weinstein (2000)

Journal of the European Mathematical Society

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We define a C 1 distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G , there is a G -invariant submanifold C 1 -close to N . The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros...

The natural operators lifting 1-forms to some vector bundle functors

J. Kurek, W. M. Mikulski (2002)

Colloquium Mathematicae

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Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators T | f T ( 0 , 0 ) ( F | f ) * transforming vector fields to functions on the dual bundle functor ( F | f ) * . Next, we study the natural operators T * | f T * ( F | f ) * lifting 1-forms to ( F | f ) * . As an application we classify the natural operators T * | f T * ( F | f ) * for some well known vector bundle functors F.

On convergence sets of divergent power series

Buma L. Fridman, Daowei Ma, Tejinder S. Neelon (2012)

Annales Polonici Mathematici

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A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family y = φ s ( t , x ) = s b ( x ) t + b ( x ) t ² + of analytic curves in ℂ × ℂⁿ passing through the origin, C o n v φ ( f ) of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series f ( φ s ( t , x ) , t , x ) converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that E = C o n v φ ( f ) if...

A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli, Giovanni Felder (2008)

Annales scientifiques de l'École Normale Supérieure

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Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n -dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology H H 2 n ( 𝒟 n , 𝒟 n * ) of the algebra of differential operators on a formal neighbourhood...

Lifting to the r-frame bundle by means of connections

J. Kurek, W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let m and r be natural numbers and let P r : f m be the rth order frame bundle functor. Let F : f m and G : f k be natural bundles, where k = d i m ( P r m ) . We describe all f m -natural operators A transforming sections σ of F M M and classical linear connections ∇ on M into sections A(σ,∇) of G ( P r M ) P r M . We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

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We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on T A M , where T A is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

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Let  X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L . We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair ( X , L ) , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic...

Grauert's line bundle convexity, reduction and Riemann domains

Viorel Vâjâitu (2016)

Czechoslovak Mathematical Journal

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We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X . This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H 0 ( X , L ) separates each point of X , then X can be realized as a Riemann domain over the complex projective...

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let π V denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an m -measurable subset of ℝⁿ with m ( A ) < . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle ( V , v ) | V G ( n , m ) , v V such that, for all P ∈ A, one has m ( n - m ) ( V G ( n , m ) | ( V , π V ( P ) ) Z ) > 0 . One can replace “for all P ∈ A” by “for...

Complete noncompact submanifolds with flat normal bundle

Hai-Ping Fu (2016)

Annales Polonici Mathematici

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Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in n + p with flat normal bundle. We prove that if the second fundamental form A of M satisfies M i | A | α < , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and M | A | d < , d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L α -norm curvature in ℝ⁷ are considered.

The Group of Invertible Elements of the Algebra of Quaternions

Irina A. Kuzmina, Marie Chodorová (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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We have, that all two-dimensional subspaces of the algebra of quaternions, containing a unit, are 2-dimensional subalgebras isomorphic to the algebra of complex numbers. It was proved in the papers of N. E. Belova. In the present article we consider a 2-dimensional subalgebra ( i ) of complex numbers with basis 1 , i and we construct the principal locally trivial bundle which is isomorphic to the Hopf fibration.

Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

Shuzo Izumi (2015)

Annales Polonici Mathematici

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Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection b from the local ring n , b onto the space Z b of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and b induces the structure of an Artinian algebra on Z b . In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈...

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let D be an open set of a Stein manifold X of dimension n such that H k ( D , 𝒪 ) = 0 for 2 k n - 1 . We prove that D is Stein if and only if every topologically trivial holomorphic line bundle L on D is associated to some Cartier divisor 𝔡 on D .

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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We prove that the problem of finding all f m -natural operators B : Q Q T A lifting classical linear connections ∇ on m-manifolds M to classical linear connections B M ( ) on the Weil bundle T A M corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all f m -natural operators C : Q ( T ¹ p - 1 , T * T * T ) transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps C M ( ) : T ¹ p - 1 M = M p - 1 T M T * M T * M T M .

The natural operators T ( 0 , 0 ) T ( 1 , 1 ) T ( r )

Włodzimierz M. Mikulski (2003)

Colloquium Mathematicae

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We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor A ( f ) : T T ( r ) M T T ( r ) M on the vector r-tangent bundle T ( r ) M = ( J r ( M , ) ) * over M. This problem is reflected in the concept of natural operators A : T | f ( 0 , 0 ) T ( 1 , 1 ) T ( r ) . For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ( T ( r ) ) and we construct explicitly a basis of this module.

Complete pluripolar graphs in N

Nguyen Quang Dieu, Phung Van Manh (2014)

Annales Polonici Mathematici

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Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that Γ g ( F ) : = ( z , g ( z ) ) : z F is complete pluripolar in N + 1 . Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that Γ g ( D ) is complete pluripolar in N + 1 . These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math....

A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let E be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form id + K where K is compact. Assume that the characteristic fiber of E has the compact approximation property. Let n be the complex dimension of X and 0 q n . Then: If V X is a holomorphic vector bundle (of finite rank) with H q ( X , V ) = 0 , then dim H q ( X , V E ) &lt; . In particular, if dim H q ( X , 𝒪 ) = 0 , then dim H q ( X , E ) &lt; .

On canonical constructions on connections

Jan Kurek, Włodzimierz Mikulski (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We study  how a projectable general connection Γ in a 2-fibred manifold Y 2 Y 1 Y 0   and a general vertical connection Θ in Y 2 Y 1 Y 0 induce a general connection A ( Γ , Θ ) in Y 2 Y 1 .

The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

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Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators T p r o j | m , n T ( 0 , 0 ) ( F | m , n ) * transforming projectable vector fields on Y to functions on the dual bundle (FY)* for any m , n -object Y. Next, under some assumption on F we study natural operators T * h o r | m , n T * ( F | m , n ) * lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators T * h o r | m , n T * ( F | m , n ) * for some vector bundle functors F on fibered manifolds.

Approximation of sets defined by polynomials with holomorphic coefficients

Marcin Bilski (2012)

Annales Polonici Mathematici

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Let X be an analytic set defined by polynomials whose coefficients a , . . . , a s are holomorphic functions. We formulate conditions on sequences a 1 , ν , . . . , a s , ν of holomorphic functions converging locally uniformly to a , . . . , a s , respectively, such that the sequence X ν of sets obtained by replacing a j ’s by a j , ν ’s in the polynomials converges to X.