Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles

Vadim V. Shurygin; Svetlana K. Zubkova

Displaying similar documents to “Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles”

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 $P^{r} : ℳ f_{m} \to ℱ ℳ$ be the rth order frame bundle functor. Let $F : ℳ f_{m} \to ℱ ℳ$ and $G : ℳ f_{k} \to ℱ ℳ$ be natural bundles, where $k = d i m (P^{r} ℝ^{m})$ . We describe all $ℳ f_{m}$ -natural operators A transforming sections σ of $F M \to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G (P^{r} M) \to P^{r} M$ . 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.

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...

Constructions on second order connections

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

Annales Polonici Mathematici

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We classify all $ℱ ℳ_{m, n}$ -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 $(Γ) : V^{A} Y \to J ² V^{A} Y$ on the vertical Weil bundle $V^{A} Y \to M$ corresponding to a Weil algebra A.

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.

A classification theorem on Fano bundles

Roberto Muñoz, Luis E. Solá Conde, Gianluca Occhetta (2014)

Annales de l’institut Fourier

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In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^{2} (X, ℤ) ≅ H^{4} (X, ℤ) ≅ ℤ$ . The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $ℙ (ℰ)$ , that allows us to obtain the cohomological invariants of $X$ and $ℰ$ . As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

On natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$

A. Ntyam, A. Mba (2009)

Annales Polonici Mathematici

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Some properties and applications of natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$ are presented.

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 $ℳ f_{m}$ -natural operators $B : Q ⟿ Q T^{A}$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_{M} (\nabla)$ 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 $ℳ f_{m}$ -natural operators $C : Q ⟿ (T ¹_{p - 1}, T * \otimes T * \otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_{M} (\nabla) : T ¹_{p - 1} M = ⨁_{M}^{p - 1} T M \to T * M \otimes T * M \otimes T M$ .

Pseudo-real principal Higgs bundles on compact Kähler manifolds

Indranil Biswas, Oscar García-Prada, Jacques Hurtubise (2014)

Annales de l’institut Fourier

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Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_{G}$ . We define pseudo-real principal $G$ -bundles on $X$ . These are generalizations of real algebraic principal $G$ -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$ -bundles. Their relationships with the usual...

The natural operators lifting projectable vector fields to some fiber product preserving bundles

W. M. Mikulski (2003)

Annales Polonici Mathematici

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Admissible fiber product preserving bundle functors F on $ℱ ℳ_{m}$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ ℳ_{m}$ all natural operators $B : T_{p r o j | ℱ ℳ_{m, n}} \to T F$ lifting projectable vector fields to F are classified.

Line bundles with partially vanishing cohomology

Burt Totaro (2013)

Journal of the European Mathematical Society

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Define a line bundle $L$ on a projective variety to be $q$ -ample, for a natural number $q$ , if tensoring with high powers of $L$ kills coherent sheaf cohomology above dimension $q$ . Thus 0-ampleness is the usual notion of ampleness. We show that $q$ -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that $q$ -ampleness is a Zariski open condition, which is not clear from the definition. ...

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

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All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r} E$ of E are given. Similar results are deduced for the r-jet prolongations $J_{v}^{r} E$ and $J^{[r]} E$ in place of $J^{r} E$ .

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 $^{m} (A) < \infty$ . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V, v) | V \in G (n, m), v \in V$ such that, for all P ∈ A, one has $^{m (n - m)} (V \in G (n, m) | (V, π_{V} (P)) \in Z) > 0$ . One can replace “for all P ∈ A” by “for...

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 $T_{| ℳ f ₙ} ⇝ T^{(0, 0)} (F_{| ℳ f ₙ}) *$ transforming vector fields to functions on the dual bundle functor $(F_{| ℳ f ₙ}) *$ . Next, we study the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ lifting 1-forms to $(F_{| ℳ f ₙ}) *$ . As an application we classify the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ for some well known vector bundle functors F.

Semistability of Frobenius direct images over curves

Vikram B. Mehta, Christian Pauly (2007)

Bulletin de la Société Mathématique de France

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Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p > 0$ . Given a semistable vector bundle $E$ over $X$ , we show that its direct image $F_{*} E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank- $p$ vector bundles $F_{*} L$ , where $L$ is a line bundle over $X$ .

On the stratification of the orbit space for the action of automorphisms on connections

Witold Kondracki, Jan Rogulski

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CONTENTSIntroduction..................................................................................................................................................5§1. Basic notions and notation.....................................................................................................................7 1.1. Automorphisms of principal bundles....................................................................................................7 1.2. Connections and parallel...

Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Lutz Hille, Markus Perling (2014)

Annales de l’institut Fourier

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Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$ . Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$ . The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective...

On the $γ$ -equivalence of semiholonomic jets

Miroslav Doupovec, Ivan Kolář (2019)

Archivum Mathematicum

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It is well known that the concept of holonomic $r$ -jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic $r$ -jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.

The gradient flow of Higgs pairs

Jiayu Li, Xi Zhang (2011)

Journal of the European Mathematical Society

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We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a Kähler surface $(M, ω)$ , and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition $(A_{0}, φ_{0})$ converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point $(A_{\infty}, φ_{\infty})$ of this functional. We also prove that the limiting Higgs pair $(A_{\infty}, φ_{\infty})$ can be extended smoothly to a vector bundle...

On lifts of projectable-projectable classical linear connections to the cotangent bundle

Anna Bednarska (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We describe all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $D : Q_{p r o j - p r o j}^{τ} ⇝ Q T^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D (\nabla)$ on cotangent bundles $T^{*} Y$ of $Y$ . We show that this problem can be reduced to finding $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $D : Q_{p r o j - p r o j}^{τ} ⇝ (T^{*}, \otimes^{p} T^{*} \otimes \otimes^{q} T)$ for $p = 2$ , $q = 1$ and $p = 3$ , $q = 0$ .

Lifting right-invariant vector fields and prolongation of connections

W. M. Mikulski (2009)

Annales Polonici Mathematici

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We describe all $_{m} (G)$ -gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^{r} P = P^{r} M \times_{M} J^{r} P$ of P → M. In other words, we classify all $_{m} (G)$ -natural transformations $J^{r} L P \times_{M} W^{r} P \to T W^{r} P = L W^{r} P \times_{M} W^{r} P$ covering the identity of $W^{r} P$ , where $J^{r} L P$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all $_{m} (G)$ -natural transformations which are similar to the Kumpera-Spencer isomorphism $J^{r} L P = L W^{r} P$ . We formulate axioms which...

Complex structures on product of circle bundles over complex manifolds

Parameswaran Sankaran, Ajay Singh Thakur (2013)

Annales de l’institut Fourier

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Let ${\bar{L}}_{i} \to X_{i}$ be a holomorphic line bundle over a compact complex manifold for $i = 1, 2$ . Let $S_{i}$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\bar{L}}_{i}$ . We construct complex structures on $S = S_{1} \times S_{2}$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\bar{L}}_{i}$ are equivariant ${(ℂ^{*})}^{n_{i}}$ -bundles satisfying some additional conditions....

Groups of $C^{r, s}$ -diffeomorphisms related to a foliation

Jacek Lech, Tomasz Rybicki (2007)

Banach Center Publications

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The notion of a $C^{r, s}$ -diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of $C^{r, s}$ -diffeomorphisms is proved. A remark on $C^{n + 1}$ -diffeomorphisms is given.

$𝒟$ -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

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We study the geometry of $𝒟$ -bundles—locally projective $𝒟$ -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of $𝒟$ -bundles; in particular, we prove that the local structure of $𝒟$ -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions...

Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections

Anna Bednarska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We classify all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\nabla$ on fibered-fibered manifolds $Y$ of dimension $(m_{1}, m_{2}, n_{1}, n_{2})$ into $r$ th order Lagrangians $A (r)$ on the fibered-fibered linear frame bundle $L^{f i b - f i b} (Y)$ on $Y$ . Moreover, we classify all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_{1}, m_{2}, n_{1}, n_{2})$ into Euler morphism $B (\nabla)$ on $L^{f i b - f i b} (Y)$ . These classifications can be expanded on...

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 $2 \leq k \leq n - 1$ . 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 $D$ .