Showing papers on "Frame bundle published in 1999"
TL;DR: In this article, a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional information included in the vector case) is presented.
Abstract: This paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional as well as multiplicity information included in the vector case). This new formalism unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles over an abstract Riemann surface) with an earlier approach of the authors (where the Riemann surface was assumed to be the normalizing Riemann surface for an algebraic curve embedded in C 2 with determinantal representation, and the vector bundles were assumed to be presented as the kernels of linear matrix pencils). The main tool is a version of the Cauchy kernel appropriate for flat vector bundles over the Riemann surface. Our formula for the interpolating bundle map (in the special case of a single zero and a single pole) can be viewed as a generalization of the Fay trisecant identity from the usual line bundle case to the vector bundle case in terms of Cauchy kernels. In particular we obtain a new proof of the Fay trisecant identity.
62 citations
TL;DR: In this article, the authors identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on a punctured RiemANN surface.
Abstract: We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L^2 complex. As a consequence, we obtain a Hard Lefschetz-type theorem.
49 citations
TL;DR: In this article, it was shown that the maximal dimension of a linear subspace of the space of symmetric matrices such that each nonzero element has even rank (m-r+1) is O(m − r+1).
Abstract: Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We use this result to solve the constant rank problem for symmetric matrices, proving that the maximal dimension of a linear subspace of the space of $m\times m$ symmetric matrices such that each nonzero element has even rank $r \ge 2$ is $m-r+1$. We explain how this result relates to the study of dual varieties in projective geometry and give some applications and examples.
48 citations
Posted Content•
TL;DR: In this article, the authors studied the integrability conditions of a manifold with Spin(9)-structure and derived the corresponding differential equation for the unique self-dual 8-form differential equation assigned to any type of spin-9-structure.
Abstract: The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame bundle of a 16-dimensional compact manifold (Stiefel-Whitney and Pontrjagin classes). We compute the homotopy groups $\pi_i (X^{84})$ of the space $X^{84}= SO(16) / Spin(9)$ for $i \le 14$. Next we introduce different geometric types of $Spin(9)$-structures and derive the corresponding differential equation for the unique self-dual 8-form $\Omega^8$ assigned to any type of $Spin(9)$-structure. Finally we construct the twistor space of a 16-dimensional manifold with $Spin(9)$-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.
44 citations
TL;DR: A complete description of all fiber product preserving bundle functors on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps is given in this article.
Abstract: We present a complete description of all fiber product preserving bundle functors on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. This result is based on several general properties of such functors, which are deduced in the first two parts of the paper.
39 citations
Posted Content•
TL;DR: In this article, the authors show that for the tautological bundle tensor L tensor the determinant bundle associated to A vanish and the space of global sections is computed in terms of $H^0(A)$ and$H^ 0(L\otimes A)$ on the moduli space of rank 2 semi-stable sheaves on the projective plane, supporting Le Potier's Strange duality conjecture.
Abstract: We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by the computation of the space of global sections of the determinant bundle on the moduli space of rank 2 semi-stable sheaves on the projective plane, supporting Le Potier's Strange duality conjecture on the projective plane.
35 citations
TL;DR: In this paper, a Kaehler structure on the bundle of nonzero tangent vectors to a Riemannian manifold of constant positive sectional curvature was obtained by a Lagrangian depending on the density energy only.
Abstract: We obtain a Kaehler structure on the bundle of nonzero tangent vectors to a Riemannian manifold of constant positive sectional curvature This Kaehler structure is determined by a Lagrangian depending on the density energy only
32 citations
TL;DR: In this paper, a line bundle over the Brownian bridge is defined by using its section, which allows us to define a Hilbert space of spinor fields over the bridge when the first Pontryaguin class of the spin bundle is equal to 0.
Abstract: We give the definition of a line bundle over the Brownian bridge by using its section. This allows us to define a Hilbert space of sections of a line bundle over the Brownian bridge associated to the transgression of a representative of an element of H3(M;Z). We consider the case of a string structure over the Brownian bridge: this allows us to define a Hilbert space of spinor fields over the Brownian bridge, when the first Pontryaguin class of the spin bundle over the manifold is equal to 0.
25 citations
01 Jan 1999
22 citations
Posted Content•
TL;DR: F-manifolds with a commutative and associative multi-plication on the tangent bundle are called F-Manifolds and are closely related to discrim-inants and Lagrange maps as discussed by the authors.
Abstract: Manifolds with a commutative and associative multi- plication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability con- dition. They are studied here. They are closely related to discrim- inants and Lagrange maps. Frobenius manifolds are F-manifolds. As an application a conjecture of Dubrovin on Frobenius manifolds and Coxeter groups is proved.
21 citations
TL;DR: In this paper, stable holomorphic vector bundles on elliptically fibered Calabi-Yau n-fold Zn were constructed in terms of F-theory compactifications on local singularities.
Abstract: We use a recently proposed construction of stable holomorphic vector bundles V on elliptically fibered Calabi-Yau n-fold Zn in terms of F-theory compactifications on local singularities to describe stability conditions on V. Specifically, the requirement that the F-theory compactification manifold is Calabi-Yau implies a a stability criterion on V which is formulated in terms of the existence of holomorphic sections of certain line bundles.
TL;DR: In this article, the authors compute wave trace invariants for Dirac type systems of differential operators acting on sections of a hermitian vector bundle over a compact manifold. But they do not define the wavetrace invariants themselves.
Abstract: In thls article, the author computes certain spectral invariants known as the wavetrace invariants for Dirac type systems of differential operators acting on sections of a hermitian vector bundle E over a compact manifold
07 Dec 1999
TL;DR: In this paper, the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold M = XO U X1 has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families over the two halves.
Abstract: It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold M = XO U X1 has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves
TL;DR: In this paper, it was shown that ML(r, cl, c2) and ML(2, c 2) are generically smooth of the expected dimension provided c 2 is large enough ([D]).
Abstract: In the 1980’s, Donaldson proved that $ and ML(2; cl, c2) are generically smooth of the expected dimension provided c2 is large enough ([D]). As a consequence he obtained some spectacular new results on classification of $ four manifolds. Since then, many mathematicians have studied the structure of the moduli spaces $ (resp. ML(r; cl, c2)) from the point of view of algebraic geometry, of topology and of differential geometry; giving very pleasant connections between these areas.
01 Jan 1999
TL;DR: A complete description of all fiber product preserving bundle functc'rs on tibered manifolds with m-dimensional bases and fiber preservin g maps with local diffeomorphisms as base maps is given in this paper.
Abstract: Ahsmrc~t: We present a complete description of all fiber product preserving bundle functc’rs on the category of tibered manifolds with m-dimensional bases and fiber preservin g maps with local diffeomorphisms as base maps. This result is based on several general properties of such functors, which are dec.uced in the tirst two part\ of the paper. K~~~~nvrt/.r: I3undle functor. Weil bundle, jet bundle, natural transformation. .41S C/C/.\ c(fic~triott: 58AO5, SXA20.
TL;DR: In this paper, Liouville distribution on the tangent bundle of a pseudo-Finsler manifold is defined and proved integrable, and geometric properties of both leaves of the Liouve distribution and the vertical distribution are analyzed.
Abstract: We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.
29 Jan 1999
Journal Article•
TL;DR: In this article, the authors prove the existence of two natural families C(+) and C(-) of curves on the projective plane P(2), such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X).
Abstract: Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X). The general curve C of any of these two families is a section of the conic bundle projection, and our approach relates such C to a maximal subbundle of a rank 2 vector bundle E(C) on C, or -- to a minimal section of the ruled surface P(E(C)). The families C(+) and C(-) correspond to the two possible types of versal deformations of ruled surfaces over curves of fixed genus g(C). As an application, we find parameterizations of J(X) and (Theta) for certain classes of Fano threefolds, and study the sets Sing(Theta) of the singularities of (Theta).
TL;DR: In this paper, the k-jet ampleness and spannedness for a vector bundle, E, on a projective manifold were studied. And different characterizations of projective space in terms of such positivity properties for E were obtained.
Abstract: We introduce and study the k-jet ampleness and the k-jet spannedness for a vector bundle, E , on a projective manifold. We obtain different characterizations of projective space in terms of such positivity properties for E . We compare the 1-jet ampleness with different notions of very ampleness in the literature.
01 Jan 1999
TL;DR: In this article, the vector bundle value version of Kohn's closed range theorem over strongly pseudoconvex CR manifolds is proved, which is a crucial step toward CR construction of the semi-universal family of normal isolated surface singularities.
Abstract: We prove the vector bundle value version of J. J. Kohn’s closed range theorem over three dimensional strongly pseudoconvex CR manifolds. That closed range theorem is a crucial step toward CR construction of the semi-universal family of normal isolated surface singularities (cf. [11]).
TL;DR: In this article, the existence of Hermitian-Einstein metric on the parabolic Higgs bundle and its stability on an open Riemann surface with a finite set of punctures is established.
Abstract: Let M be an open Riemann surface with a finite set of punctures, a complete Poincare-like metric is introduced near the punctures and the equivalence between the stability of an indecomposable parabolic Higgs bundle, and the existence of a Hermitian-Einstein metric on the bundle is established.
01 Jan 1999
TL;DR: In this paper, the second-order connections on a fibered manifold Y are studied in the sense of C. Ehresmann's connection from Y into the second nonholonomic jet extension of Y.
Abstract: . We study the second order connections in the sense of
C. Ehresmann. On a fibered manifold Y, such a connection is a
section from Y into the second non-holonomic jet prolongation
of Y. Our main aim is to extend the classical theory to the
functional bundle of all smooth maps between the fibers over
the same base point of two fibered manifolds over the same
base. This requires several new geometric results about the
second order connections on Y, which are deduced in the first
part of the paper.
TL;DR: In this article, the sets of compatible connections are determined and a canonical connection is defined, and the structure of electromagnetic type on a vector bundle is introduced and studied for the metric case.
Abstract: Structures of electromagnetic type on a vector bundle are introduced and studied. The metric case is also defined and studied. The sets of compatible connections are determined and a canonical connection is defined.
TL;DR: In this article, the authors give an effective bound d0 in positive characteristic when E is a rank-2 vector bundle on a surface: if d ≥ d0 the restriction E|C of E to a general member C ∈ |O(d)| is semistable.
Abstract: Let E be a rank-r torsion free sheaf on a normal projective variety of dimension n ≥ 2 defined over an algebraically closed field k. Assume that E is semistable with respect to a very ample line bundle O(1): Namely, if we set μ(F ) = (c1(F ) · O(1)n−1) for a subsheaf F of E, μ(E) ≥ μ(F ) holds for all subsheaf F of E. A problem of finding a condition when the restriction E|C to a member C ∈ |O(d)| is semistable on C has been considered by several authors ([1], [3], [6], [7], [8]): Maruyama [6] proved that if r < n then E|C is semistable for general C ∈ |O(d)| for every d ≥ 1; Mehta and Ramanathan [7] proved that there exists an integer d0 such that if d ≥ d0 then E|C is semistable for general C ∈ |O(d)|; Flenner [3] proved that if k is of characteristic 0 and d satisfies ( d+n d )−d−1 d > (O(1)n) · max( r 2−1 4 , 1) then E|C is semistable for general C ∈ |O(d)|. In other direction, in characteristic 0, Bogomolov [1] and Moriwaki [8] obtained an effective bound d0 for some special restriction E|C to be semistable. The purpose here is to give an effective bound d0 in positive characteristic when E is a rank-2 vector bundle on a surface: If d ≥ d0 the restriction E|C of E to a general member C ∈ |O(d)| is semistable. Our result is the following.
TL;DR: In this article, the equivalence classes of almost principal bundles with a fixed base B and a fixed structure group G are investigated, along with its subgroup of equivalence class of principal G-bundles over B, and also the groups of classes of these bundles with respect to the morphisms of the category C of decomposable bundles.
Abstract: The class A of bundles with the following properties is investigated: each bundle in A is the composition of a regular cover and a principal bundle (over the covering space) with Abelian structure group; the standard fibre G of this decomposable bundle is a Lie group; the bundle has an atlas with multivalued transition functions taking values in the group G. The equivalence class of such an atlas will be called an almost principal bundle structure. The group of equivalence classes of almost principal bundles with a fixed base B and a fixed structure group G is computed, along with its subgroup of equivalence classes of principal G-bundles over B, and also the groups of equivalence classes of these bundles with respect to the morphisms of the category C of decomposable bundles. A base and an invariant are found for an almost principal bundle that is not isomorphic to a principal bundle even in the category C. Applications are considered to the variational problem with fixed ends for multivalued functionals.