Symplectic Vector Bundle Research Articles

HECKE CORRESPONDENCE FOR SYMPLECTIC BUNDLES WITH APPLICATION TO THE PICARD BUNDLES

We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let [Formula: see text] be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg (E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on [Formula: see text], which is a projective bundle whose fiber over any point [Formula: see text] is ℙ(H0(X,E)). We prove that this projective bundle is stable.

Residues of Chern-Maslov classes

We describe a localization theory for Maslov classes associated with two Lagrangian subbundles in a real symplectic vector bundle and give a definition of the residue of the Maslov classes. We also compute explicitly the residue of the first Maslov class in the case that the non-transversal set of the two Lagrangian subbundles have codimension 1.

NOTE ON NORMAL EMBEDDING

It was shown by L. Polterovich ([3]) that if L is a totally real submanifold of a symplectic manifold <TEX>$(M,\omega)$</TEX> and L is parallelizable then L is normal. So we try to find an answer to the question of whether there is a compatible almost complex structure J on the symplectic vector bundle <TEX>$TM<TEX>$\mid$</TEX>_{L}$</TEX> such that <TEX>$TL{\cap}JTL=0$</TEX> assuming L is normal and parallelizable. Although we could not reach an answer, we observed that the claim holds at the vector space level. And related to the question, we showed that for a symplectic vector bundle <TEX>$(M,\omega)$</TEX> of rank 2n and <TEX>$E=E_1{\bigoplus}E_2$</TEX>, where <TEX>$E=E_1,E_2$</TEX>are Lagrangian subbundles of E, there is an almost complex structure J on E compatible with <TEX>${\omega}$</TEX> and <TEX>$JE_1=E_2$</TEX>. And finally we provide a necessary and sufficient condition for a given embedding into a symplectic manifold to be normal.

Partially hyperbolic geodesic flows are Anosov

We prove that if a Z or R -action by symplectic linear maps on a symplectic vector bundle E has a weakly dominated invariant splitting E= S⊕ U with dim U=dim S, then the action is hyperbolic. In particular, contact and geodesic flows with a dominated splitting with dim S=dim U are Anosov. To cite this article: G. Contreras, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 585–590.

Projective Structure, Symplectic Connection and Quantization

Let X be a connected Riemann surface equipped with a projective structure \(\mathfrak{p}\). Let E be a holomorphic symplectic vector bundle over X equipped with a flat connection. There is a holomorphic symplectic structure on the total space of the pullback of E to the space of all nonzero holomorphic cotangent vectors on X. Using \(\mathfrak{p}\), this symplectic form is quantized. A moduli space of Higgs bundles on a compact Riemann surface has a natural holomorphic symplectic structure. Using \(\mathfrak{p}\), a quantization of this symplectic form over a Zariski open subset of the moduli space of Higgs bundles is constructed.

Residue classes of Lagrangian subbundles and Maslov classes

For Lagrangian subbundles with singularities in symplectic vector bundles, explicit formulas of relation between their residue classes and Maslov classes outside singularities are obtained. Then a Lagrangian subbundle with singularity is constructed where all possible Maslov classes are nonzero but residue classes vanish for dimension &gt; 2 &gt; 2 . Moreover, a Lagrangian immersion with singularity is constructed, where the similar property for the associated Maslov classes and residue classes is shown.

Chern-Simons-Maslov classes of some symplectic vector bundles

Let E 0 , J 0 {E_0},\;{J_0} , and L 0 {L_0} be the symplectic 2 n 2n -vector bundle, the compatible complex operator, and the Lagrangian subbundle that are determined by the U ( n ) U(n) -extension of the principal O ( n ) O(n) -bundle U ( n ) → U ( n ) / O ( n ) U(n) \to U(n)/O(n) . We compute the Chern-Simons-Maslov class μ 1 ( E 0 , J 0 , L 0 ) {\mu ^1}({E_0},{J_0},{L_0}) . Then for a trivial symplectic 2 n 2n -bundle E E , a compatible complex operator J J , and a Lagrangian subbundle L L , we compute Chern-Simons-Maslov classes μ h ( E , J , L ) {\mu ^h}(E,J,L) under some condition on the base space of E E .

BIRATIONAL GEOMETRY OF MODULI VARIETIES OF VECTOR BUNDLES OVER $ \mathbf P^2$

The fields of rational functions on the moduli varieties of stable (symplectic) vector bundles over the projective plane are described. As a consequence the rationality of some series of moduli varieties is proved.

Classifying pairs of lagrangians in a hermitian vector space

A new elementary geometric proof, exploiting the positive curvature of complex projective space, of a basic lemma in the theory of lagrangian pairs in hermitian vector space is presented. Applications to the classification of such pairs and to symplectic vector bundles possessing a pair of lagrangian subbundles are given.

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C ∗-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.

On the pairing of polarizations

IfF is a positive Lagrangian sub-bundle of a symplectic vector bundle (E, ω) we show by elementary means that the Chern classes ofF are determined by ω. The notions of metaplectic structure for (E, ω), metalinear structure forF and square root ofKF, the canonical bundle ofF are shown to be essentially the same. IfF andG are two positive Lagrangian sub-bundles with\(F \cap \bar G = D^\mathbb{C} \), we define a pairing ofKF andKG into the bundleD−2(D) of densities of order −2 onD. This is the square of Blattner's half-form pairing and so characterizes the latter up to a sign.