Lagrangian Subbundles Research Articles

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Counting maximal Lagrangian subbundles over an algebraic curve

Let C be a smooth projective curve and W a symplectic bundle over C. Let LQe(W) be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves E⊂W of degree e. We give a closed formula for intersection numbers on LQe(W). As a special case, for g≥2, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [14].

Doubled aspects of Vaisman algebroid and gauge symmetry in double field theory

The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analog of the Drinfel’d double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel’d double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian sub-bundles (L,L̃) in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for (L,L̃) to be a Lie bialgebroid. The analysis provides a foundation toward the “coquecigrue problem” for the gauge symmetry in DFT.

Non-defectivity of Grassmannian bundles over a curve

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

Chern–Weil Maslov index and its orbifold analogue

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surfacewith boundary, which consists of symplectic vector bundle onand a Lagrangian subbundle on @� as well as its gen- eralization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold inte- rior singularites, via curvature integral, and find also an analogous topological definition in these cases.

Semi-basic 1-forms and courant structure for metrizability problems

The metrizability of sprays, particularly symmetric linear connections, is studied in terms of semi-basic 1-forms using the tools developed by Bucataru and Dahl in [2]. We introduce a type of metrizability in relationship with the Finsler and projective metrizability. The Lagrangian corresponding to the Finsler metrizability as well as the Bucataru{Dahl characterization of Finsler and projective metrizability are expressed by means of the Courant structure on the big tangent bundle of TM. A byproduct of our computations is that a at Riemannian metric, or generally an R-at Finslerian spray, yields two complementary, but not orthogonally, Dirac structures on TbigTM. These Dirac structures are also Lagrangian subbundles with respect to the natural almost symplectic structure of TbigTM.

Pseudo-Dirac structures

A Dirac structure is a Lagrangian subbundle of a Courant algebroid, L⊂E, which is involutive with respect to the Courant bracket. In particular, L inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, W⊂E, together with a pseudo-connection on its sections, satisfying a natural integrability condition. As a consequence of the definition, W will be a Lie algebroid. Allowing non-isotropic subbundles of E incorporates non-skew tensors and connections into Dirac geometry. Novel examples of pseudo-Dirac structures arise in the context of quasi-Poisson geometry, Lie theory, generalized Kähler geometry, and Dirac Lie groups, among others. Despite their greater generality, we show that pseudo-Dirac structures share many of the key features of Dirac structures. In particular, they behave well under composition with Courant relations.

A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve

A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

Equivariant symplectic homology of Anosov contact structures

We show that the differential in positive equivariant symplectic homology or linearized contact homology vanishes for non-degenerate Reeb flows with a continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several applications are given, including obstructions to the existence of these flows and abundance of periodic orbits for contact forms representing an Anosov contact structure.

DIRAC STRUCTURES FROM LIE INTEGRABILITY

We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

Lagrangian subbundles of symplectic bundles over a curve

AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

The Geometry of Systems of Third Order Differential Equations Induced by Second Order Regular Lagrangians

A system of third order differential equations, whose coefficients do not depend explicitly on time, can be viewed as a third order vector field, which is called a semispray, and lives on the second order tangent bundle. We prove that a regular second order Lagrangian induces such a semispray, which is uniquely determined by two associated Poincare-Cartan one-forms. To study the geometry of this semispray, we construct a horizontal distribution, which is a Lagrangian subbundle for an associated Poincare-Cartan two-form. Using this semispray and the associated nonlinear connection we define dynamical covariant derivatives of first and second order. With respect to this, the second order dynamical derivative of the Lagrangian metric tensor vanishes.

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.

Symplectic twist maps without conjugate points

For sequences of symplectic twist maps without conjugate points, an invariant Lagrangian subbundle is constructed. This allows one to deduce that absence of conjugate points is a rare property in some classes of maps.

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.

Loops of Lagrangian submanifolds and pseudoholomorphic discs

We examine three invariants of exact loops of Lagrangian submanifolds that are modelled on invariants introduced by Polterovich for loops of Hamiltonian symplectomorphisms. One of these is the minimal Hofer length in a given Hamiltonian isotopy class. We determine the exact values of these invariants for loops of projective Lagrangian planes. The proof uses the Gromov invariants of an associated symplectic fibration over the 2-disc with a Lagrangian subbundle over the boundary.

Lagrangian subbundles and codimension 3 subcanonical subschemes

We show that a Gorenstein subcanonical codimension 3 subscheme Z⊂X=ℙN, N≥4, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately and conversely. We extend this result to singular Z and all quasi-projective ambient schemes X under the necessary hypothesis that Z is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of D. Buchsbaum and D. Eisenbud [6] and says that Z is Pfaffian. We also prove codimension 1 symmetric and skew-symmetric analogues of our structure theorems.

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.

Residue Classes of Lagrangian Subbundles and Maslov Classes

Residue Classes of Lagrangian Subbundles and Maslov Classes

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 .