arXiv research

A dual pair is constructed for contact groups, linking submanifolds and orbits.

Introduces contact dual pairs using line bundles.

Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.

Study symplectic spinors and Frobenius structures on manifolds.

Weak dual pairs defined in Dirac-Jacobi geometry, proving equivalence and leaf correspondence theorems.

Defines duals of higher vector bundles for Lie 2-groupoids.

We generalize the notions of dual pair and polarity introduced by S. Lie and A. Weinstein in order to accommodate very relevant situations where the application of these ideas is desirable. The new notion of polarity is designed to deal with the loss of smoothness caused by the presence of singularities that are encoun…

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C$ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…

We study isomorphism classes of symplectic dual pairs P <- S -> P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P, these Morita self-equivalences of P form a group Pic(P) under a natural `…

We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic do…

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit red…

The paper explores symplectic geometry of Cartan-Hartogs domains.

We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces Jan Steven's list [Manuscripta Math. 1993] of the numbers of $P$-resolutions of each singularities. We then compute the dimensions and Miln…

Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection \nabla^A, and Dirac operator \D^A on spinor fields. Let σ: W^+ --> Λ^+ be the natural squaring map, taking self-d…

We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.

New flows introduced for symplectic geometry.

S-dual of Hamiltonian spaces connects to Langlands duality.

New metrics found on toric LCS manifolds.

We prove Gray--Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a one-form is replaced by the conditi…

The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,φ,Λ), with the calibration 3-form φand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. …

We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $\geq 4$. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-produc…

We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic…

The paper finds symplectic mapping class relations using pencil pairs.

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either …

Study of meromorphic connections and their spectral duals in $\mathfrak{gl}_3(\mathbb{C})$.

The paper classifies 10 antipodal pairings of self-dual maps.

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squ…

This paper is a rigorous study of two dual pairs of momentum maps arising in the context of fluid equations whose configuration Lie group is the group of automorphism of a trivial principal bundle, generically called here non-abelian fluids. It is shown that the actions involved are mutually completely orthogonal, whic…

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic stru…

In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the S…

New Einstein manifolds from symplectic systems.

This paper is a rigorous study of the dual pair structure of the ideal fluid and the dual pair structure for the $n$-dimensional Camassa-Holm (EPDiff) equation, including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads natura…

Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

The paper studies dual pairs of generic conformally flat hypersurfaces in 4-space.

The purpose of this paper is to introduce Harvey-Lawson manifolds and review the construction of certain mirror dual Calabi-Yau submanifolds inside a G_2 manifold. More specifically, given a Harvey-Lawson manifold HL, we explain how to assign a pair of tangent bundle valued 2 and 3-forms to a G_2 manifold (M,HL, \varph…

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…

Integrates Manin pairs to simplify Poisson and symplectic groupoid constructions.

We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is loca…

This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs …

We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the cond…

In this paper we show that there exists a family of simply connected, symplectic 4-manifolds such that the (Poincare dual of the) canonical class admits both connected and disconnected symplectic representatives. This answers a question raised by Fintushel and Stern.

This paper proves an $C^{\infty}$ closing lemma for Hamiltonian flows on symplectic 4-manifolds.

It is shown that the cotangent bundle of a matched pair Lie group is itself a matched pair Lie group. The trivialization of the cotangent bundle of a matched pair Lie group are presented. On the trivialized space, the canonical symplectic two-form and canonical Poisson bracket are explicitly written. Various symplectic…

We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of $G_2$-manifolds. We find a large class …

In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along c…

We prove that certain dual pairs of Calabi-Yau manifolds have matching orbifold Euler characteristics.