arXiv research

Constructs symplectic surface bundles with positive signatures.

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

Study fills nonorientable surfaces' cotangent bundles uniquely.

In this paper, we prove the existence of certain symplectic conifold transitions on all $CP^{1}$-bundles over symplectic 4--manifolds, which generalizes Smith, Thomas and Yau's examples of symplectic conifold transitions on trivial $CP^{1}$-bundles over Kähler surfaces. Our main result is to determine the diffeomorphis…

Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.

The paper studies symplectic surface bundles and their characteristic classes.

This paper constructs a Weinstein trisection for a surface bundle.

Existence of symplectic structure shown on Seiberg-Witten moduli space product of Riemann surfaces.

Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.

Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.

Stein and Weinstein structures are described for disk cotangent bundles of surfaces.

This paper constructs symplectic surfaces in 4-manifolds with transversal intersections.

Twists Gromov and Lefschetz invariants for symplectic and fibered 3-manifolds.

We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors.…

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures $Ω_{Ψ_0}$ on the moduli space, parametrised by $Ψ_0$, a section o…

We discuss the existence and non-existence of cobordisms between symplectic surface bundles over the circle.

We obtain an explicit formula for the symplectic form over the double quotient with help of the Green function of a Riemann surface.

In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over the 2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which is not Stein. (3) The circle bundle over a genus n>1 surface with euler number e=-1 ad…

For non-degenerate surfaces in $R^4$, a distinguished transversal bundle called affine normal plane bundle was proposed in [Nomizu-Vrancken]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper we characterize those surfa…

The paper studies signature cocycles on mapping class groups and symplectic groups.

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle…

The paper constructs a canonical connection on bundles over Riemann surfaces and relates it to the theta divisor.

The Liouville symplectic form connects various moduli spaces in algebraic geometry.

We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motiv…

This paper establishes a correspondence between Einstein-Hermitian metrics and stability for generalized holomorphic vector bundles over symplectic type generalized Kahler manifolds.

Symplectic structure found on moduli space of framed Higgs bundles.

The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.

Fintushel-Stern surgery on fibered knots in torus bundles yields symplectic manifolds.

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particul…

For any closed oriented surface F of genus at least three, we prove the existence of foliated F-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form on the fiber. We relate the cohomolog…

Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.

Introduces group-valued momentum maps for symplectic fiber bundles.

Teichmüller space realized as symplectic quotient.

Paper constructs moduli spaces of Higgs bundles and connects them to Teichmüller space structures.

We give examples of compact symplectic manifolds with disconnected contact type boundary in dimension $4n$ for any $n\geq 1$. The example is given by a subset of the tangent bundle of a compact quotient of the complex hyperbolic space endowed with the canonical symplectic form plus a generalized magnetic field and its …

Generalizes Kawai theorem for orbifold Riemann surfaces.

There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at …

Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.

Survey of bundle gerbes in geometry, field theory, and quantization.

New geometries derived from symplectic Monge-Ampère structures.

Researchers calculate Hofer-Zehnder capacity for twisted tangent bundles over surfaces.

Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent …

Constructs a new mathematical structure for Riemann surfaces with projective structures.

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality…

The paper constructs a moduli space for opers and proves its symplectic properties.

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_ρ}{\mathcal …

The Goldman symplectic form is trivialized for a class of surfaces.

We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character variet…