arXiv research

We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the…

Study geometric quantization on K3 surfaces, showing spectral convergence.

On a compact symplectic manifold $(X,ω)$ with a prequantum line bundle $(L,\nabla,h)$, we consider the one-parameter family of $ω$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are several researches which show that the holomorphic sections of t…

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Ou…

Estimates holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds.

Let $π\colon (M,ω)\to B$ be a non-singular Lagrangian torus fibration on a complete base $B$ with prequantum line bundle $\bigl(L,\nabla^L\bigr)\to (M,ω)$. Compactness on $M$ is not assumed. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,ω)$ invariant along the fiber of $π$, let $D$ be …

Quantizes $b$-symplectic toric manifolds using $T$-modules.

In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this…

New surgery operation preserves monotonicity of Lagrangians.

We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map.

Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of …

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of m…

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.

Connecting ideas of geometric formulation of quantum mechanics with new results in symplectic geometry a new approach to geometrical quantization procedure is proposed. As a first result we verify that the correspondence between "classical" Poisson bracket and "quantum" one takes place.

This paper proposes a new way to quantize classical mechanical systems. Here we use ALAG - programme to construct moduli space of half weighted Bohr - Sommerfeld lagrangian cycles of fixed volume which is our quantum phase space. "Dynamical correspondence" principle makes possible to prove that this ALAG - quantization…

We introduce the notion of an isotropic quantum state associated with a Bohr-Sommerfeld manifold in the context of Berezin-Toeplitz quantization of general prequantized symplectic manifolds, and we study its semi-classical properties using the off-diagonal expansion of the Bergman kernel. We then show how these results…

We study the half-form Kaehler quantization of a smooth symplectic toric manifold $(X,ω)$, such that $[ω/2π]-c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}})$ and is nonnegative. We define the half-form corrected quantization of $(X,ω)$ to be given by holomorphic sections of a certain hermitian line bundle $L\rightarrow X$ with Ch…

We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. By using a co…

These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and d…

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In o…

Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.

In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a co…

Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.

Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.

We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families…

Introduces Darboux-Lie derivative for fiber bundles.

The paper proves non-existence of positive scalar curvature on certain fiber bundles.

New examples show not all homology fiber bundles are topological.

This paper develops a geometric framework for Wilson surfaces in higher gauge theory.

Study shows simplicial volume of certain fiber bundles is zero.

New exotic 4-manifolds found with fiber bundles.

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…

We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a characterisation of fibered surfaces with zero signature.

Constructs a family to handle unstable fibers on complex surfaces.

The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.

Groups with Property (T) have fiber products with Property (T).

Boundary of fiber convex domains is a cohomological sphere.

Study complex surfaces fibered over Teichmüller curves with Veech fibers.

In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of exa…

A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first …

Describes 3-manifolds by families of singularly fibered surfaces.

Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.

Paper connects homologically fibered links to solutions of equations.

Defines products for fibered corners manifolds, generalizing resolutions.

Study simplicial volume in fiber bundles with connected groups.

New research finds infinitely many hyperbolic knots not almost-fibered.