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arXiv1997

Moment maps and non-compact cobordisms

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of the normal bundles of the connected components of the fixed points set. We use this to give simple new proofs of two formulas: Guillemin's topological version of the abelian Jeffrey-Kirwan localization, and the Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measure.

dg-ga↗
arXiv1997

Equivariant Holomorphic Morse Inequalities III: Non-Isolated Fixed Points

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the connected components of the fixed-point set and the whole manifold. We also investigate the consequences in geometric quantization, especially in the context of symplectic cutting.

dg-ga↗
arXiv1998

Moduli Spaces of Stable Polygons and Symplectic Structures on $\bar{M}_{0,n}$

In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\bfmit M}_{{\bf r}, ε}$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space $\bar{\cM}_{0,n}$ of the Deligne-Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data leading toward to a family of (classes of) symplectic (Kähler) forms. To some degree, ${\bfmit M}_{{\bf r}, ε}$ may be considered as symplectic counterparts of $\bar{\cM}_{0,n}$ and Kapranov's Chow quotient construction of $\bar{\cM}_{0,n}$. All these together brings up a new tool to study the Kähler topology of $\bar{\cM}_{0,n}$.

dg-ga↗
arXiv1997

On Distortion and Thickness of Knots

What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measures, and show that the distortion thickness is upper semi-continuous in the C^0 topology, suggesting that shortest curves of thickness 1 should exist in each knot class.

dg-ga↗
arXiv1997

Gluing formulae for Donaldson invariants for connected sums along surfaces

We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in four-manifolds, Preprint) about the relationship of the basic classes of two four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are embedded Riemann surfaces of genus $g \geq 2$ and self-intersection zero (and representing odd homology classes) with the basic classes of the manifold X which appears as a connected sum along the surfaces (supposing this latter one is of simple type). This is also expressed as constraints in the basic classes of X. The result is in accordance with the results on Seiberg-Witten invariants (Morgan, Szabo and Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture).

dg-ga↗
arXiv1997

Donaldson invariants for connected sums along surfaces of genus 2

We relate the Donaldson invariants of two four-manifolds $X_i$ with embedded Riemann surfaces of genus 2 and self-intersection zero with the invariants of the manifold X which appears as a connected sum along the surfaces. When the original manifolds are of simple type with $b_1=0$ and $b^+>1$, X is of simple type with $b_1=0$ and $b^+>1$ as well, and the relationship between the invariants is expressed as constraints in the basic classes for X. Also we give some applications. For instance, if $X_i$ have both $b_1=0$ then X is of simple type with $b_1=0$, $b^+>1$, and has no basic classes evaluating zero on the Riemann surface. Finally, we prove that any four-manifold with $b^+>1$ and with an embedded surface of genus 2, self-intersection zero and representing an odd homology class, is of finite type of second order.

dg-ga↗
arXiv2013

Moduli spaces of PU(2)-monopoles

The goal of this article was the S^1-equivariant transversality-problem and the compactification-problem for the moduli spaces of (perturbed) PU(2)-monopoles. A substantially improved version entitled "Moduli spaces of PU(2)-monopoles (revised version)" which gives simpler, clearer proofs of the transversality results, has been published on arxiv in June 99 and appeared in Asian J. Math, see Moduli spaces of PU(2)-Monopoles, Asian J. Math. Vol. 4, No. 2 (2000), 391-436.

dg-ga↗
arXiv1997

Locally conformal flat Riemannian manifolds with constant principal Ricci curvatures and locally conformal flat C-spaces

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4,5,6,7 and 8. It is shown that any n-dimensional $(4\leq n \leq 8)$ locally conformal flat Riemannian manifold with constant principal Ricci curvatures is a Riemannian locally symmetric space.

dg-ga↗
arXiv1998

Dual Teichm\" uller spaces

We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support respectively. We show explicitly that the latter spaces are asymptotically isomorphic to the former. We discuss briefly quantisation of Teichm\" uller spaces and some other application of the constructed approach. The paper does not require any preliminary knowledge of the subject above the Poincar\' e uniformisation theorem.

dg-ga↗
arXiv1997

Gerstenhaber algebras and BV-algebras in Poisson geometry

The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylinski operator of a Poisson manifold and its modular class. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.

dg-ga↗
arXiv1997

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.

dg-ga↗
arXiv1997

Classical Mechanics of Spinning Patricle in a Curved Space

An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian formalism for this system and possible method for its quantization are discussed. It is shown that the Hamilton equations coincide with the Papapetrou equations for spinning test-particle in general relativity.

dg-ga↗
arXiv2000

The Seiberg-Witten theory of homology 3-spheres

In this thesis we study the Seiberg-Witten theory of an oriented homology 3-sphere. The goal is to extract topological invariants - the Seiberg-Witten invariants - by counting the solutions to the Seiberg-Witten equations on the manifold. The first question we consider is whether the Seiberg-Witten invariants depend on the geometric or analytic data involved in their definition. In the first main result of this thesis, we completely determine the dependence of the Seiberg-Witten invariants on the data involved in their definition. In particular, we show that even for the simplest manifold, the 3-sphere $S^3$, the Seiberg-Witten invariants take infinitely many different values. The rest of this thesis is devoted to understanding the Seiberg-Witten invariants in a specific geometric setting - the surgery setting. In that context we prove a gluing formula, which identifies the Seiberg-Witten invariants as certain ``homological intersection numbers''.

dg-ga↗
arXiv1997

On Guichard's nets and Cyclic systems

In the first part, we give a self contained introduction to the theory of cyclic systems in n-dimensional space which can be considered as immersions into certain Grassmannians. We show how the (metric) geometries on spaces of constant curvature arise as subgeometries of Moebius geometry which provides a slightly new viewpoint. In the second part we characterize Guichard nets which are given by cyclic systems as being Moebius equivalent to 1-parameter families of linear Weingarten surfaces. This provides a new method to study families of parallel Weingarten surfaces in space forms. In particular, analogs of Bonnet's theorem on parallel constant mean curvature surfaces can be easily obtained in this setting.

dg-ga↗
arXiv1998

Transports along Paths in Fibre Bundles. III. Consistency with Bundle Morphisms

The general problem for consistency between arbitrary transports along paths in fibre bundles and bundle morphisms between them is formulated and investigated. The special case of one fibre bundle, its morphism and transport along paths acting in it is considered. The consistency between linear transports along paths in a vector bundle and a Hermitian structure in it is studied.

dg-ga↗
arXiv1997

Lie-Rinehart algebras, Gerstenhaber algebras, and B-V algebras

For a Lie-Rinehart algebra (A,L), generators for the Gerstenhaber algebra Λ_A L correspond bijectively to right (A,L)-connections on A in such a way that B-V structures correspond to right (A,L)-module structures on A. When L is projective as an A-module, given an exact generator \partial, the homology of the B-V algebra (Λ_A L,\partial) coincides with that of L with coefficients in A with respect to the right (A,L)-module structure determined by \partial. When L is also of finite rank n, there are bijective correspondences between (A,L)-connections on Λ_A^nL and right (A,L)-connections on A and between left (A,L)- module structures on Λ_A^nL and right (A,L)-module structures on A. Hence there are bijective correspondences between (A,L)-connections on Λ_A^n L and generators for the Gerstenhaber bracket on Λ_A L and between (A,L)-module structures on Λ_A^n L and B-V algebra structures on Λ_A L. The homology of such a B-V algebra (Λ_A L,\partial) coincides with the cohomology of L with coefficients in Λ_A^n L, for the left (A,L)-module structure determined by \partial. Some applications are discussed.

dg-ga↗