arXiv research

In this article we extend cutting and blowing up to the nonrational symplectic toric setting. This entails the possibility of cutting and blowing up for symplectic toric manifolds and orbifolds in nonrational directions.

Functor connects symplectic and contact structures via cutting and blowups.

We define spin-c prequantization of a symplectic manifold to be a spin-c structure and a connection which are compatible with the symplectic form. We describe the cutting of an S^1-equivariant spin-c prequantization. The cutting process involves a choice of a spin-c prequantization for the complex plane. We prove that …

A symplectic cut of a manifold M with a Hamiltonian circle action is a symplectic quotient of M x C. If M is Kaehler then, since C is Kaehler, the cut space is Kaehler as well. The symplectic structure on the cut is well understood. In this paper we describe the complex structure (and hence the metric) on the cut. We t…

The ``symplectic cut'' construction [Lerman] produces two symplectic orbifolds $C_-$ and $C_+$ from a symplectic manifold $M$ with a Hamiltonian circle action. We compute the rational cohomology ring of $C_+$ in terms of those of $M$ and $C_-$.

We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…

By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.

We obtain estimates on the character of the cohomology of an $S^1$-equivariant holomorphic vector bundle over a Kaehler manifold $M$ in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of $M$. In particular, we prove and extend inequalities conjectured by Wu and Zhang. The proof is bas…

We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly, the other gives metric deformations. We…

We describe a contact analog of the symplectic cut construction. As an application we show that the group of contactomorphisms for a particular overtwisted contact structure on the three sphere contains countably many nonconjugate two tori.

We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cu…

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbi…

New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.

We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define $k$-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic $k$ 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-r…

In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry b…

The paper extends symplectic techniques to generalized complex geometry.

Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.

This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operatio…

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing prop…

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete …

Let $(M,ω)$ be a connected symplectic manifold on which a connected Lie group $G$ acts properly and in a Hamiltonian fashion with moment map $μ:M \lra \mf g^*$. Our purpose is investigate multiplicity-free actions, giving criteria to decide a multiplicity freenes of the action. As an application we give the complete cl…

Introduces a new method for symplectic reduction along submanifolds.

Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic m…

We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension $4n$ with a tri-Hamiltonian action of a torus of dimension $n$, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyp…

Classifies almost-toric systems in four dimensions.

A construction is introduced for modifying hyperkaehler manifolds with tri-Hamiltonian circle action, that in favourable situations increases the second Betti number by one. This is based on the symplectic cut construction of Lerman. In 4 or 8 dimensions the construction may be interpreted as adding a D6-brane. A numbe…

In this paper, using the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along…

New method to encode Weinstein 4-manifolds using multisections with divides.

The paper calculates Seiberg-Witten invariants and finds non-symplectic diffeomorphisms.

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is giv…

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-poin…

In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over…

NeuralCut learns to select cutting planes by looking ahead, outperforming traditional methods.

Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.

The paper derives a formula for Chow weights of toric blow-ups.

NeVI-Cut uses neural networks to efficiently propagate uncertainty without feedback.

Paper connects probability density cuts to graph theory eigenfunctions.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …

A closed 3-form $H \in Ω^3_0(M)$ defines an extension of $Γ(TM)$ by $Ω^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian redu…

Stability of cut locus under metric perturbations in compact Riemannian manifolds.

New equivalence relation for links using cut-diagrams.

Study shows convergence rates for Cheeger cuts on data clouds.

Unified framework for differentiable graph partitioning with probabilistic cuts.

Stochastic cutting planes improve data-driven optimization speed.

We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…