arXiv research

Develops deformation theory for symplectic foliations using $L_\infty$-algebras.

Study on symplectic structures and their deformations.

The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.

Deformation quantization yields a new moment map on symplectic diffeomorphisms.

Degenerate twistor deformations of Kähler manifolds are also Kähler.

We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. By a fact, we can deform a given symplectic structure $ω$ to a new symplectic structure $ω_t$ parametrized by some element $t$ in $Λ^2\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of a Lie group $G$. Moreover,…

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies.…

Extends Gromov non-squeezing to locally conformally symplectic structures.

Symplectic coordinates found on projective structures on orbifolds.

Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.

Solves a problem about deforming symplectic forms on a Klein bottle.

The paper studies deformations of symplectic forms and Lagrangian submanifolds.

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformati…

New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.

Affine deformations of cotangent groupoids

We explain the geometric origin of the $L_{\infty}$-algebra controlling deformations of pre-symplectic structures.

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we call Koszul $L_\infty$-algebra. This $L_\infty$-algebra is a cousin of the Koszul…

New method for quantizing symplectic manifolds with Lagrangian bundles.

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic $4$-manifolds made out of elliptic surfaces are in fact symplectically deformation equivalent. We also show that blow-up points can be traded from o…

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic …

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove…

The paper studies deformations of Lagrangian fibrations on symplectic manifolds.

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[ω]])$ depending on symplectic deformation equivalence class $[[ω]]$. We first prove tha…

Deforms orbits in Lie algebras to Lagrangian submanifolds.

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the $L_\infty$-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended de…

Study symplectic fillings of sandwiched singularities.

Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.

The study shows boundedness of certain fibered varieties in algebraic geometry.

In this paper we state an analog of Calabi's conjecture proved by Yau. The difference with the classical case is that we propose deformation of the complex structure, whereas the complex Monge--Ampère equation describes deformation of the Kähler (symplectic) structure.

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the ch…

The paper studies deformations of Lagrangian submanifolds using algebraic tools.

We describe the space of Poisson bivectors near a log-symplectic structure up to small diffeomorphisms.

The paper explores symplectic foliations and their leaves on manifolds.

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fun…

Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.

Deform moment map on symplectic connections using star product algebras.

The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.

The paper generalizes hyperkahler metrics near Lagrangian submanifolds.

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstruc…

Symplectic forms match on circle pattern space.

B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flatte…

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

In the first part of this article we provide a geometrically oriented approach to the theory of orbispaces which originally had been introduced by Chen. We explain the notion of a vector orbibundle and characterize the good sections of a reduced vector orbibundle as the smooth stratified sections. In the second part of…

The paper extends Gromov's non-squeezing theorem to deformed symplectic forms.

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the $L_{\infty}$-algebras governing …