arXiv research

Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.

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.

We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symple…

New symplectic 4-manifolds with non-negative signatures are constructed using complex surfaces and quotients.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

Study compact symplectic solvmanifolds' hard Lefschetz property.

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a …

Smooth resolutions found for quotient of R^2 by infinite discrete groups.

Let $K$ be a compact group. For a symplectic quotient $M_λ$ of a compact Hamiltonian Kähler $K$-manifold, we show that the induced complex structure on $M_λ$ is locally invariant when the parameter $λ$ varies in $\mathrm{Lie}(K)^*$. To prove such a result, we take two different approaches: (i) by using the complex geom…

Let $S^1$ act on a symplectic manifold in a Hamiltonian fashion with momentum map $Ψ$. Fix a value $a$ of $Ψ$. There is a question of whether the symplectic quotient at $a$ is diffeomorphic to the orbit space of some proper Lie group action. We prove under mild assumptions that this only occurs if the symplectic quotie…

Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.

The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.

We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic vari…

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon, to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a fo…

New symplectic annular Khovanov homology connects knot theory to Floer homology.

Formula calculates Riemann-Roch number for singular symplectic quotients.

Let $M$ be a symplectic manifold, equipped with a Hamiltonian action of a torus $T$. We give an explicit formula for the rational cohomology ring of the symplectic quotient $M//T$ in terms of the cohomology ring of $M$ and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain …

When a complex semisimple group $G$ acts holomorphically on a Kähler manifold $(X,ω)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $ω$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with quotient of the zero-set of the m…

Study on braid group quotients by congruence subgroups.

Hyperkahler quotients by non-free actions are typically highly singular, but are remarkably still partitioned into smooth hyperkahler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which induce the hyperkahler struc…

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectom…

This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to give some topological interpretations and characterization of Chow quotient which have the advantage to be more intuitive and geometric. This is to be done over the field of complex numbe…

In this paper we determine conditions of existence of an induced Riemannian structure on the symplectic quotient of a symplectic and Riemannian manifold following the action of a Lie group acting upon it in a hamiltonian way with equivariant momentum mapping.

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

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g…

Simplified presentation of symplectic fillings of lens spaces.

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to q…

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to …

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued connections…

Let n>1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property.

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

New computations show symplectic groups and mapping class groups have different properties regarding torsion.

We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to "lift" a symplectic structure from a symplectic manifold to the bundle of frames. This is a by-product of a failed attempt to prov…

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian act…

SU(n)-structures derived from Kähler manifolds with torus actions.

Paper withdrawn - lemma 4.1 was false. Quite a lot of changes need to be made!

The paper studies Lie algebroid and groupoid quotients with forms, applying to Poisson and Dirac structures.

We generalize the hyperkaehler quotient construction to the situation where there is no group action preserving the hyperkaehler structure but for each complex structure there is an action of a complex group preserving the corresponding complex symplectic structure. Many (known and new) hyperkaehler manifolds arise as …

The paper constructs contact-hyperbolic manifolds with large automorphism groups.

We show that certain submanifolds of generalized complex manifolds ("weak branes") admit a natural quotient which inherits a generalized complex structure. This is analog to quotienting coisotropic submanifolds of symplectic manifolds. In particular Gualtieri's generalized complex submanifolds ("branes") quotient to sp…

Assume $(X, ω)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $μ$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics…

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blo…

Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtai…

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…

Develops a new method for equivariant Lagrangian Floer homology using symplectic homotopy quotients.