arXiv research

The Hitchin component Hit_n(S) of a closed surface S is a preferred component of the character variety X_PSL_n(R)(S) consisting of homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R)(S), whose elements enjoy remarkable geometric and dynamical properties. We consider a certain type of deformations…

The paper defines and calculates Reidemeister torsion for a specific class of representations.

Explicit computation of symplectic form for $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component.

Goldman parametrizes the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g-10$ coordinates are canonical. We prove that the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admi…

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an h…

The paper derives formulas for symplectic volume forms on surface representation varieties.

Symplectic coordinates found on projective structures on orbifolds.

Paper constructs moduli spaces of Higgs bundles and connects them to Teichmüller space structures.

Study shows infinite volumes of moduli spaces for certain groups.

Random representations of surface groups approach asymptotic freeness in large $n$ limit.

We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…

Infinite volume found in the thick part of $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space.

We study a particular class of representations from the fundamental groups of punctured spheres $Σ_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple clos…

Study infinite symplectic forms on ruled surfaces.

Symplectic forms match on circle pattern space.

The paper explores spaces of Kähler and symplectic forms on 4-manifolds.

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…

Smooth symplectic manifolds can be approximated by PL symplectic manifolds.

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

A symplectic form has a primitive with nowhere vanishing $β$.

Characterizes density-valued symplectic forms on multisymplectic manifolds.

The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.

The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions $n=4r-2$. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…

The paper classifies symplectic forms on R^4 and determines invariants under symplectomorphisms.

Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.

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…

Method resolves 4D symplectic orbifolds using complex geometry.

The paper constructs symplectic forms on 4-manifolds using branched coverings and holomorphic line bundles.

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

Symplectic forms from two phase spaces are proven equivalent.

We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…

A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural r…

Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.

We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a $2n$-dimensional manifold. In the $\mathbb C$-analytic category this set consists of the Martinet hypersurface $Σ_2$, the restriction of the singular symplectic form $ω$ to $TΣ_2$ and the kern…

We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.

The paper defines and proves a new property for symplectic manifolds.

The paper explores complex geometries of 3-forms on symplectic 6-manifolds.

New method constructs symplectic structures on 4-manifolds from trisections.

New action-angle coordinates found for singular symplectic manifolds.

The paper explores symplectic foliations and their leaves on manifolds.

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

Study symplectic forms on manifolds to find Lagrangian pinwheels that can be separated.

We study left invariant contact forms and left invariant symplectic forms on Lie groups. We give the classification of all symplectic structures on nilpotent Lie algebras up the dimension 6.

Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition

The study proves symplectic quandles cannot have good involutions.

Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin-Vilkoviski operator.

Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.

We analyze symplectic forms on six dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four dimensional nilradical.