Related papers: Singularities and Poisson geometry of certain repr…

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…

We characterize the image of the Poisson transform on any distinguished boundary of a Riemannian symmetric space of the noncompact type by a system of differential equations. The system corresponds to a generator system of a two sided…

In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are…

In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety $X$ via derived algebraic geometry. We prove that if $X$ is a Calabi-Yau variety of dimension $d$ then this…

This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the…

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction…

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties.…

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on R^2 are investigated under suitable continuity restrictions on cochains. The zeroth, first, and second cohomology spaces in…

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex.)…

We define a kind of moduli space of nested surfaces and mappings, which we call a comparison moduli space. We review examples of such spaces in geometric function theory and modern Teichmueller theory, and illustrate how a wide range of…

Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P\to \Sigma, over a compact oriented surface, \Sigma, carries a Poisson structure. If we…

In this paper we consider structures of complex Poisson brackets on the space of smooth functions in a $n$-dimensional complex manifold generated by the $(1,1)$-form $d=\partial+\overline{\partial}$-closed and non-degenerate (with…

These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharov's…

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli…

Let G be a split semisimple algebraic group with trivial center. Let S be a compact oriented surface, with or without boundary. We define {\it positive} representations of the fundamental group of S to G(R), construct explicitly all…

The main purpose of the paper is to study hyperkahler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of…

We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of…

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson…

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…