Related papers: Poisson structure on the moduli spaces of sheaves …

We develop a theory of quasimaps to a moduli space of sheaves $M$ on a surface $S$. Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic…

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with…

A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from…

In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the…

We prove that the weight-two Hodge structure of moduli spaces of torsion-free sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we assume the first Chern class is primitive). We prove the moduli space is an…

On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus…

We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely…

Poisson structures of divisor-type are those whose degeneracy can be captured by a divisor ideal, which is a locally principal ideal sheaf with nowhere-dense quotient support. This is a large class of Poisson structures which includes all…

The $\text{PSL}(4,\mathbb{R})$ Hitchin component of a closed surface group $\pi_1(S)$ consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of $S$. We prove that the leaves of the…

We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the…

We introduce the concept of partial Poisson structure on a manifold $M$ modelled on a convenient space. This is done by specifying a (weak) subbundle $T^{\prime}M$ of $T^{\ast}M$ and an antisymmetric morphism $P:T^{\prime}M\rightarrow TM$…

Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B \supset T$. For an arbitrary automorphism $\theta$ of $G$, we introduce a holomorphic Poisson structure $\pi_\theta$ on $G$ which is…

We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G\rr P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in…

We prove that any holomorphic Poisson manifold has an open symplectic leaf which is a pseudo-K\"ahler submanifold, and we define an obstruction to study the equivariance of momentum map for tangential Poisson action. Some properties of…

Let S be a ruled surface without sections of negative self-intersection. We classify the irreducible components of the moduli stack of torsion-free sheaves of rank 2 sheaves on S. We also classify the irreducible components of the…

We study complex projective surfaces admitting a Poisson structure. We prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.

We classify all the quadratic Poisson structures on $so^*(4)$ and $e^*(3)$, which have the same foliation by symplectic leaves as the canonical Lie-Poisson tensors. The separated variables for the some of the corresponding bi-integrable…

Let X be a smooth projective variety over C. We find the natural notion of semistable orthogonal bundle and construct the moduli space, which we compactify by considering also orthogonal sheaves, i.e. pairs (E,\phi), where E is a torsion…

We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space $M_\sigma(\mathbf{v})$ of $\sigma$-semistable objects of class $\mathbf{v}$ for a generic stability condition…

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a…