Related papers: Pre-quantization of the Moduli Space of Flat G-Bun…

Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $\Lambda$ is graded by lengths of paths. The paper addresses the classifiability, via…

We shall prove that a moduli space of flat irreducible Lie algebroid connections over a compact manifold has locally a natural structure of a smooth differentiable space. This is a generalization of some well known results for the moduli…

We suggest a general method of computation of the homology of certain smooth covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces…

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable connections, and of holomorphic integrable…

Let $X$ be a complex surface obtained as the quotient of the complex Euclidean space $\mathbb{C}^2$ by a discrete subgroup of rank $3$. We investigate the cohomology group $H_0^1(X, E)$ with compact support for a unitary flat line bundle…

We show that the moduli space of semistable G-bundles on an elliptic curve for a reductive group G is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This…

We develop a unified framework based on topological crossed modules for various lifting obstructions for $\Gamma$-kernels. It allows us to identify actions, cocycle actions and $\Gamma$-kernels up to their natural equivalence relations with…

Let $G$ be an algebraic group and $\Gamma$ a finite subgroup of automorphisms of $G$. Fix also a possibly ramified $\Gamma$-covering $\widetilde{X} \to X$. In this setting one may define the notion of $(\Gamma,G)$-bundles over…

We show that the moduli space of Ricci flat metrics of unit volume (including orbifold metrics) on a K3 surface is simply connected and that it has the same rational cohomology as the automorphism group of the K3 lattice $(-E_8)^{\oplus…

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as…

In this note we identify two complex structures (one is given by algebraic geometry, the other by gauge theory) on the set of isomorphism classes of holomorphic bundles with section on a given compact complex manifold. In the case of line…

In this note, we revisit the quantization of Lie bialgebras described by the second author, placing it in the more general framework of the quantization of moduli spaces developed in our previous work. In particular, we show that embeddings…

The moduli space of slope-stable vector bundles on a normal projective variety over an algebraically closed field of characteristic $p\geq 0$ is stratified with respect to the decomposition type. On a smooth projective curve of genus at…

Let $X$ be a smooth, connected complex projective curve of genus at least $2$. A Higgs coherent system is an augmented bundle $(E,V)$, where $E$ is a holomorphic vector bundle, and $V$ is a linear subspace of the spaces of Higgs bundles of…

We construct a smooth Deligne-Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m,…

Let G be a connected Lie group, G^d the underlying discrete group, and BG, BG^d their classifying spaces. Let R denote the radical of G. We show that all classes in the image of the canonical map in cohomology H^*(BG,R)->H^*(BG^d,R) are…

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over…

When there is a family of complex structures on the phase space, parametrized by a set $S$, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we…

Let $\mathcal C$ be a smooth irreducible projective curve over the complex numbers and let $G$ be a simple simply-connected complex algebraic group. Let $\mathfrak M=\mathfrak M(G,\mathcal C)$ be the moduli space of semistable principal…

A projective moduli space of pairs (C,E) where E is a slope- semistable torsion free sheaf of uniform rank on a Deligne- Mumford stable curve C is constructed via G.I.T. There is a natural SL x SL action on the relative Quot scheme over the…