Related papers: The moduli space of complex two dimensional associ…
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 present a study of the homological algebra of bimodules over $A_\infty$-algebras endowed with an involution. Furthermore we introduce a derived description of Hochschild homology and cohomology for involutive $A_\infty$-algebras.
It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a 6-dimensional complex ball. In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the…
Associated to any subspace arrangement is a "De Concini-Procesi model", a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne-Mumford compactification of the moduli space of…
We develop a ready-to-use comprehensive theory for (super) 2-vector bundles over smooth manifolds. It is based on the bicategory of (super) algebras, bimodules, and intertwiners as a model for 2-vector spaces. We discuss symmetric monoidal…
We complete Mori's program for Kontsevich's moduli space of degree 2 stable maps to Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial…
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…
The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell's well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing…
In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some…
Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics…
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex…
This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be…
We give infinite series of groups Gamma and of compact complex surfaces of general type S with fundamental group Gamma such that 1) Any surface S' with the same Euler number as S, and fundamental group Gamma, is diffeomorphic to S. 2) The…
This informal note collects key results and open problems on the (co)homology of the Deligne-Mumford moduli spaces of real marked rational curves. The open problems are both of topological nature, aiming to investigate the (co)homology of…
We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric…
We study the compactification of the moduli space of a certain class of rank-two irregular connections on the Riemann sphere, presenting one double pole and two simple poles. To construct the compactification explicitly, we identify a class…
Second-order automorphic forms are similar to the usual automorphic forms but have a weaker automorphy condition. We answer a question of Zagier and find the dimensions of spaces of holomorphic, even weight, second-order forms. We also…
In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus…
In this paper, we compute the cohomology ring of all homology split polyhedral product spaces and the cohomology algebra over a field of all polyhedral product spaces. As an application, we give two polyhedral product spaces such that all…
We study the moduli space of discrete, faithful, type-preserving representations of the modular group $\mathbf{PSL}(2,\mathbb{Z})$ into $\mathbf{PU}(3,1)$. The entire moduli space $\mathcal{M}$ is a union of…