Related papers: Espace de Modules Marques des Surfaces Projectives…

We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(\pi_1(\Sigma),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$…

This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the…

We introduce the concept of a strongly t-logarithmic t-generating set for a Z[t,t^{-1}]-module, which enables us to prove that a large class of soluble groups are not almost convex. We also prove some results about dead-end depth.

Let $g$ be a finite dimensional simple Lie algebra. Denote by $\mathcal B$ the category of all bounded weight $g$-modules, i.e. those which are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of…

In 2006, Kenyon and Okounkov computed the moduli space of Harnack curves of degree $d$ in $\mathbb{C}\mathbb{P}^2$. We generalize to any projective toric surface some of the techniques used there. More precisely, we show that the moduli…

In response to a question raised by Belolipetsky and the first author, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to…

We give new contributions to the existence problem of canonical surfaces of high degree. We construct several families (indeed, connected components of the moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical map has…

Let $X$ be a smooth, irreducible, projective algebraic surface, and let $\alpha \in \mathbb{Q}[m]_{>0}$ be a polynomial. In this paper, we determine topological and geometric properties of the moduli space of $\alpha$-stable coherent…

Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely…

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral…

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of…

We examine moduli spaces of locally homogeneous surfaces of Type~$\mathcal{B}$ with torsion where the symmetric Ricci tensor is non-degenerate. We also determine the space of affine Killing vector fields in this context.

Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily…

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a…

We construct a connected, irreducible component of the moduli space of minimal surfaces of general type with $p_g=q=2$ and $K^2=5$, which contains both examples given by Chen-Hacon and the first author. This component is generically smooth…

The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli…

For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we…

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected.…

Let $\Delta_n$ and $Q_n$ denote the regular $n$-simplex of side length $\sqrt{2}$ embedded in $\mathbb{R}^{n+1}$ and the volume one cube in $\mathbb{R}^n$, respectively. We derive a closed-form formula for the hyperplane volume projections…

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli…