Related papers: An Isomorphism Between Moduli Spaces of Abelian Va…

We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$…

We prove an explicit formula for the Bergman kernel of polarized abelian varieties. As applications, we show that if two positive line bundles represent the same first Chern class and have identical Bergman kernel functions for some tensor…

In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the moduli space of curves with an…

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n…

The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2…

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack $\mathcal K^{\text{abs}}_g$ of abstract Kummer varieties and the second one is the stack $\mathcal K^{\text{em}}_g$ of…

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…

In this article, we study the variation of the Gieseker and Uhlenbeck compactifications of the moduli spaces of Mumford-Takemoto stable vector bundles of rank 2 by changing polarizations. Some {\it canonical} rational morphisms among the…

Let A be a unital C* algebra with involution * represented in a Hilbert space H, G the group of invertible elements of A, U the unitary group of A, G^s the set of invertible selfadjoint elements of A, Q={e in G : e^2 = 1} the space of…

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…

In this paper we give a bound on the dimension of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. This…

We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational…

There are many equivalent ways to describe the p-torsion of a principally polarized abelian variety in characteristic p. We briefly explain these methods and then illustrate them for abelian varieties A of arbitrary dimension g in several…

Kuga and Satake associate with every polarized complex K3 surface (X,L) a complex abelian variety called the Kuga-Satake abelian variety of (X,L). We use this construction to define morphisms between moduli spaces of polarized K3 surface…

We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other…

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the…

The degree of a curve $C$ in a polarized abelian variety $(X,\lambda)$ is the integer $d=C\cdot\lambda$. When $C$ generates $X$, we find a lower bound on $d$ which depends on $n$ and the degree of the polarization $\lambda$. The smallest…

We investigate the Tits buildings of the paramodular groups with or without canonical level structure, respectively. These give important combinatorical information about the boundary of the toroidal compactification of the moduli spaces of…

Pareschi showed that an ample divisor of degree $d$ on a simple abelian variety of dimension $g$ has mild singularities when $d<g$. We extend his result to general polarized abelian varieties. We also show that ample divisors of degree $3$…

Let X be a nonsingular projective algebraic curve of genus g\ge3. We consider the moduli space M of stable bundles of fixed determinant with rank n and degree d coprime and d>n(2g-2). There is a universal bundle on XxM and we consider the…