相关论文: The topology of $A_g$ and its compactifications

We survey results on compact Clifford-Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact…

For $g\ge 2$ and $n\ge 0$, let $\mathcal{H}_{g,n}\subset \mathcal{M}_{g,n}$ denote the complex moduli stack of $n$-marked smooth hyperelliptic curves of genus $g$. A normal crossings compactification of this space is provided by the theory…

For g>2 we study the cohomology classes in the closure of a stratum of abelian differentials defined by the boundary strata of codimension one. As an application, we find an explicit stratification of the spin moduli space for an odd spin…

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic…

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in…

We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a…

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian…

The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation…

Projective structures on topological surfaces support the structure of 2d CFTs with a degree of technical simplification. We propose a complex analytic space $\mathcal{P}_g$ biholomorphic to $T^*_{(1,0)} \mathcal{M}_g$ as a candidate moduli…

We study the weight 11 part of the compactly supported cohomology of the moduli space of curves $M_{g,n}$, using graph complex techniques, with particular attention to the case $n = 0$. As applications, we prove new nonvanishing results for…

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 give a summary of joint work with Michael Thaddeus that realizes toroidal compactifcations of split reductive groups as moduli spaces of framed bundles on chains of rational curves. We include an extension of this work that covers Artin…

We show that Martin Olsson's compactification of moduli space of polarized abelian varieties in \cite{ols08} can be interpreted in terms of KSBA stable pairs. We find that there is a canonical set of divisors $S(K_2)$ associated with each…

We investigate the $\mathcal F$-Borel complexity of topological spaces in their different compactifcations. We provide a simple proof of the fact that a space can have arbitrarily many different complexities in different compactifications.…

Let ${\mathcal M}_g$ be the moduli space of compact connected Riemann surfaces of genus $g\geq 2$ and let $\widehat{{\mathcal M}_g}$ be its Deligne-Mumford compactification, which is stratified by the topological type of the stable Riemann…

Following work of Rieffel, we define the Cayley compactification of an abelian group with specified generating set. We investigate its structure using methods from discrete geometry and commutative algebra.

An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is…

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincare' decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in…

We define and investigate modulation invariant spaces on a locally compact abelian group $G$ with respect to a closed subgroup of the dual group $\widehat{G}$. Using a range function approach, we establish a characterization of modulation…

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…