Related papers: The non-existence of stable Schottky forms

By Koebe's retrosection theorem, every closed Riemann surface of genus $g \geq 2$ is uniformized by a Schottky group. Marden observed that there are Schottky groups that are not classical ones, that is, they cannot be defined by a suitable…

For fixed $k<g$ and a family of polarized abelian varieties of dimension $g$ over $\mathbb{R}$, we give a criterion for the density in the parameter space of those abelian varieties over $\mathbb{R}$ containing a $k$-dimensional abelian…

For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this…

We prove that the moduli space $\mathcal{M}_g$ of smooth curves of genus $g$ is the union of $g-1$ affine open subsets for every $g$ with $2 \le g \le 5$, as predicted by an intriguing conjecture of Eduard Looijenga.

For every integer $g \geq 1$ we define a universal Mumford curve of genus $g$ in the framework of Berkovich spaces over $\mathbb{Z}$. This is achieved in two steps: first, we build an analytic space $\mathcal{S}_g$ that parametrizes marked…

We show that for g > 2k+2 the k-rational homotopy type of the moduli space M_{g,n} of algebraic curves of genus g with n punctures is independent of g, and the space M_{g,n} is k-formal. This implies the existence of a limiting rational…

For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has…

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the…

The spectral side of the (conjectural) Betti geometric Langlands correspondence concerns sheaves on the character stack of an algebraic curve; in particular, the categories in question are manifestly invariant under deformations of the…

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set…

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…

In this paper, we study moduli spaces of sextic curves with simple singularities. Through period maps of K3 surfaces with ADE singularities, we prove that such moduli spaces admit algebraic open embeddings into arithmetic quotients of type…

Since the sixties it is well known that there are no non-trivial closed holomorphic $1$-forms on the moduli space $\mathcal{M}_g$ of smooth projective curves of genus $g>2$. In this paper, we strengthen such result proving that for $g\geq…

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider…

In this paper we consider unramified coverings of the moduli space $\mathcal{M}_g$ of smooth projective complex curves of genus $g$. Under some hypothesis on the branch locus of the finite extended map to the Deligne-Mumford…

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with…

We study the moduli space ${V}_4\mathcal{M}_{g}$ of Klein four covers of genus $g$ curves and its natural compactification. This requires the construction of a related space which has a choice of basis for the Klein four group. This space…

The tautological Chow ring of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from…

We investigate the moduli space ${\mathcal P}_g$ of smooth complex projective curves of genus $g$ equipped with a projective structure. When $g\, \geq\, 3$, it is shown that this moduli space ${\mathcal P}_g$ does not admit any nonconstant…

Let $X$ be a smooth complex elliptic curve and $G$ a connected reductive affine algebraic group defined over $\mathbb C$. Let ${\mathcal M}_X(G)$ denote the moduli space of topologically trivial algebraic $G$--connections on $X$, that is,…