Related papers: Simultaneous normalization of period map and affin…

I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a…

For every compact Kaehler manifold we give a canonical extension of Griffith's period map to generalized deformations, intended as solutions of Maurer-Cartan equation in the algebra of polyvector fields. Our construction involves the notion…

We show that the moduli space of regular affine vortices, which are solutions of the symplectic vortex equation over the complex plane, has the structure of a smooth manifold. The construction uses Ziltener's Fredholm theory results [31].…

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmueller space of such Riemann surfaces and demonstrate that in…

In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichm\"{u}ller space. We prove that the holonomy map from the…

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…

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented…

We describe a period map for those simply connected Enriques surfaces in characteristic 2 whose canonical double cover is K3. The moduli stack for these surfaces has a Deligne-Mumford quotient that is an open substack of a $\mathbb…

We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}_{alg}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures…

Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.

Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and…

We study the moduli space of pairs consisting of a smooth cubic surface and a smooth hyperplane section, via a Hodge theoretic period map due to Laza, Pearlstein, and the second named author. The construction associates to such a pair a…

It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.

It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.

The moduli space for polarized hyperk\"ahler manifolds of $\mathrm{K3}^{[m]}$-type or $\mathrm{Kum}_m$-type with a given polarization type is not necessarily connected, which is a phenomenon that only happens for $m$ large. The period map…

We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so called higher complex structure we use the punctual Hilbert scheme of the plane. The moduli space of higher…

This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.

A fine moduli space is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$…

We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…