Related papers: Geometrically connected components of Lubin-Tate d…

The Lubin-Tate moduli space $X_{0}^{\text{rig}}$ is a $p$-adic analytic open unit polydisc which parametrizes deformations of a formal group $H_{0}$ of finite height defined over an algebraically closed field of characteristic $p$. It is…

Let $O$ be the ring of power series in one variable over a finite field, with $K$ its fraction field. We introduce the notion of a "formal $K$-vector space"; this is a certain kind of $K$-vector space object in the category of formal…

Suppose we are given a profinite group $G$ acting on a formal moduli stack $\mathcal{M}$, and we want to understand the group action, and compute cohomology related to this group action. How can we do it? This prolegomenon surveys two…

We study the action on the deformation space of a formal group by the maximal finite subgroup $G$ of its automorphisms, at the first height where the group has nontrivial $p$-torsion for odd $p$. We show given this group $G$ there is a…

We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is…

The geometry of the Lubin-Tate space of deformations of a formal group is studied via an \'etale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of…

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let $G$ be a Lie group acting on a space…

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on…

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected…

In this paper, we treat moduli spaces of parabolic connections. We take \'etale coverings of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation for each…

Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the…

In this PhD thesis, we give a new geometric approach to higher Teichm\"uller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The…

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the…

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for…

We show that the Kisin varieties associated to simple $\phi$-modules of rank $2$ are connected in the case of an arbitrary cocharacter. This proves that the connected components of the generic fiber of the flat deformation ring of an…

We consider arrangements of n connected codimensional one submanifolds in closed d-dimensional manifold M. Let f be the number of connected components of the complement in M to the union of submanifolds. We prove the sharp lower bound for f…

Let X be a compact connected Riemann surface of genus g > 0 equipped with a nonzero holomorphic 1-form. Let M denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g-1)+1; it is a complex symplectic manifold. Using…

The aim of this article is the proof of the following result: Let M be a connected manifold endowed with a regular Cartan geometry modelled on the boundary X of the d-dimensional real (resp. complex, resp. quaternionic, resp. octonionic)…

If $X$ is a variety with an additional structure $\xi$, such as a marked point, a divisor, a polarization, a group structure and so forth, then it is possible to study whether the pair $(X,\xi)$ is defined over the field of moduli. There…

For a representation of a finite group $G$ on a complex vector space $V$ we determine when a holomorphic $\binom{p}{q}$-tensor field on the principle stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant tensor field…