Related papers: Holonomy map fibers of $\mathbb{C}{\rm P}^1$-struc…

Let $C$ be a smooth elliptic curve embedded in a smooth complex surface $X$ such that $C$ is a leaf of a suitable holomorphic foliation of $X$. We investigate complex analytic properties of a neighborhood of $C$ under some assumptions on…

Let X be a right Hilbert C*-module over A. We study the geometry and the topology of the projective space P(X) of X, consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of…

We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb…

The non-abelian Hodge correspondence maps a polystable $\mathrm{SL}(2,\mathbb{R})$-Higgs bundle on a compact Riemann surface $X$ of genus $g\geq2$ to a connection which, in some cases, is the holonomy of a branched hyperbolic structure. On…

Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We…

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and…

The moduli space of projective structures on a compact oriented surface $\Sigma$ has a holomorphic symplectic structure, which is constructed by pulling back, using the monodromy map, the Atiyah--Bott--Goldman symplectic form on the…

We study the category of holonomic $\mathscr{D}_{X}$-modules for a quasi-compact, quasi-separated, smooth rigid analytic variety $X$ over the field $\mathbb{C}(\!(t)\!)$. In particular, we prove finiteness of the de Rham cohomology for such…

In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets…

The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of…

This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then…

We prove that structured vector bundles whose holonomies lie in GL(N,C), SO(N,C), or Sp(2N,C) have structured inverses. This generalizes a theorem of Simons and Sullivan.

Consider all moduli points corresponding with polarized abelian varieties in characteristic p such that the associated quasi-polarized p-divisible group is geometrically isomorphic with a given one. This defines a subset C of the moduli…

We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2…

We study the relationship between singular holomorphic foliations in $(\mathbb{C}^{2},0)$ and their separatrices. Under mild conditions we describe a complete set of analytic invariants characterizing foliations with quasi-homogeneous…

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this…

We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete…

We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat…

We study the compactness problem for moduli spaces of holomorphic supercurves which, being motivated by supergeometry, are perturbed such as to allow for transversality. We give an explicit construction of limiting objects for sequences of…

We study holonomy representations admitting a pair of supplementary faithful sub-representations. In particular the cases where the sub-representations are isomorphic respectively dual to each other are treated. In each case we have a…