Related papers: Shafarevich mappings and period mappings

Let $M$ be a hyperbolizable $3$-manifold with boundary, and let $\chi_0(M)$ be a component of the $PSL_2\mathbb{C}$-character variety of $M$ that contains the convex co-compact characters. We show that the peripheral map…

We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow $\psi_{t}$ of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a…

Let $f : X \rightarrow B$ be a proper flat dominant morphism between two smooth quasi-projective complex varieties $X$ and $B$. Assume that there exists an integer $l$ such that all closed fibres $X_b$ of $f$ satisfy $CH_j(X_b) = \Q$ for…

Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is,…

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of…

Given a period map defined over a quasi-projective variety, we construct a completion with rich geometric and Hodge-theoretic meaning. This result may be regarded as an analog of Mumford's toroidal compactification for locally symmetric…

Let $X$ be a smooth quasi-projective algebraic surface and let $\Delta_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{\Delta_n}$ and their invariants…

Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for…

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be…

We extend Siu's and Sampson's celebrated rigidity results to non-compact domains. More precisely, let $M$ be a smooth quasi-projective variety with universal cover $\tilde M$ and let $\tilde X$ be a symmetric space of non-compact type, a…

Let $\cal{F}$ be a regular codimension 1 holomorphic foliation on a compact K\" ahler manifold. One assumes in addition that $\cal{F}$ possesses a transverse invariant positive current. The aim of this paper is to establish the following…

Let T be a compact torus and X a nice compact T-space (say a manifold or variety). We introduce a functor assigning to X a "GKM-sheaf" F_X over a "GKM-hypergraph" G_X. Under the condition that X is equivariantly formal, the ring of global…

Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time they have very strong "extra" extension properties in…

Let X be a compact Moishezon manifold which becomes projective after blowing up a smooth subvariety $Y \subset X$. We assume also that there exists a proper map $\rho :X \to X'$ onto a projective variety X' with $\rho(Y)$ a point, such that…

Suppose $X$ is a smooth quasiprojective variety over $\cc$ and $\rho : \pi _1(X,x) \to SL(2,\cc)$ is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then $\rho$ factors through a map $X\to Y$ with $Y$ either a…

Let $X=U/K$ be a compact Hermitian symmetric space, and let $\sE$ be a $U$-homogeneous Hermitian vector bundle on $X$. In a previous paper, we showed that the space of nearly holomorphic sections is well-adapted for harmonic analysis in…

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is…

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold…