Related papers: Proper discs in Stein manifolds avoiding complete …

Let $E_i$ be an oriented circle bundle over a closed oriented aspherical $n$-manifold $M_i$ with Euler class $e_i\in H^2(M_i;\mathbb{Z})$, $i=1,2$. We prove the following: (i) If every finite-index subgroup of $\pi_1(M_2)$ has trivial…

We consider a finite \'etale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton…

We prove that the Bergman space of a Stein manifold separates points whenever its Bergman metric is well defined and has non-positive constant holomorphic sectional curvature. This, combined with earlier proved results, shows that a Stein…

Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to…

For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and…

It is proved that a Stein manifold acted on by a connected compact Lie group is spherical if and only if there exists an antiholomorphic involution preserving each orbit of the action. This involution can be chosen equivariant with respect…

Let X be a complex nonsingular affine algebraic variety, K a holomorphically convex subset of X, and Y a homogeneous variety for some complex linear algebraic group. We prove that a holomorphic map f:K-->Y can be uniformly approximated on K…

In this paper, we prove the following result : let X be a complex manifold, hyperbolic for the Carath\'eodory distance and let U be an open set relatively compact in X. Then, there exists k<1 such that we get, for the Carath\'eodory…

Let $X$ be a compact metric space which is locally absolutely retract and let $\phi: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence of $n$…

Given N points in the plane $P_1 P_2...P_N$ and a location $\Omega$, the union of discs with diameters $[\Omega P_i], i = 1, 2,...N$ covers the convex hull of the points. The location $\Omega_s$ minimizing the area covered by the union of…

We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are homotopic rel $\partial$, concordant and coincide near their boundaries, yet are not properly…

By studying periodic points for rational maps on $\bm{C}^d$ with $p$ invariants, we show that they form an invariant variety of dimension $p$ if the periodicity conditions are `fully correlated', and a set of isolated points if the…

Given a bounded n-connected domain in the plane bounded by non-intersecting Jordan curves, and given one point on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping of the domain onto the unit disc that…

We give an explicit example of a fibration $f \colon X \to Y$ between smooth projective varieties whose "orbifold base" $\Delta_f$ in the sense of Campana has the property that the induced morphism $X \to (Y, \Delta_f)$ is not a morphism of…

Consider a complex one dimensional foliation on a complex surface near a singularity $p$. If $\mathcal{I}$ is a closed invariant set containing the singularity $p$, then $\mathcal{I}$ contains either a separatrix at $p$ or an invariant real…

We consider several natural sets of curves associated to a given Teichm\"uller disc, such as the systole set or cylinder set, and study their coarse geometry inside the curve graph. We prove that these sets are quasiconvex and agree up to…

Let $G$ be a reductive complex Lie group and $K$ be a maximal compact subgroup of $G$. Let $X$ be a reduced Stein $G$-space and $Y$ be a $G$-elliptic manifold. We prove the following parametric equivariant Oka principle. The inclusion of…

For every topological group G one can define the universal minimal compact G-space X=M_G characterized by the following properties: (1) X has no proper closed G-invariant subsets; (2) for every compact G-space Y there exists a G-map X-->Y.…

In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain…

We show: If a bounded domain in a Stein space covers a compact complex space, it must be smooth. This give a negative answer to a question of Koll\'ar. Furthermore, we deduce some related results.