Related papers: On total reality of meromorphic functions
Let $M\subset \mathbb C^n$ be a real analytic hypersurface, $M'\subset \mathbb C^N$ $(N\geq n)$ be a strongly pseudoconvex real algebraic hypersurface of the special form and $F$ be a meromorphic mapping in a neighborhood of a point $p\in…
The subject of this paper is the bounded level curves of a meromorphic function $f$ with domain $G$ such that each component of $\partial{G}$ consists of a level curve of $f$. (A primary example of such a function being a ratio of finite…
We prove that every 4-dimensional oriented handlebody without 3- and 4-handles can be modified to admit infinitely many exotic smooth structures, and moreover prove that their genus functions are pairwise equivalent. We furthermore show…
It was shown by A. Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then ${\rm deg}(\varphi_{|K_M|})\leq 36$. The first example of a surface with canonical degree 36 was…
We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.
Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical…
We show that a general plane curve of degree at least 4 is uniquely determined by the full set of its bitangent lines. This problem has an elementary solution for degree at least 5, and the paper is almost entirely devoted to curves of…
Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$…
A trivial automorphism of the Boolean algebra $\mathcal P(\mathbb N) / \mathrm{Fin}$ is an automorphism induced by the action of some function $\mathbb N \rightarrow \mathbb N$. In models of forcing axioms all automorphisms are trivial, and…
We prove several reality properties for finite simple orthogonal groups. For any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are strongly real in the simple groups $\mathrm{P}\Omega^{\pm}(4m+2,q), m \geq 1$,…
We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is…
We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with…
We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…
The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…
For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of…
We determine the maximum number of rational points on a curve over $\mathbb{F}_2$ with fixed gonality and small genus.
The main result establishes an estimate for the growth of a real meromorphic function $f$ on the unit disc $\Delta$ such that: (i) at least one of $f$ and $1/f$ has finitely many poles and non-real zeros in $\Delta$; (ii)~$f^{(k)}$ has…
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…
The objective of this paper is to further study the anabelian object referred to as \emph{pointed virtual curves}. Building upon previous work that investigated these fundamental-group-theoretic pullbacks of Galois sections in the…
In this paper, we prove that if the area functional of a surface $\Sigma^2$ in a symplectic manifold $(M^{2n},\bar{\omega})$ has a critical point or has a compatible stable point in the same cohomology class, then it must be…