arXiv subjects
Publications and source records attributed to Tien-Cuong Dinh.
Let f_1, f_2 be holomorphic endomorphisms of P^k, of degrees d_1\geq 2, d_2\geq 2. Assume that f_1\circ f_2=f_2\circ f_1 and that d_1^{n_1}\not =d_2^{n_2} for all integers n_1, n_2. We then show that f_j are critically finite. Moreover there is an orbifold (P^k,n) such that f_1, f_2 are coverings of (P^k,n). In the P^2 case we give the list of commuting pairs satisfying the above conditions.
We determine all couples of commuting polynomial endomorphisms of C^2 that extends to holomorphic endomorphisms of P^2 and that have disjoint sequences of iterates.
Let E be a compact set of C of positive logarithmic capacity. Let us suppose that for every polynomial $P\not=id$ we have $P^{-1}(E)\not=E$. Then for all no constant polynomials f and g such that $f^{-1}(E)=g^{-1}(E)$ we have f=g.
Nous demontrons qu'une sous-variete reelle, compacte, orientee et lisse Γde dimension $2p-1\geq 3$ de CP^n est le bord d'un sous-ensemble analytique s'il existe une variete reelle $V\subset G(n-p+2,n+1)$ de codimension 1 satisfaisant les conditions suivantes pour tout $ν\in V$ 1. La reunion $\bigcup_{ν\in V} P^{n-p+1}_ν$ recouvre un ouvert dense de Γ. 2. Le (n-p+1)-plan $P^{n-p+1}_ν$ intersecte Γtransversalement. 3. $Γ\cap P^{n-p+1}_ν$ est le bord d'une surface de Riemann dans $P^{n-p+1}_ν$. 4. Aucun ouvert non vide de $Γ\cap P^{n-p+1}_ν$ n'est reel analytique. Pour la preuve, nous utilisons et demontrons le resultat suivant: pour toute surface de Riemann a bord rectifiable (eventuellement reductible et singuliere) S d'une variete complexe, $\bar S$ admet un systeme fondamental de voisinages de Stein. Il existe Γreelle algebrique verifiant 1-3, qui n'est pas bord d'un sous-ensemble analytique.
We give a wedge removability theorem for metrically thin sets of two codimensional Hausdorff null measure. This removability theorem combined with the wedge removability theorem of Merker for closed subsets of two codimensional manifolds, gives a CR-meromorphic extension theorem in the greater codimensional case.
Let $D\subset\subset\mathbb{C}^n$ be a complex manifold of dimension $p\geq 2$ with $\C^2$ boundary in $\mathbb{C}^n$. Let $f$ be a $\C^1$ function on $bD$ and $V$ a generic and large enough family of complex $(n-p+1)$-planes. Let suppose that for $ν\in V$, no connected component of $bD\cap \mathbb{C}^{n-p+1}_ν$ is "almost" real analytic and that $f$ extends holomorphically in $D\cap\mathbb{C}^{n-p+1}_ν$. Then $f$ extend as a holomorphic function in $D$. In a special case, this result gives a partial answer to a conjecture of Globevnik-Stout. By generalizing the theorem of Harvey-Lawson, we prove a Morera type theorem for the boundary problem in $\mathbb{C}^n$ which answer to a problem asked by Dolbeault and Henkin.