In this paper, we consider local holomorphic mappings f: M\to M' between real algebraic CR generic manifolds (or more generally, real algebraic sets with singularities) in the complex euclidean spaces of different dimensions and we search necessary and sufficient conditions for f to be algebraic. These conditions appear to exclude two certain flatness of M and of M'. From the point of view of CR geometry, a real analytic CR manifold M can be flat in essentially to ways, being biholomorphic to a product M_1\times Δ^k, k\geq 1, by a polydisc (algebraic degeneracy), or to a product M_1\times I^l,l\geq 1 by a real cube (transversal degeneracy), in a neighborhod of a Zariski generic point. We also require that the CR manifold M is minimal in the sense of Tumanov at a generic point. Our first result provides a characterization of mappings with positive transcendence degree k. Such maps have the property that near a Zariski generic point f(p) in M', there exists a k-algebraically degenerate real algebraic set X'' which contains f(M) and which is contained in M'. This solves the algebraic mapping problem for a minimal source M completely. Our second main result is the construction of canonical foliations by Segre surfaces of the extrinsic complexification of M. We prove in particular that a holomorphic function defined in a neighborhood of a minimal M is algebraic if and only if its restriction to each Segre surface of M is algebraic. We also show by an example that the double reflection foliation in the spirit of tangential CR derivations does not yield a characterization of positivity of transcendence degree of holomorphic mappings.
In this note, our purpose is to establish shortly the algebraicity of a holomorphic mapping between real algebraic CR manifolds under a double reflection condition which generalizes the classical single reflection. A complete study of various double reflection conditions is also provided.
We give conditions under which a germ of a holomorphic mapping in $\Bbb C^N$, mapping an irreducible real algebraic set into another of the same dimension, is actually algebraic. Let $A\subset \bC^N$ be an irreducible real algebraic set. Assume that there exists $\po \in A$ such that $A$ is a minimal, generic, holomorphically nondegenerate submanifold at $\po$. We show here that if $H$ is a germ at $p_1 \in A$ of a holomorphic mapping from $\bC^N$ into itself, with Jacobian $H$ not identically $0$, and $H(A)$ contained in a real algebraic set of the same dimension as $A$, then $H$ must extend …
We prove that a germ of a holomorphic map $f$ between $C^n$ and $C^{n'}$ sending one real-algebraic submanifold $M\subset C^n$ into another $M'\subset C^{n'}$ is algebraic provided $M'$ contains no complex-analytic discs and $M$ is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.
Contents 1. Algebraicity criterion: statement 2. Proof of the algebraicity criterion. 3. Pseudoeffectivity and movable classes. 4. Harder-Narasimhan filtrations and pseudo-effectivity. 5. Pseudo-effectivity of relative canonical bundles. 6. Rational curves and non-pseudoeffectivity of the canonical/cotangent bundles. 7. Birational stability of the cotangent bundle 8. Shafarevich-Viehweg conjecture. 9. Numerically trivial foliations. 10. The Beauville-Bogomolov-Yau decomposition 11. Some questions 11.1. Pseudoeffectivity of determinants. 11.2. Variations on Abundance. 11.3. Special manifolds. R…
We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.
In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.
We define a new local invariant (called degeneracy) associated to a triple (M,M',H), where M and M' are real submanifolds of C^N and C^N', respectively, and H: M->M' is either a holomorphic map, a formal holomorphic map, or a smooth CR-map. We use this invariant to find sufficient conditions under which finite jet dependence, convergence and algebraicity results hold.
T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.
We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves contained in C^2 with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of "key forms" - a 'glob…
We consider the algebraic degeneracy of holomorphic curves from a point of view of meromorphic vector fields. Employing the notion of Jocabian sections introduced by W. Stoll, we establish a Second Main Theorem type inequality. As consequences, several algebraic degeneracy theorems are obtained for holomorphic curves into a complex projective variety.
We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle on an algebraic curve. Using this, we study the asymptotic distribution of the zeros of random sequences of sections in these spaces.
We show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into a sphere in $\bC^{N+1}$ for any $N$. In fact, we show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into any compact, strictly pseudoconvex, real algebraic hypersurface.
We present a geometric proof of the theorem saying that holomorphic maps from Runge domains to affine algebraic varieties admit approximation by Nash maps. Next we generalize this theorem.
We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.
We show the following algebraicity result for a complex projective variety $X$ with big representation of $π_1$ into a semi-simple algebraic group: There exists a proper subvariety $Z \subset X$ such that for any algebraic curve $C$, any holomorphic map $γ:\, C \to X$ with $γ(C) \not\subset Z$ is induced from an algebraic morphism. As an application, we prove pseudo-Brody hyperbolicity of certain varieties with big reductive representations of $π_1$.
We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of $\C^N$ and a real-algebraic CR-submanifold of $\C^{N'}$.
C-holomorphic functions defined on algebraic sets and having algebraic graphs can be considered as a complex counterpart of regulous functions introduced recently in real geometry. This note is a part of our study on the subject; we prove herein some effective Nullstellensätze.
Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.
We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.
We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.
We show that a $2k$-current $T$ on a complex manifold is a real holomorphic $k$-chain if and only if $T$ is locally real rectifiable, $d$-closed and has $\mathcal{H}^{2k}$-locally finite support. This result is applied to study homology classes represented by algebraic cycles.
The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.
We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.
