We associate to any germ of an analytic variety a Lie algebra of tangent vector fields, the {\it tangent algebra}. Conversely, to any Lie algebra of vector fields an analytic germ can be associated, the {\it integral variety}. The paper investigates properties of this correspondence: The set of all tangent algebras is characterized in purely Lie algebra theoretic terms. And it is shown that the tangent algebra determines the analytic type of the variety.
Under very general conditions it is shown that if $A$ is a uniform algebra generated by real-analytic functions, then either $A$ consists of all continuous functions or else there exists a disc on which every function in $A$ is holomorphic. This strengthens several earlier results concerning uniform algebras generated by real-analytic functions.
We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.
We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.
It was once conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$, and if each point of $X$ is a peak point for $A$, then $A = C(X)$. This peak-point conjecture was disproved by Brian Cole in 1968. Here we establish a peak-point theorem for uniform algebras generated by real-analytic functions on real-analytic varieties, generalizing previous results of the authors and John Wermer.
We introduce a cohomology, called extendable cohomology, for abstract complex singular varieties based on suitable differential forms. Beside a study of the general properties of such a cohomology, we show that, given a complex vector bundle, one can compute its topological Chern classes using the extendable Chern classes, defined via a Chern-Weil type theory. We also prove that the localizations of the extendable Chern classes represent the localizations of the respective topological Chern classes, thus obtaining an abstract residue theorem for compact singular complex analytic varieties. As …
The main result of [HL] (Annals of Math. 102 (1975), 223--290) gives a characterization of the boundaries of complex subvarieties in C^n. An application of this result concerning pseudoconvexity, namely Theorem 10.4 in [HL], is amplified.
We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H. We deal with the problem by cutting with a family of complex hyperplanes in the fashion of [2] and applying the first named author's result for the compact case [13].
Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of $\mathbb C^n$, $n\geq 2$, consider its (random) zero variety $Z(f_L)$. We study the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ inside a pseudo-hyperbolic ball of radius $r$. We first express this variance as an integral of a positive function in the unit disk. Then we study its asymptotic behaviour as $L\to\infty$ and as $r\to 1^{-}$. Both the results and the proofs generalise to the ball those given by Jeremiah Buckley for the unit disk.
The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi…
We study singular curves from analytic point of view. We give completely analytic proofs for the Serre duality and a generalized Abel's theorem. We also reconsider Picard varieties, Albanese varieties and generalized Jacobi varieties of singular curves analytically. We call an Albanese variety considered as a complex Lie group an analytic Albanese variety. We investigate them in detail. For a non-singular curve (a compact Riemann surface) $X$, there is the relation between the meromorphic function fields on $X$ and on its Jacobi variety $J(X)$. We try to extend this relation to the case of sin…
We show that biholomorphic automorphisms of a real analytic CR manifold can be considered as (pointwise) Lie symmetries of a holomorphic second order PDE system defining its Segre family. This allows to use general methods of the geometric PDE theory in order to study the CR geometry of real analytic manifolds.
Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of the pair $(f,X)$, the following relative analog of the well known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce-Roberts numbers) is equivalent to the relative quasihomogeneity of the pair $(f,X)$, i.e. to the existence of a coordinate system such that both $f$ and $X$ are quasih…
We prove that every smooth CR manifold $M\subset\subset \C^n$, of hypersurface type, has a complex strip-manifold extension in $\C^n$. If $M$ is, in addition, pseudoconvex-oriented, it is the "exterior" boundary of the strip. In turn, the strip extends to a variety with boundary $M$ (Rothstein-Sperling Theorem); in case $M$ is contained in a pseudoconvex boundary with no complex tangencies, the variety is embedded in $\C^n$. Altogether we get: $M$ is the boundary of a variety (Harvey-Lawson Theorem); if $M$ is pseudoconvex oriented the singularities of the variety are isolated in the interior;…
If $f$ is a non-Archimedean analytic curve in a projective variety $X$ embedded in $P^N$ and if $D_1,\dots,D_q$ are hypersurfaces of $P^N$ in general position with $X,$ then we prove the defect relation: $$ \sum_{j=1}^q δ(f,D_j) \le \dim X. $$
If $V$ is an analytic set in a pseudoconvex domain $Ω$, we show there is always a pseudoconvex domain $G \subseteq Ω$ that contains $V$ and has the property that every bounded holomorphic function on $V$ extends to a bounded holomorphic function on $G$ with the same norm. We find such a $G$ for some particular analytic sets. When $Ω$ is an operhedron we show there is a norm on holomorphic functions on $V$ that can always be preserved by extensions to $Ω$.
In this paper, we study the amoeba volume of a given $k-$dimensional generic analytic variety $V$ of the complex algebraic torus $(\C^*)^n$. When $n\geq 2k$, we show that $V$ is algebraic if and only if the volume of its amoeba is finite. In this precise case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the $k-$linear spaces will be given.
Using the classical S.Lie method we obtain a complete description of infinitesimal symmetries of a holomorphic PDE system defining the Segre family of a real analytic hypersurface. This gives a new proof of some well known results of CR geometry.
In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which have been motivated by classical works of Darboux, Poincaré and Painlevé on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincaré and Painlevé problem of bounding the degree and the genus of analytic varieties invariant by holomorphic foliations and Pfaff systems. We shall discuss the general idea…
The subject considered in this paper has, at least, three points of interest. Suppose that we have a sequence of one-dimensional analytic varieties in a domain in $\Bbb C^n$. The cluster of this sequence consists from all points in the domains such that every neighbourhood of such points intersects with infinitely many different varieties. The first question is: what analytic properties does the cluster inherit from varieties? We give a sufficient criterion when the cluster contains an analytic disk, but it follows from examples of Stolzenberg and Wermer that, in general, clusters can contain …
In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-dimensional spherical polyhedron, then each irreducible component of the variety is algebraic. This gives a converse of a theorem of Bieri and Groves and generalizes a result proven in \cite{MN2-11}. More precisely, if the dimension of the ambient space is at least twice of the dimension of the generic analytic subvariety, then these properties are equivalent to the volume of the amoeba of the subvariety being finite.
The systematic study of neighborhoods of analytic varieties was started by H. Grauert in his celebrated article 1962. In that article he considers a manifold X and a negatively embedded submanifold A subset X. He introduces n-neighborhood, n in N, of A and studies when an isomorphism of two n-neighborhoods can be extended to an isomorphism of (n+1)-neighborhoods. He observes that obstructions to this extension problem lie in the first cohomology group of certain sheaves involving the normal bundle of A in X. Using a version of Kodaira vanishing theorem he shows that for a large n these cohomol…
By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre theory of coherent analytic sheaves.
It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski's class $\mathscr{Q}_G$ has a realisation as a good quotient, and that every complete algebraic variety in $\mathscr{Q}_G$ is unirational with finitely generated Cox ring and at worst rational singular…
Iterated Segre mappings of real analytic generic submanifolds in complex space have been an essential tool in the study of holomorphic, formal, and CR mappings between such manifolds. In this paper we present a theory of iterated Segre mappings for smooth or formal submanifolds. In particular, we characterize finite type in terms of the rank of the iterated Segre mappings. The approach given here is new, simplifies proofs, and extends results previously known for the real analytic case.
For a lattice $Γ$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $Γ$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of $Γ$ at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bund…
Arquile varieties are zerosets of polynomial, algebraic, analytic, or formal equations f(t,y_1,...,y_m) = 0 with solutions y(t) = (y_1(t),...,y_m(t)) in affine m-space over an algebraic, convergent or formal power series ring k<t>, k{t}, or k[[t]]. As such they generalize the concept of the arc space of an algebraic variety. In the article, the geometry of arquile varieties is studied in detail. Among other things, it is shown that, after a suitable stratification, their singularities, once defined appropriately, are confined to a finite dimensional part. The main technique to do this is to co…
The purpose of this paper is to explore Nevanlinna theory of the entire curve $\exh_A f:=(\exp_Af,f):\C \to A \times \Lie(A)$ associated with an entire curve $f: \C \to \Lie(A)$, where $\exp_A:\Lie(A)\to A$ is an exponential map of a semi-abelian variety $A$. Firstly we give a Nevanlinna theoretic proof to the {\em analytic Ax-Schanuel Theorem} for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z)-f(0) \in \Lie(A)\iso \C^n$ ar…
Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or $\mathrm{dim}(X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial}$ complex, denoted here $\overlineð_{\mathrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\mathrm{dim}(\mathrm{sing}(X))=0$, is proved also for $\overlineð_{\mathrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overli…
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formu…
