Analytic Subvariety Research Articles

Croissance de la trace d'un courant positif fermé sur les plans complexes de [formula omitted

Let T be a closed positive current of dimension p in C n and H an analytic subvariety of C n of dimension n − q ( 1 ⩽ q ⩽ p ) . We denote by 〈 T , H 〉 the slice of T in H if it exists. We consider the functions on ] 0 , + ∞ [ defined by n T ( r ) = r − 2 p ∫ | z | ⩽ r T ∧ β p and n T ( H , r ) = r − 2 ( p − q ) ∫ | z | ⩽ r 〈 T , H 〉 ∧ β p − q . The aim of this paper is to establish relationship between the growth of these functions. We prove results generalizing those of L. Gruman in [Séminaire Lelong–Skoda 1981–1983, Lecture Notes in Math., vol. 1028, Springer-Verlag, Berlin, 1983, pp. 125–162] and B. Molzan, B. Shiffman and N. Sibony in [Math. Ann. 257 (1981) 43].

Sur la transformation d'Abel–Radon de courants localement résiduels

The aim of this Note is to give a generalisation of the following theorem of Henkin and Passare: let Y be an analytic subvariety of pure codimension p in a linearly p-concave domain U, and ω a meromorphic q-form ( q>0) on it; if the Abel–Radon transform A(ω∧[Y]) , which is meromorphic on U ∗ , has a meromorphic prolongation to U ∗ , then Y extends to an analytic subvariety of U , and ω to a meromorphic form on it. The problem is to show the analogous statement when we replace ω∧[ Y] by a current α of a more general type, called locally residual. We give the proof if α is of bidegree ( N,1), or ( q+1,1), 0< q< N in the particular case where A(α)=0 . We conclude with some applications of the theorem. To cite this article: B. Fabre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on $\mathcal \{A\}(\mathbb \{R\}^4)$

The local Phragmen-Lindelof condition for analytic subvarieties of C n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hormander has shown. Here, necessary geometric conditions for this Phragmen-Lindelof condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in C 3 . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on R 4 .

Truncated microsupport and holomorphic solutions of D-modules

We study the truncated microsupport SS k of sheaves on a real manifold. Applying our results to the case of F=R Hom D ( M, O) , the complex of holomorphic solutions of a coherent D -module M , we show that SS k ( F) is completely determined by the characteristic variety of M . As an application, we obtain an extension theorem for the sections of H j ( F), j< d, defined on an open subset whose boundary is non-characteristic outside of a complex analytic subvariety of codimension d. We also give a characterization of the perversity for C -constructible sheaves in terms of their truncated microsupports.

A {̄∂}∂–Poincaré lemma for forms near an isolated complex singularity

Let X X be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let η \eta be a smooth form of type ( 1.1 ) (1.1) defined on X ∖ { 0 } X \setminus \{0\} . The main result of this note is a criterion for solubility of the equation ∂ ¯ ∂ u = η \bar {\partial }\partial u = \eta . This implies a criterion for triviality of a Hermitian– holomorphic line bundle ( L , h ) → X ∖ { 0 } (L,h)\to X\setminus \{0\} in a neighbourhood of the origin.

Fatou maps in ℙn dynamics

We study the dynamics of a holomorphic self‐map f of complex projective space of degree d &gt; 1 by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics of f are a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified by f (and therefore any hyperbolic periodic point attracts a point of the critical set of f). We also show that Fatou components are hyperbolically embedded in ℙn and that a Fatou component which is attracted to a taut subset of itself is necessarily taut.

Volumes of complex analytic subvarieties of Hermitian symmetric spaces

We give lower bounds of volumes of k -dimensional complex analytic subvarieties of certain naturally defined domains in n -dimensional complex space forms of constant (positive, zero, or negative) holomorphic sectional curvature. For each 1 ≤ k ≤ n , the lower bounds are sharp in the sense that these bounds are attained by k -dimensional complete totally geodesic complex submanifolds. Such lower bounds are obtained by constructing singular potential functions corresponding to blow-ups of the Kähler metrics involved. Similar lower bounds are also obtained in the case of Hermitian symmetric spaces of noncompact type. In this case, the lower bounds are sharp for those values of k at which the Hermitian symmetric space contains k -dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature.

Projective resolutions of coherent sheaves and descent relations between branes

We notice that, for branes wrapped on complex analytic subvarieties, the algebraic-geometric version of K-theory makes the identification between brane-antibrane pairs and lower-dimensional branes automatic. This is because coherent sheaves on the ambient variety represent gauge bundles on subvarieties, and they can be put in exact sequences (projective resolutions) with sheaves corresponding to vector bundles on the pair; this automatically gives a D(p-2) as a formal difference of bundles on the Dp - D\bar p pair, both belonging to the Grothendieck group of coherent sheaves of the ambient.

On Hartshorne's problem for compact [formula omitted]-analytic surfaces M with κ( M)=−∞

Let M be a compact C -analytic surface, let Γ⊂ M be a compact analytic subvariety and let X:= M\\ Γ. The following two problems will be considered: Assume that X does not contain any compact curve and that Γ is an irreducible compact curve in X such that Γ 2≥0 (respectively assume that the analytic cohomology groups H 1(X,Ω p)=0 , for all 0≤ p≤2). Is X always Stein? It is our main purpose here to provide an affirmative answer to those two problems, provided M is a (minimal) compact C -analytic surface such that its Kodaira dimension κ( M)=−∞. Also the affine structure of such Stein surfaces will be discussed.

Rigid analytic flatificators

Let K be an algebraically closed field endowed with a complete non-archimedean norm. Let f : Y → X be a map of K-affinoid varieties. We prove that for each point x ∈ X, either f is flat at x, or there exists, at least locally around x, a maximal locally closed analytic subvariety Z ⊂ X containing x, such that the base change f−1(Z)→ Z is flat at x, and, moreover, g−1(Z) has again this property in any point of the fibre of x after base change over an arbitrary map g : X′ → X of affinoid varieties. If we take the local blowing up π : X → X with this centre Z, then the fibre with respect to the strict transform f of f under π, of any point of X lying above x, has grown strictly smaller. Among the corollaries to these results we quote, that flatness in rigid analytic geometry is local in the source; that flatness over a reduced quasi-compact rigid analytic variety can be tested by surjective families; that an inclusion of affinoid domains is flat in a point, if it is unramified in that point.

Subvarieties of ℂn with non-extendable automorphisms

Abstract We investigate algebraic and analytic subvarieties of ℂn with automorphisms which cannot be extended to the ambient space.

Deformations of trianalytic subvarieties of hyperkähler manifolds

Let M be a compact complex manifold equipped with a hyperkahler metric, and X be a closed complex analytic subvariety of M. In alg-geom 9403006, we proved that X is trianalytic (i.e., complex analytic with respect to all complex structures induced by the hyperkahler structure), provided that M is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of X are trianalytic and naturally isomorphic to X as complex analytic varieties. We show that this isomorphism is compatible with the metric induced from M. Also, we prove that the Douady space of complex analytic deformations of X in M is equipped with a natural hyperkahler structure.

On the problems of Hartshorne and Serre for some -analytic surfaces

authorAbstract Let M be a compact -analytic surface, let Γ ⊂ M be a compact analytic subvariety, and let X := Mx00393;. We are interested in the following two problems: Assume that X does not contain any compact curve and that Γ is an irreducible compact curve such that Γ2 ≥ 0 (resp. assume that the analytic cohomology groups H1 (X, Ωp) = 0, for all 0 ≤ p ≤ 2). Is X always Stein? It is our main purpose here to provide an affirmative answer to those two problems, provided M is either a (minimal) ruled surface or a non-algebraic compact surface. Also, the affine structure of such Stein surfaces will be discussed.

Locally stable holomorphic maps and their application to a global moduli problem for some kinds of analytic subvarieties

Locally stable holomorphic maps and their application to a global moduli problem for some kinds of analytic subvarieties

Heights of projective varieties and positive Green forms

Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil, Schmidt, Nesterenko, Philippon, and Faltings. Several of their properties are proved, including lower bounds and an arithmetic Bézout theorem for the height of the intersection of two projective varieties. New estimates for the size of (generalized) resultants are derived. Among the analytic tools used in the paper are “Green forms” for analytic subvarieties, and the existence of positive Green forms is discussed.

General Integral Representation of the Holomorphic Functions on the Analytic Subvariety

Henkin and Ramirez obtained an integral representation of holomorphic functions for strictly pseudoconvex domains in Range and Siu gave a generalization of Henkin-Ramirez's formula to the domains in C with piecewise smooth strictly pseudoconvex boundaries Sommer proved an integral formula of Weil type for analytic polyhedra in C. Sergeev and Henkin also obtained an integral representation for the strictly pseudoconvex ployhedra. Stout and Hatziafratis have respectively proved integral formulas for strictly pseudoconvex domains in codimension-one and codimension-m complex submanifolds of The formula which was given by Stout is valid not only for nonsingular hyper surf aces, but also for certain subvarieties which may possess sufficiently restricted singular points. Hatziafratis' work is based on the results of Stout. In this paper we derive integral formulas which include all the above ([!]-[?]) integral formulas for holomorphic functions. The papers of Stout, Hatziafratis and the author are most relevant references to this work.

Currents, metrics and Moishezon manifolds

A compact complex manifold M is Moishezon if and only if there exists an integral closed positive (1, 1)-cument ω such that ω ≥ Eσ and ω is smooth outside an analytic subvariety

On locally trivial families of analytic subvarieties with locally stable parametrizationsof compact complex manifolds

On locally trivial families of analytic subvarieties with locally stable parametrizationsof compact complex manifolds

On Verdier's specialization formula for Chern classes

Let M be a complex manifold of complex dimension m, and let X C M be a proper analytic subvariety. Let D C ~ be the unit disk, and f : X ~ D a proper analytic morphism. Put X~: = f l(s), s e D. We know that, for r > 0 sufficiently small, the inclusion Xo C f I(D,) is a homotopy equivalence. Thus for small nonzero s e D we may define the specialization map on homology a. :H,(X~)~H.(Xo) as the composition ~, H,(X~) , H , ( f I(D,)) ~ n,(Xo).

Moduli spaces of quadratic differentials

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μπ, on the unit hypersurface, V1(π), for the Teichmuller metric. Μπ is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be Lp(Μπ), 0 <p < 1, over the moduli space V1(π)/γ (g, n). In particular, Μπ(V1(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur.