Complex Analytic Subvariety Research Articles

On the Volumes of Analytic Varieties Near the Diagonal for a Compact Quotient of the Polydisk

We give sharp lower bounds for the volumes of one-dimensional complex analytic subvarieties in factorwise geodesic tubular neighborhoods of the Cartesian self-product of a compact quotient of the polydisk. As an application, we obtain a sufficient condition for the pluri-canonical bundles of a compact quotient of the polydisk to satisfy the property $$(N_p)$$ on linear syzygies.

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

We study the extension problem for germs of holomorphic isometries f: (D;x_0) \to (\Omega;f(x_0)) up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics ds_D^2 on D and ds_\Omega^2 on \Omega . Our main focus is on boundary extension for pairs of bounded domains (D,\Omega) such that the Bergman kernel K_D(z,w) extends meromorphically in (z,\overline w) to a neighborhood of \overline D \times D , and such that the analogous statement holds true for the Bergman kernel K_{\Omega}(\zeta,\xi) on \Omega . Assuming that (D;ds_D^2) and (\Omega;ds_\Omega^2) are complete Kähler manifolds, we prove that the germ of map f extends to a proper holomorphic isometric embedding such that Graph (f) extends to a complex-analytic subvariety on some neigborhood of \overline D \times \overline{\Omega} . In the event that the Bergman kernel K_D(z,w) extends to a rational function in (z;\overline w) and the analogue holds true for the Bergman kernel K_{\Omega}(\zeta,\xi) , we show that Graph (f) extends to an affine-algebraic variety. Our results apply especially to pairs (D,\Omega) of bounded symmetric domains in their Harish-Chandra realizations. When D is the complex unit ball B^n of dimension n \ge 2 , we obtain a new rigidity result which guarantees the total geodesy of the map under certain conditions. On the other hand, we construct examples of holomorphic isometries of the unit disk into polydisks which are not totally geodesic, answering in the negative a conjecture of Clozel-Ullmo's.

Syzygies of compact complex hyperbolic manifolds

We give a sufficient condition for the pluri-canonical bundles of a compact complex hyperbolic manifold to satisfy the property ( N p ) (N_p) on linear syzygies in terms of the hyperbolic injectivity radius. In the process, we obtain sharp lower bounds for the volumes of one-dimensional complex analytic subvarieties in geodesic tubular neighborhoods of the Cartesian self-product of a compact complex hyperbolic manifold.

Analyticity of compact complements of complete Kähler manifolds

Let X be a Stein manifold, dim ℂ X ≥ 2, K a compact subset of X, and Ω an open subset of X containing K such that Ω\K is connected. Suppose that Ω\K carries a complete Kahler metric of bounded bisectional curvature, and locally of finite volume near K. If K admits a Stein neighborhood V, V ⊆ Ω, such that V\K is connected and H 2 (V, ℝ) = 0, then K is a complex analytic subvariety of X, hence reduced to a finite number of points.

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface M ⊂ C N M\subset \mathbb {C}^N , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p ∈ M p\in M the local real-analytic CR automorphisms of M M fixing p p can be parametrized real-analytically by their ℓ p \ell _p jets at p p . As a direct application, we derive a Lie group structure for the topological group Aut ⁡ ( M , p ) \operatorname {Aut}(M,p) . Furthermore, we also show that the order ℓ p \ell _p of the jet space in which the group Aut ⁡ ( M , p ) \operatorname {Aut}(M,p) embeds can be chosen to depend upper-semicontinuously on p p . As a first consequence, it follows that given any compact real-analytic hypersurface M M in C N \mathbb {C}^N , there exists an integer k k depending only on M M such that for every point p ∈ M p\in M germs at p p of CR diffeomorphisms mapping M M into another real-analytic hypersurface in C N \mathbb {C}^N are uniquely determined by their k k -jet at that point. Another consequence is the following boundary version of H. Cartan’s uniqueness theorem: given any bounded domain Ω \Omega with smooth real-analytic boundary, there exists an integer k k depending only on ∂ Ω \partial \Omega such that if H : Ω → Ω H\colon \Omega \to \Omega is a proper holomorphic mapping extending smoothly up to ∂ Ω \partial \Omega near some point p ∈ ∂ Ω p\in \partial \Omega with the same k k -jet at p p with that of the identity mapping, then necessarily H = Id H=\textrm {Id} . Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

Remarks on the Alexander-Wermer theorem for curves

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact we establish a slightly stronger version which applies to a wider class of boundary 1-cycles. Arguments here are based on the Hahn-Banach Theorem and some geometric measure theory. Several ingredients in the original proof have been eliminated. 1. The Alexander-Wermer Theorem We present a different proof of the Alexander-Wermer Theorem [AW], [W2] for curves which uses the Hahn-Banach Theorem and techniques of geometric measure theory. Several ingredients of the original proof are eliminated, such as the reliance on the result in [HL1] that if a curve satisfies the moment condition, then it bounds a holomorphic 1-chain. The arguments given here have been adapted to study the analogous problem in general projective manifolds (cf. [HL3,4,5]). Our arguments will also apply to a more general class of curves which we now introduce. ∗Partially supported by the N.S.F. 54 F. R. HARVEY H. B. LAWSON, JR Definition 1.1. Let X be a complex manifold and suppose there exists a closed subset Σ(Γ) of Hausdorff 1-measure zero and an oriented, properly embedded C-submanifold of X−Σ(Γ) with connected components Γ1,Γ2, .... If, for given integers n1, n2, ..., the sum Γ = ∑∞ k=1 nkΓk defines a current of locally finite mass in X which is d-closed (i.e., without boundary), and if sptΓ has only a finite number of connected components, then Γ will be called a scarred 1cycle (of class C) in X. By a unique choice of orientation on each Γk we may assume each nk > 0. Exemple 1.2. Any real analytic 1-cycle is automatically a scarred 1-cycle (of class C) – see [F, p. 433]. Definition 1.3. By a positive holomorphic 1-chain with boundary Γ we mean a sum V = ∑∞ k=1 mk[Vk] with mk ∈ Z + and Vk an irreducible 1dimensional complex analytic subvariety of X − sptΓ such that V has locally finite mass in X and, as currents in X,

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.

Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

The classical Schwarz reflection principle states that a continuous map f between real-analytic curves M and M ′ in C that locally extends holomorphically to one side of M , extends also holomorphically to a neighborhood of M in C. It is well-known that the higher-dimensional analog of this statement for maps f : M → M ′ between real-analytic CR-submanifolds M ⊂ C andM ′ ⊂ C ′ does not hold without additional assumptions (unlessM andM ′ are totally real). In this paper, we assume that f is C-smooth and that the target M ′ is real-algebraic, i.e. contained in a real-algebraic subset of the same dimension. If f is known to be locally holomorphically extendible to one side of M (when M is a hypersurface) or to a wedge with edge M (when M is a generic submanifold of higher codimension), then f automatically satisfies the tangential CauchyRiemann equations, i.e. it is CR. On the other hand, if M is minimal, any CR-map f : M → M ′ locally extends holomorphically to a wedge with edge M by Tumanov’s theorem [Tu88] and hence, in that case, the extension assumption can be replaced by assuming f to be CR. Local holomorphic extension of a CR-map f : M → M ′ may clearly fail when M ′ contains an (irreducible) complex-analytic subvariety E ′ of positive dimension and f(M) ⊂ E . Indeed, any nonextendible CR-function on M composed with a nontrivial holomorphic map from a disc in C into E ′ yields a counterexample. Our first result shows that this is essentially the only exception. Denote by E ′ the set of all points p ∈ M ′ through which there exist irreducible complex-analytic subvarieties of M ′ of positive dimension. We prove: Theorem 1.1. Let M ⊂ C and M ′ ⊂ C ′ be connected real-analytic and real-algebraic CRsubmanifolds respectively. Assume that M is minimal at a point p ∈ M . Then for any C-smooth CR-map f : M → M , at least one of the following conditions holds: (i) f extends holomorphically to a neighborhood of p in C ; (ii) f sends a neighborhood of p in M into E . If M ′ is a real-analytic hypersurface, the set E ′ consists exactly of those points that are not of finite type in the sense of D’Angelo [D82] (see Lempert [L86] for the proof) and, in particular,

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.

Approximation and convergence properties of formal CR-maps

Let M⊂ C N be a minimal real-analytic CR-submanifold and M′⊂ C N′ a real-algebraic subset through points p∈ M and p′∈ M′ respectively. We show that that any formal (holomorphic) mapping f:( C N,p)→( C N′,p′) , sending M into M′, can be approximated up to any given order at p by a convergent map sending M into M′. If M is furthermore generic, we also show that any such map f, that is not convergent, must send (in an appropriate sense) M into the set E′⊂M′ of points of D'Angelo infinite type. Therefore, if M′ does not contain any nontrivial complex-analytic subvariety through p′, any formal map f sending M into M′ is necessarily convergent. To cite this article: F. Meylan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 671–676.

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.

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.

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.

Holomorphic automorphisms and cancellation theorems

Let X be a complex analytic space of positive dimension and A a complex analytic subvariety of X. We call A a direct factor of X if there exist a complex analytic space B and a biholomorphic mapping f: A × B → X such that, for some b ∊ B, f(a, b) = a on A, and a complex analytic space X to be primary if X has no direct factor, not equal to X itself, of positive dimension.

On subvarieties of a Hermitian manifold

We make use of the variational properties of the geodesic distance function of a Riemannian manifold and the technique of the “blowing-up” ( σ \sigma -process) on a complex manifold to derive the nonexistence of compact complex analytic subvarieties in a simply connected, complete Hermitian manifold with nonpositive sectional curvature.

An Application of Logic to Analysis

Let F be a complex analytic subvariety of an open subset of Cn and p ϵ V let be the germs at p of holomorphic, weakly holomorphic, infinitely differentiable, and k times continuously differentiate functions respectively. Spallek [15] has shown that for any p £ V there exists an integer such that , generalizing the result of Malgrange [12] that .In [14], Siu proved Spallek's result from a more sheaf theoretic point of view and showed the minimal integer function is bounded on compact sets. Bloom [7] reproved Malgrange's result by using differential operators on varieties.

On the Zariski tangent space to a complex analytic variety

Let (X, (9) be a complex space and x a point of X. T(X, x), the Zariski tangent space to X at x, is defined as the space of (U-linear derivations from the local ring (9 x to (U. As such it is an algebraic object. T(X, x) is a vector space and it is the minimal dimensional vector space into which a sufficiently small neighborhood of x in X may be embedded. In the case when X is a complex analytic subvariety of an open subset of (I~", Whitney [6, 7] has defined certain cones associated to the germ of X at x. These cones are subsets of the Zariski tangent space T (X, x). The largest of the Whitney cones Cs(X, x) is, in general, a proper subset of the Zariski tangent space. In fact, the vector subspace of T(X, x) spanned by Cs(X, x) is, in general, only a proper subspace of T(X, x). The cone Cs(X, x) is defined as the limiting positions of certain sequences of vectors. The purpose of this article is to define, via a geometric procedure analogous to that used to define Cs(X, x), an increasing sequence of cones Dj, I(X, x) in T(X, x), with D 1 ,! (X, x) -C s ( X , x) and for some integer Jl, Dj,,1 (X, x) = T(X, x). The minimal values Jl for which Di,,1 (X, x) is equal to T(X, x) will depend on the singularity of X at x. There are hig;ler order tangent spaces Tk(X, x) (see Definition 1.2) associated to the germ of X at x and we will define a similar sequence of cones in each one of these.

Volumes of Images of Varieties in Projective Space and In Grassmannians

If V is a complex analytic subvariety of pure dimension k in the unit ball in ${{\mathbf {C}}^n}$ which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of $(n - k)$-complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.

Volumes of images of varieties in projective space and in Grassmannians

If V is a complex analytic subvariety of pure dimension k in the unit ball in C n {{\mathbf {C}}^n} which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of ( n − k ) (n - k) -complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.

On the compactification of a Stein surface

Let M and S be complex manifolds of the same dimension, with M open and S compact. We shall say that S is a compactification of M if S contains a closed complex analytic subvariety F such that S-F is biholomorphically equivalent to M. It seems natural to ask what manifolds can occur as compactifications if various conditions are imposed on M. In particular, if M is a Stein manifold, what conclusions may be drawn concerning S? As a small first step toward answering this question, we take up the case of dimension two, and prove the following: Theorem. Let S be a compact, complex analytic surface, and let F be a closed subvariety of S such that S - F is a Stein manifold. Then either (i) S is an algebraic surface, or (if) the first Betti number bl(S)= 1 and S admits no nonconstant merornorphic function. The proof, which is a simple application of KODAIRA'S theory of surfaces ([4], [5]), will be given in the next section. First, however, we will show by an example that alternative (if) above does indeed occur. To this end, let W = C 2 - (0, 0), and let g : W-+ W be the mapping