dilatation and order of contact for holomorphic self maps of strongly convex domains let d be a bounded strongly convex domain and let f be a holomorphic self map of d in this paper we introduce and study the dilatation alpha f of f defined if f has no fixed points in d as the usual boundary dilatation coefficient of f at its wolff point or if f has some fixed points in d as the ratio of shrinking of the kobayashi balls around a fixed point of f in particular we show that the map alpha defined as alpha f mapsto alpha f in 0 1 is lower semicontinuous among other things this allows us to study the limits of a family of holomorphic self maps of d in the case of an inner fixed point the dilatation is an intrinsic measure of the order of contact of f d to partial d finally using complex geodesics we define and study a directional dilatation which is a measure of the shrinking of the domain along a given direction again results of semicontinuity are given and applied to a family of holomorphic self maps

tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

the complex monge ampere equation zoll metrics and algebraization let m be a real analytic riemannian manifold an adapted complex structure on tm is a complex structure on a neighborhood of the zero section such that the leaves of the riemann foliation are complex submanifolds this structure is called entire if it may be extended to the whole of tm we prove here that the only real analytic zoll metric on the n sphere with an entire adapted complex structure on tm is the round sphere using similar ideas we answer a special case of an algebraization question raised by the first author characterizing some stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex monge ampere equation the result presented here is an extension to higher dimensions of an observation attributed to w stoll for the case of riemann surfaces

singular monge ampere foliations this paper generalizes results of lempert and szoke on the structure of the singular set of a solution of the homogeneous monge ampere equation on a stein manifold their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution in addition their requirement that the square of the solution be c 3 everywhere is replaced by a smoothness condition on the blowup of the singular set under these conditions the singular set is shown to inherit a finsler metric which in the real analytic case uniquely determines the solution of the monge ampere equation these results are proved using techniques from contact geometry

divisors of polynomial matrices theory and applications in this note divisors g s of a given polynomial matrix n s are discussed the note deals with the following problems extraction of a greatest right left divisor of n s computation for a given state space system c a b a state space feedback making the system c a bf b maximally or appropriately observable and determination for a given system c a b o w o a disturbance rejection feedback f the developed methods are very simple from a computational point of view and can be easily implemented in the software via existing computer library procedures

canonical complex structures associated to connections and complexifications of lie groups the notion of adapted complex structure is extended from riemannian manifolds to general koszul connections the case of the canonical connection of a lie group and the levi civita connection of a pseudo riemannian manifold is studied

defocus map estimation from a single image based on two parameter defocus model defocus map estimation dme is highly important in many computer vision applications nearly all existing approaches for dme from a single image are based on a one parameter defocus model which does not allow for the variation of depth over edges in this paper a novel two parameter model of defocused edges is proposed for dme from a single image we can estimate the defocus amounts for each side of the edges through this proposed model and the confidence that the edge is a pattern edge where the depth remains the same over the edge can be generated then we modify the tv l1 algorithm for structure texture decomposition by taking advantage of this confidence to eliminate pattern edges while preserving structural ones finally the defocus amounts estimated at the edge positions are used as initial values and the structure component is employed as a guidance in the following laplacian matting procedure to avoid the influence of pattern edges on the final defocus map experiment results show that the proposed method can effectively eliminate the influence of pattern edges compared with the state of art method furthermore the estimated defocus map is feasible in applications of depth estimation and foreground background segmentation

the pluricomplex poisson kernel for strongly convex domains let d be a bounded strongly convex domain in the complex space of dimension n for a fixed point p epsilon partial derivative d we consider the solution of a homogeneous complex monge ampere equation with a simple pole at p we prove that such a solution enjoys many properties of the classical poisson kernel in the unit disc and thus deserves to be called the pluricomplex poisson kernel of d with pole at p in particular we discuss extremality properties such as a generalization of the classical phragmen lindelof theorem relations with the pluricomplex green function of d uniqueness in terms of the associated foliation and boundary behaviors finally using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions

an inexact parallel splitting augmented lagrangian method for large system of linear equations parallel iterative methods are powerful in solving large systems of linear equations les the existing parallel computing research results focus mainly on sparse systems or others with particular structure most are based on parallel implementation of the classical relaxation methods such as gauss seidel sor and aor methods which can be efficiently carried out on multiprocessor system in this paper we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts then we convert the original problem les into a monotone linear variational inequality problem vi with separable structure finally an inexact parallel splitting augmented lagrangian method is proposed to solve the variational inequality problem vi to avoid dealing with the matrix inverse operator we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is o n 2 in addition the proposed method does not require any special structure of system of les under consideration convergence of the proposed methods in dealing with two and three separable operators respectively is proved numerical computations are provided to show the applicability and robustness of the proposed methods

non parametric blur map regression for depth of field extension real camera systems have a limited depth of field dof which may cause an image to be degraded due to visible misfocus or too shallow dof in this paper we present a blind deblurring pipeline able to restore such images by slightly extending their dof and recovering sharpness in regions slightly out of focus to address this severely ill posed problem our algorithm relies first on the estimation of the spatially varying defocus blur drawing on local frequency image features a machine learning approach based on the recently introduced regression tree fields is used to train a model able to regress a coherent defocus blur map of the image labeling each pixel by the scale of a defocus point spread function a non blind spatially varying deblurring algorithm is then used to properly extend the dof of the image the good performance of our algorithm is assessed both quantitatively using realistic ground truth data obtained with a novel approach based on a plenoptic camera and qualitatively with real images

sur l existence de solutionsc 1 1 n c 1 1 etc 2 radiales pour des quations de type monge ampre complexe dans la boule unit de n on considere dans la boule unite de c n le probleme de dirichlet pour une equation non lineaire de type monge ampere on demontre l existence d une solution radiale

boundary behavior of infinitesimal generators in the unit ball we prove a julia wolff caratheodory type theorem for infinitesimal generators on the unit ball in n moreover we study jets expansions at the boundary and give necessary and sufficient conditions on such jets for an infinitesimal generator to generate a group of automorphisms of the ball

sur l existence et la regularite de solutions radiales pour des equations de type monge ampere complexe on etablit des resultats d existence et de regularite pour l equation de type monge ampere complexe suivante dans la boule de c n det u ij f z u z xu z z b u b 0 ou f est une fonction positive et radiale en la premiere variable dans b r 2 et x est le champ radial regulier x i 1 2n x i xi

linearly recursive sequences and operator polynomials abstract operator polynomials i e polynomials over endomorphism algebras of vector spaces are studied in the framework of polynomial modules and their comodule duals i e the comodules of linearly recursive sequences the subcomodules which replace the concept of jordan matrix pairs contain enough information to recover the polynomials up to unit factors

estimating spatially varying defocus blur from a single image estimating the amount of blur in a given image is important for computer vision applications more specifically the spatially varying defocus point spread functions psfs over an image reveal geometric information of the scene and their estimate can also be used to recover an all in focus image a psf for a defocus blur can be specified by a single parameter indicating its scale most existing algorithms can only select an optimal blur from a finite set of candidate psfs for each pixel some of those methods require a coded aperture filter inserted in the camera in this paper we present an algorithm estimating a defocus scale map from a single image which is applicable to conventional cameras this method is capable of measuring the probability of local defocus scale in the continuous domain it also takes smoothness and color edge information into consideration to generate a coherent blur map indicating the amount of blur at each pixel simulated and real data experiments illustrate excellent performance and its successful applications in foreground background segmentation

characterization of strictly convex domains biholomorphic to a circular domain recently parabolic exhaustions have been used successfully to classify complex spaces stoll 6 and 7 burns 2 p wong 9 here we define parabolic exhaustions for strictly convex domains in c and give necessary and sufficient conditions for such a domain to be biholomorphically equivalent to a circular domain or even to the ball the results presented in this note are part of the ph d thesis the author is completing under the direction of professor w stoll details and more implications will appear at a later date a strictly convex domain d c c is a domain for which there exists a defining function whose real hessian is strictly positive on tx dd for all x g dd let a be the open unit disk in c and let s be the unit sphere in c let d c c be a strictly convex domain and p e d be any point for b g s lempert 4 constructs an extremal map f d b a d which is holomorphic with f 0 6 v and f 0 6 x where 1 x 0 is the length of b in the infinitesimal kobayashi metric of d at p these conditions determine f d b uniquely and the map extends smoothly to an embedding f u b s d also f a x s d is of class c and surjective one and only one function r d r exists such that r f z b z for z b g a x s then r is a continuous exhaustion of d positive and of class c on d p also r 1 on 3d we refer to r as the lempert exhaustion of d at p

monge ampere measures on subvarieties in this article we address the question whether the complex monge ampere equation is solvable for measures with large singular part we prove that under some conditions there is no solution when th

extremal analytic discs with prescribed boundary data this paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain d in cn we prove that for any two distinct points p q in dd respectively p e dd and a vector v such that f lv e tp dd and v p p 2 vji j p 0 where v p is the outward normal to dd at p there exists an extremal analytic disc passing through p q if dd e ck k 3 respectively l p l v if dd e ck k 14 consequently we can foliate d with these extremal analytic discs introduction in this paper we always assume that d is a bounded strictly linearly convex domain in cn see 1 for definitions let u denote the open unit disc in c an extremal analytic disc of d is a holomorphic map from u into d such that the kobayashi distance of any two points in f u is realized by lempert in 6 7 has proved the existence uniqueness and boundary regularity of such maps with data given at interior points following 7 we shall use the term e mapping for the extremal analytic disc it is also known 6 proposition 5 that if dd is of class ck k 3 then any e mapping of d is in ck 2 u

regularity of the complex monge ampere equation for radially symmetric functions of the unit ball soit m l operateur de monge ampere complexe dans c n soit b la boule unite dans c n etant donnee f 0 sur b on cherche une fonction plurisousharmonique u c 2 b telle que 1 mu f dans b et 2 u 0 sur bb

the dirichlet problem with prescribed interior singularities abstract in this paper we solve the nonlinear dirichlet problem uniquely for functions with prescribed asymptotic singularities at a finite number of points and with arbitrary continuous boundary data on a domain in r n the main results apply in particular to subequations with a riesz characteristic p 2 it is shown that without requiring uniform ellipticity the dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the riesz kernel j k p x x j where k p x 1 x p 2 for 2 p log x if p 2 at any prescribed finite set of points x 1 x k in the domain and any finite set of positive real numbers 1 k this sharpens a previous result of the authors concerning the discreteness of high density sets of subsolutions uniqueness and existence results are also established for finite type singularities such as j x x j 2 p for 1 p 2 the main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of armstrong sirakov smart in the uniformly elliptic case

maximal complexifications of certain riemannian homogeneous manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

implicit ode solvers with good local error control for the transient analysis of markov models obtaining the transient probability distribution vector of a continuous time markov chain ctmc using an implicit ordinary differential equation ode solver tends to be advantageous in terms of run time computational cost when the product of the maximum output rate of the ctmc and the largest time of interest is large in this paper we show that when applied to the transient analysis of ctmcs many implicit ode solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1 norm of the error this allows the development of implementations of those ode solvers for the transient analysis of ctmcs that can be more efficient and more accurate than more standard implementations

a unified proof for the convergence of jacobi and gauss seidel methods we present a new unified proof for the convergence of both the jacobi and the gauss seidel methods for solving systems of linear equations under the criterion of either a strict diagonal dominance of the matrix or b diagonal dominance and irreducibility of the matrix these results are well known the proof for criterion a makes use of gersgorin s theorem while the proof for criterion b uses taussky s theorem that extends gersgorin s work hence the topic is interesting for teaching purposes

maximal complexifications of certain homogeneous riemannian manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

monge ampere equations and moduli spaces of manifolds of circular type abstract a bounded manifold of circular type is a complex manifold m of dimension n admitting a bounded exhaustive real function u defined on m minus a point x o so that a it is a smooth solution on m x o to the monge ampere equation d d c u n 0 b x o is a singular point for u of logarithmic type and e u extends smoothly on the blow up of m at x o c d d c e u 0 at any point of m x o this class of manifolds naturally includes all smoothly bounded strictly linearly convex domains and all smoothly bounded strongly pseudoconvex circular domains of c n a set of modular parameters for bounded manifolds of circular type is considered in particular for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form it is also shown that the class of normalizing maps for an n dimensional manifold m is a new holomorphic invariant with the following property it is parameterized by the points of a finite dimensional real manifold of dimension n 2 when m is a non convex circular domain while it is of dimension n 2 2 n when m is a strictly linearly convex domain new characterizations of the circular domains and of the unit ball are also obtained

comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains we prove that for a strongly pseudoconvex domain d subset mathbb c n the infinitesimal caratheodory metric g c z v and the infinitesimal kobayashi metric g k z v coincide if z is sufficiently close to bd and if v is sufficiently close to being tangential to bd also we show that every two close points of d sufficiently close to the boundary and whose difference is almost tangential to bd can be joined by a unique up to reparameterization complex geodesic of d which is also a holomorphic retract of d the same continues to hold if d is a worm domain as long as the points are sufficiently close to a strongly pseudoconvex boundary point we also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed this has consequences for the behavior of the squeezing function

the asymptotic expansion of a cr invariant and grauert tubes cr manifolds the abstract models of real hypersurfaces in complex manifolds are 2n 1 dimensional manifolds m with a codimension one subbundle h of the tangent bundle which carries a complex structure the cr refers to cauchy riemann because for m c c n l the subbundle h consists of induced cauchy riemann operators there is a wealth of geometry and analysis associated with these structures especially when the cr manifolds are strictly pseudoconvex for example two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are cr equivalent a fundamental problem in cr geometry is to find computable invariants associated with the cr structures the global cr invariant we will consider in this paper is the chem simons type invariant discovered by bums and epstein b e 1 it is a real valued global cr invariant of a compact 3dimensional strictly pseudo convex cr manifold whose holomorphic tangent bundle is trivial cheng and lee independently found this invariant and extend the definition of b e invariant to a relative invariant on an arbitrary compact 3 dimensional cr manifold cf c l we will evaluate this g asymptotically on the boundary of small grauert tubes before posing the question in a more precise form we will first say a few workds about grauert tubes let x be a real analytic manifold then every coordinate patch u c r n can be thickened to obtain an open set r c c n since the coordinate changes of x are real analytic maps by taking power series expansions and by shrinking c u to get convergence they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition

symmetries and other transformations of the complex monge ampere equation on definit des transformations tangentes en utilisant la structure symplectique holomorphe du fibre cotangent holomorphe

abel averages and holomorphically pseudo contractive maps in banach spaces abstract a class of maps in a complex banach space is studied which includes both unbounded linear operators and nonlinear holomorphic maps the defining property which we call pseudo contractivity is introduced by means of the abel averages of such maps we show that the studied maps are dissipative in the spirit of the classical lumer phillips theorem for pseudo contractive holomorphic maps we establish the power convergence of the abel averages to holomorphic retractions

complex monge ampere equations and totally real submanifolds abstract we study the dirichlet problem for complex monge ampere equations in hermitian manifolds with general non pseudoconvex boundary our main result theorem 1 1 extends the classical theorem of caffarelli kohn nirenberg and spruck in c n we also consider the equation on compact manifolds without boundary attempting to generalize yau s theorems in the kahler case as applications of the main result we study some connections between the homogeneous complex monge ampere hcma equation and totally real submanifolds and a special dirichlet problem for the hcma equation related to donaldson s conjecture on geodesics in the space of kahler metrics

modular data and regularity of monge ampere exhaustions and of kobayashi distance regularity properties of intrinsic objects for a large class of stein manifolds namely of monge ampere exhaustions and kobayashi distance is interpreted in terms of modular data the results lead to a construction of an infinite dimensional family of convex domains with squared kobayashi distance of prescribed regularity properties a new sharp refinement of stoll s characterization of mathbb c n is also given

resultant operators and the bezout equation for analytic matrix functions i this paper is a continuation of 4 it contains our results on resultant operators for a family of matrix functions that are analytic in a finitely connected domain the proofs are based on the results about the bezout equation obtained in 4 it is assumed that the reader is familiar with the concepts and results from 4 let f denote a cauchy contour in the complex plane consisting of several non intersecting simple smooth closed contours which form the positively oriented boundary of a finitely connected bounded domain a throughout this chapter we assume for convenience that 0 e a the general case can be obtained by a shift in the complex plane let a a be regular n x n matrix functions which are analytic in a and continuous in a u r and assume that c a n r 3 the paper consists of two sections in section 1 we show that the role of the resultant operator of the matrix functions a a is played by the operator r l t l 1 p co defined by

common eigenvalues divisors and multiples of matrix polynomials a review abstract the main objective of this paper is to review some results obtained in the last decade using methods of spectral theory and concerning common multiples common divisors and common eigenvalues for matrix polynomials this includes discussion of resultant and bezoutian matrices for matrix polynomials and descriptions of their kernels in terms of spectral data some extensions to operator polynomials whose coefficients are bounded linear operators on a hilbert space are also included

differential geometry of real monge ampere foliations on etudie les feuilletages qui proviennent de la condition de monge ampere recquerant le hessien d une fonction definie sur une variete affine d avoir un rang inferieur au maximal

bezoutian and schur cohn problem for operator polynomials abstract a generalized bezout operator bezoutian for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials the location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle infinite dimensional version of the schur cohn problem is expressed via the inertia of a suitable bezoutian an application to the geometric dichotomy problem for difference equations with operator coefficients is given as well

defocus blur invariant scale space feature extractions we propose modifications to scale space feature extraction techniques scale invariant feature transform sift and speeded up robust features surfs that make the feature detection and description invariant to defocus blur specifically the scale space blob detection relies on the second derivative responses of images our analysis of circular defocus blur which sufficiently approximates a real camera blur kernel and its effect on scale space blob detection suggests that fourth derivative and not the usual second derivative is optimal for detecting the blurred blobs while multi scale descriptors of blurred blobs are effective at establishing correspondences between the blurred images the proposed defocus blur invariant dbi scale space feature extraction techniques which we refer to as dbi sift and dbi surf do not require image deblurring nor blur kernel estimation meaning that their accuracy does not depend on the quality of image deblurring we offer empirical evidence of blur invariance by establishing interest point correspondences between sharp or blurred reference images and blurred target images

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

algebraic integrability of the confluent neumann system in this paper we study the neumann system which describes the harmonic oscillator of arbitrary dimension constrained to the sphere in particular we will consider the confluent case where two eigenvalues of the potential coincide which implies that the system has s1 symmetry we will prove complete algebraic integrability of the confluent neumann system and show that its flow can be linearized on the generalized jacobian torus of some singular algebraic curve the symplectic reduction of s1 action will be described and we will show that the general rosochatius system is a symplectic quotient of the confluent neumann system where all the eigenvalues of the potential are double

monge ampere foliations with singularities at the boundary of strongly convex domains let d c n be a bounded strongly convex domain with smooth boundary we consider a monge ampere type equation in d with a simple pole at the boundary using the lempert foliation of d in extremal discs we construct a solution u whose level sets are bound aries of horospheres among other things we show that the biholomorphisms between strongly convex domains are exactly those maps which preserves our solution

unobservable subspaces and common roots of several polynomials abstract for an m n matrix polynomial p c is called an eigenvalue if there exists 0 x c n such that p x 0 in this case x is an eigenvector of p corresponding to the eigenvalue the pair x is called a root of p given a set of matrix polynomials p1 p2 p1 of sizes m1 n m2 n m1 n and a monic n n polynomial q a criterion is derived for the existence of a common root of these polynomials this is done by constructing a specific unobservable subspace having its dimension equal to the number of common eigenvalues of these polynomials counted with their common multiplicities these notions are related to the lengths of the common maximal jordan chains in the particular case where q t ti a the above subspace is spanned by the jordan chains which are common to a p1 p2 p1

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix