Germs Of Holomorphic Functions Research Articles

Multipliers on Spaces of Holomorphic Functions

We consider multipliers on the space of holomorphic functions of one variable H(Omega ),,Omega subset mathbb {C} open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to Omega and Omega is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on H(Omega ). We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when Omega is convex.

Cohomology of contact loci

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean’s spectral sequence converging to the Floer cohomology of the $m$‑th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.

Homologically trivial integrable deformations of germs of holomorphic functions

Homologically trivial integrable deformations of germs of holomorphic functions

Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

A solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers is provided for general classes of domains of finite type in Cn, that include the so-called special domains given by finite and infinite sums of squares of absolute values of holomorphic functions. Also included is a more general class of domains recently discovered by M. Fassina [23]. More generally, for any smoothly bounded pseudoconvex domain we introduce an invariantly defined associated sheaf S of C-subalgebras of holomorphic function germs, that combined with a result of Fassina, reduces the existence of effective subelliptic estimates at p to a purely algebraic geometric question of controlling the multiplicity of S.Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as Q∘Γ, where Γ is constructed from pre-multipliers and Q is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the holomorphic Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic geometric techniques by Y.-T. Siu [54,55] and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo [16,8].The proposed procedures are purely algebraic and as such can be of wider interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.

Analytic deformations of pencils and integrable one-forms having a first integral

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family [Formula: see text] of integrable one-forms [Formula: see text] defined in a neighborhood [Formula: see text] of the initial singular point, and parametrized by the disc [Formula: see text]. The initial foliation is defined by [Formula: see text]. The second type, more restrictive, is given by an integrable holomorphic one-form [Formula: see text] defined in the product [Formula: see text]. Then, the initial foliation is defined by the slice restriction [Formula: see text]. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by [Formula: see text] for some germ of holomorphic function [Formula: see text] at the origin [Formula: see text]. We assume that the germ [Formula: see text] is irreducible and that the typical fiber of [Formula: see text] is simply-connected. This is the case if outside of a dimension [Formula: see text] analytic subset [Formula: see text], the analytic hypersurface [Formula: see text] has only normal crossings singularities. We then prove that, if cod sing [Formula: see text] then the (germ of the) developing foliation given by [Formula: see text] also exhibits a holomorphic first integral. For the general case, i.e. cod sing [Formula: see text], we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations [Formula: see text], of a local pencil [Formula: see text], for [Formula: see text]. For dimension [Formula: see text] we consider [Formula: see text]. For dimension [Formula: see text] we assume some generic geometric conditions on [Formula: see text] and [Formula: see text]. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type [Formula: see text] with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form [Formula: see text] with some additional properties, provided that for [Formula: see text] the axes remain invariant for the foliations [Formula: see text].

A Torelli type theorem for exp-algebraic curves

An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $\HH = \{ [h_1], \cdots, [h_n] \}$, with poles of orders $d_1, \cdots, d_n \geq 1$ at points $p_1, \cdots, p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $\Omega^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ with exponential singularities at the points $p_1, \cdots, p_n$ of types $[h_1], \cdots, [h_n]$, i.e., near $p_i$ any $f \in \OO_{\HH}$ is of the form $f = ge^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $\omega \in \Omega^0_{\HH}$ is of the form $\omega = \alpha e^{h_i}$ for some germ of meromorphic $1$-form). For any $\omega \in \Omega^0_{\HH}$ the completion of $S'$ with respect to the flat metric $|\omega|$ gives a space $S^* = S' \cup \RR$ obtained by adding a finite set $\RR$ of $\sum_i d_i$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_1(S^*, \RR ; \C)$ with the deRham cohomology group defined by $H^1_{dR}(S, \HH) := \Omega^0_{\HH}/d\OO_{\HH}$. There is a degree zero line bundle $L_{\HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $\Omega^0_{\HH}$ and the space $W_{\HH}$ of meromorphic $L_{\HH}$-valued $1$-forms which are holomorphic on $S'$, so that $H_1(S^*, \RR ; \C)$ maps to a subspace $K_{\HH} \subset W^*_{\HH}$. We show that the exp-algebraic curve $(S, \HH)$ is determined uniquely by the pair $(L_{\HH},\, K_{\HH} \subset W^*_{\HH})$.

Mixed tête-à-tête twists as monodromies associated with holomorphic function germs

T\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed t\^ete-\`a-t\^ete graphs provide a generalization which define mixed t\^ete-\`a-t\^ete twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed t\^ete-\`a-t\^ete twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of t\^ete-\`a-t\^ete twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities.

Classification of ℤ3-Equivariant Simple Function Germs

The present paper deals with the classification of multivariate holomorphic function germs that are equivariant simple under representations of cyclic groups. We obtain a complete classification of such function germs of two and three variables for all possible nontrivial ℤ3-actions. Our main classification methods generalize those used for the classification of simple germs in the nonequivariant case.

A Remark on Two Notions of Order of Contact

We recall two measurements of the order of contact of an ideal in the ring of germs of holomorphic functions at a point, and we provide a class of examples in which they differ.

On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

We study a family of double confluent Heun equations that are lin-earizations of nonlinear equations on two-torus modeling the Joseph-son effect in superconductivity. They have the form LE = 0, where L = L λ,µ,n is a family of differential operators of order two acting on germs of holomorphic functions in one complex variable. They depend on parameters λ, n ∈ R, µ > 0, λ + µ 2 ≡ const > 0. We show that for every b ∈ C and n ∈ R satisfying a certain non-resonance condition and every parameter values λ, µ there exists a unique entire function f ± : C → C (up to multiplicative constant) such that z −b L(z b f ± (z ±1)) = d 0± + d 1± z for some d 0± , d 1± ∈ C. The latter d j,± are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those parameter values for which the monodromy operator of the corresponding Heun equation has given eigenvalues. This yields the description of the non-integer level curves of the rotation number of the family of equations on two-torus as a function of parameters. In the particular case, when the monodromy is parabolic (has multiple eigenvalue), we get the complete description of those parameter values that correspond to the boundaries of the phase-lock areas: integer level sets of the rotation number, which have non-empty interiors.

Hadamard Multipliers on Spaces of Holomorphic Functions

Several representation theorems of multipliers are derived: in terms of analytic functionals, and germs of holomorphic functions. The co-induced topology via the representation theorems is discussed. As an application of the representation theorems the density of non-invertible multipliers is proved. Moreover, Euler differential operators are distinguished among all multipliers.

On the Completeness of Orbits of a Pommiez Operator in Weighted (LF)-Spaces of Entire Functions

We describe cyclic vectors for a Pommiez operator on a weighted (LF)-space E of entire functions. The full description is obtained where E is the Laplace transform of the strong dual of the space of all germs of holomorphic functions on a convex locally closed set in the complex plane.

Reduction of a family of ideals

In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions. As a corollary we generalize the result of A. Ploski \cite{ploski} on the semicontinuity of the Łojasiewicz exponent in a multiplicity-constant deformation.

The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

Let $$f$$ and $$g$$ be holomorphic function-germs vanishing at the origin of complex analytic germs of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ $$f\bar{g}$$ has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ $$f\bar{g}$$ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs of the form $$f\bar{g}$$ defined in a complex surface germ, and we prove an A’Campo-type formula for the zeta function of its monodromy.

Topological equisingularity of hypersurfaces with 1-dimensional critical set

We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Lê numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey [13]). It is worth to remark that this does not happen for higher dimensional critical set [5]. We use these topological triviality results to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnoldʼs series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre.

Hilbert function, generalized Poincaré series and topology of plane valuations

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincare series, the generalized Poincare series, and the generalized semigroup Poincare series. The Hilbert function and the generalized Poincare series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincare series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.

Topological convolution algebras

In this paper we introduce a dual of reflexive Fréchet counterpart of Banach algebras of the form ⋃p∈NΦp′ (where the Φp′ are (dual of) Banach spaces with associated norms ‖⋅‖p), which carry inequalities of the form ‖ab‖p⩽Ap,q‖a‖q‖b‖p and ‖ba‖p⩽Ap,q‖a‖q‖b‖p for p>q+d, where d is preassigned and Ap,q is a constant. We study the functional calculus and the spectrum of the elements of these algebras. We then focus on the particular case Φp′=L2(S,μp), where S is a Borel semi-group in a locally compact group G, and multiplication is convolution. We give a sufficient condition on the measures μp for such inequalities to hold. Finally we present three examples, one is the algebra of germs of holomorphic functions in zero, the second related to Dirichlet series and the third in the setting of non-commutative stochastic distributions.

\mu -constant monodromy groups and marked singularities

μ-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo ±id. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these moduli spaces are connected. The relation with Torelli type problems is discussed and a new global Torelli type conjecture for marked singularities is formulated. All conjectures are proved for the simple and 22 of the 28 exceptional singularities.

On holomorphic domination, I

Let X be a separable Banach space and u : X → R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h : X → Z with u ( x ) < ‖ h ( x ) ‖ for x ∈ X . As a consequence we find that the sheaf cohomology group H q ( X , O ) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q ⩾ 1 . As another consequence we prove that if f is a C 1 -smooth ∂ ¯ -closed ( 0 , 1 ) -form on the space X = L 1 [ 0 , 1 ] of summable functions, then there is a C 1 -smooth function u on X with ∂ ¯ u = f on X.

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

Let \mathrm{M} be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d . We find an improved local geometric condition on \mathrm{M} which guarantees, at a point p on \mathrm{M} , that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p . Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition, but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d &gt; 1 , there are additional phenomena which are invisible when d = 1 .