Germs Of Functions Research Articles

Characteristic cohomology II: Matrix singularities

AbstractFor a germ of a variety , a singularity of “type ” is given by a germ , which is transverse to in an appropriate sense, such that . In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for a hypersurface), and complement and link (for the general case). It captures the cohomology of inherited from and is given by subalgebras of the cohomology for for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences for Milnor fibers and for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.In this paper, we apply these methods in the case denotes any of the varieties of singular complex matrices, which may be either general, symmetric, or skew‐symmetric (with even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds” for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general complex matrices.Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size ” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ containing such a kite map germ of size . Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.

Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

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 quasihomogeneous with respect to the same positive rational weights.

An implementation of the Suwa method for computing first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations

The Suwa method for computing versal unfoldings of holomorphic singular foliations is considered from the point of view of computational complex analysis. Based on the theory of Grothendieck local duality on residues, an effective algorithm of computing a first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations is obtained. As an application of our approach, we give an effective method for computing universal unfoldings of germs of meromorphic functions.

Cryptorchidism and puberty.

Cryptorchidism is the condition in which one or both testes have not descended adequately into the scrotum. The congenital form of cryptorchidism is one of the most prevalent urogenital anomalies in male newborns. In the acquired form of cryptorchidism, the testis that was previously descended normally is no longer located in the scrotum. Cryptorchidism is associated with an increased risk of infertility and testicular germ cell tumors. However, data on pubertal progression are less well-established because of the limited number of studies. Here, we aim to review the currently available data on pubertal development in boys with a history of non-syndromic cryptorchidism-both congenital and acquired cryptorchidism. The review is focused on the timing of puberty, physical changes, testicular growth, and endocrine development during puberty. The available evidence demonstrated that the timing of the onset of puberty in boys with a history of congenital cryptorchidism does not differ from that of non-cryptorchid boys. Hypothalamic-pituitary-gonadal hormone measurements showed an impaired function or fewer Sertoli cells and/or germ cells among boys with a history of cryptorchidism, particularly with a history of bilateral cryptorchidism treated with orchiopexy. Leydig cell function is generally not affected in boys with a history of cryptorchidism. Data on pubertal development among boys with acquired cryptorchidism are lacking; therefore, more research is needed to investigate pubertal progression among such boys.

Topology of Fold Map Germs from mathbb R^3 to mathbb R^5

Let f:(mathbb R^3,0)rightarrow (mathbb R^5,0) be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of f with a small enough sphere S^4_epsilon centered at the origin in mathbb R^5. If f is of fold type, we define a labeled tree associated to its link and prove that is a complete topological invariant for it. As an application we obtain the complete topological classification of map germs contained in the mathcal {A}^2-class (x,y,z^2,xz,0).

The Bruce–Roberts Numbers of a Function on an ICIS

Abstract We relate a number of invariants of a function germ with isolated singularity over an isolated complete intersection singularity (ICIS), generalizing our previous results, [17, 18], as a consequence we present a new relatively elementary proof of Greuel’s theorem that for a weighted homogeneous ICIS the Milnor number is equal to Tjurina number. With this relation, we are able to prove that the relative logarithmic characteristic variety is Cohen–Macaulay at any point and the logarithmic characteristic variety is Cohen–Macaulay at any point not in $X\times \{0\}$.

On real analogues of the Poincaré series

AbstractThere exist several equivalent equations for the Poincaré series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poincaré series of a collection of valuations in the real setting (i.e., on the ring of germs of functions on a real analytic variety), compute them for the case of one curve valuation on the plane and discuss some of their properties.

Fibrations of tamely composable maps

We study composed map germs with respect to their local fibrations. Under most general conditions, inspired by the tameness condition that was introduced recently, we prove the existence of singular tube fibrations, and we determine the topology of the fibres.

On the topology of the transversal slice of a quasi-homogeneous map germ

AbstractWe consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$ . We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$ , denoted by $\gamma_f$ , in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ $g\,{:}\,(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $\gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.

Morse numbers of function germs with isolated singularities

Abstract A set of Morse numbers is associated with a holomorphic function germ with stratified isolated singularity, extending the classical Milnor–Morse number to the setting of a singular base space.

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.

Bott-Chern hypercohomology and bimeromorphic invariants

AbstractThe aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.

On the Computation of the Codimension of Map Germs Using the Lie Algebra Associated with a Restricted Left–Right Group

The codimension is an important invariant, which measures the complexity of map germs and play an important role in classification and recognition problems. The restricted A-equivalence was introduced to obtain a classification of reducible curves. The aim was to classify simple parameterized curves with two components, one of them being smooth with respect to the A-equivalence in characteristic p. In characteristic 0, the corresponding classification was given by Kolgushkin and Sadykov. The aim of this article is to present an algorithm to compute the codimension of germs of singularities under a restricted left–right equivalence (A-symmetry). We also give the implementation of this algorithm in the computer algebra system singular.

Recognition and Implementation of Contact Simple Map Germs from (ℂ2, 0) → (ℂ2, 0)

The classification of contact simple map germs from (C2,0)→(C2,0) was given by Dimca and Gibson. In this article, we give a useful criteria to recognize this classification of contact simple map germs of holomorphic mappings with finite codimension. The recognition is based on the computation of explicit numerical invariants. By using this characterization, we implement an algorithm to compute the type of the contact simple map germs without computing the normal form and also give its implementation in the computer algebra system Singular.

A characterization and implementation of corank one map germs from $ 2 $-space to $ 3 $-space in the computer algebra system SINGULAR

<abstract><p>The classification and the geometry of corank one map germs from $ ({\mathbb{C}}^2, 0) \rightarrow ({\mathbb{C}}^3, 0) $ have been studied by Mond <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>]</sup>. In this paper we characterize the classification of map germs of corank at most $ 1 $, in terms of certain invariants. Moreover, by using this characterization, we develop an algorithm to compute the type of map germs with out computing the normal form. Also, we give its implementation in the computer algebra system SINGULAR <sup>[<xref ref-type="bibr" rid="b15">15</xref>]</sup>.</p></abstract>

ON SINGULAR MAPS WITH LOCAL FIBRATION

We discuss the most general condition under which a singular local tube fibration exists. We give an application to composition of map germs.

The effect of residual acetamiprid and dichlorvos in greenhouse cucumber on liver function and testicular germ cells on laboratory mice

The effect of residual acetamiprid and dichlorvos in greenhouse cucumber on liver function and testicular germ cells on laboratory mice

Thom condition and monodromy

We give the definition of the Thom condition and we show that given any germ of complex analytic function f:(X,x)rightarrow ({mathbb {C}},0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that fin {mathfrak {m}}_{X,x}^2, where {mathfrak {m}}_{X,x} is the maximal ideal of {mathcal {O}}_{X,x}. This result generalizes a well-known theorem of the second named author when X is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A’Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

MiMIC analysis reveals an isoform specific role for Drosophila Musashi in follicle stem cell maintenance and escort cell function

The Drosophila ovary is regenerated from germline and somatic stem cell populations that have provided fundamental conceptual understanding on how adult stem cells are regulated within their niches. Recent ovarian transcriptomic studies have failed to identify mRNAs that are specific to follicle stem cells (FSCs), suggesting that their fate may be regulated post-transcriptionally. We have identified that the RNA-binding protein, Musashi (Msi) is required for maintaining the stem cell state of FSCs. Loss of msi function results in stem cell loss, due to a change in differentiation state, indicated by upregulation of Lamin C in the stem cell population. In msi mutant ovaries, Lamin C upregulation was also observed in posterior escort cells that interact with newly formed germ cell cysts. Mutant somatic cells within this region were dysfunctional, as evidenced by the presence of germline cyst collisions, fused egg chambers and an increase in germ cell cyst apoptosis. The msi locus produces two classes of mRNAs (long and short). We show that FSC maintenance and escort cell function specifically requires the long transcripts, thus providing the first evidence of isoform-specific regulation in a population of Drosophila epithelial cells. We further demonstrate that although male germline stem cells have previously been shown to require Msi function to prevent differentiation this is not the case for female germline stem cells, indicating that these similar stem cell types have different requirements for Msi, in addition to the differential use of Msi isoforms between soma and germline. In summary, we show that different isoforms of the Msi RNA-binding protein are expressed in specific cell populations of the ovarian stem cell niche where Msi regulates stem cell differentiation, niche cell function and subsequent germ cell survival and differentiation.

Singularities of mappings on ICIS and applications to Whitney equisingularity

We study germs of analytic maps f:(X,S)→(Cp,0), when X is an icis of dimension n<p. We define an image Milnor number, generalizing Mond's definition, μI(X,f) and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs ft:(Cn,S)→(Cn+1,0) with isolated instabilities in terms of the constancy of the μI⁎-sequences of ft and the projections π:D2(ft)→Cn, where D2(ft) is the icis given by double point space of ft in Cn×Cn. The μI⁎-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.