Invariants Of Singularities Research Articles

The higher Du Bois and higher rational properties for isolated singularities

Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a k k -rational isolated singularity is k k -Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the k k -Du Bois and k k -rational singularities in terms of standard invariants of singularities. In particular, we show that k k -Du Bois singularities are ( k − 1 ) (k-1) -rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.

K-th singular locus moduli algebras of singularities and their derivation Lie algebras

In this paper, we introduce a series of new invariants for singularities. A new conjecture about the non-existence of negative weight derivations of the new k-th singular locus moduli algebras for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture in some cases.

DERIVATIONS OF LOCAL k-TH HESSIAN ALGEBRAS OF SINGULARITIES

In our previous work, we introduced a series of new derivation Lie algebras Lk(V) associated to an isolated hypersurface singularity (V,0). These are new analytic invariants of singularities. Here, we investigate L2(V) for fewnomial isolated singularities and obtain the formula of λk(V) (i.e., the dimension of Lk(V)) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for L2(V). This is a continuation of our previous work (Math. Z. 298:3-4 (2021), 1813–1829). We proposed two new conjectures for τk(V) and λk(V) and we prove these conjectures for a large class of singularities.

Multiplicity Along Points of a Radicial Covering of a Regular Variety

We study the maximal multiplicity locus of a variety X over a field of characteristic p>0 that is provided with a finite surjective radicial morphism δ:X→V, where V is regular, for example, when X⊂An+1 is a hypersurface defined by an equation of the form Tq−f(x1,…,xn)=0 and δ is the projection onto V:=Spec(k[x1,…,xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V)⊂K(X). We denote by Fd(X)⊂X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities x∈Fd(X) with a good behavior property under blowups X′→X along regular centers included in Fd(X), which we call invariants with the pointwise inequality property. A finite radicial morphism δ:X→V as above will be expressed in terms of an OVq-submodule M⊆OV. A blowup X′→X along a regular equimultiple center included in Fd(X) induces a blowup V′→V along a regular center and a finite morphism δ′:X′→V′. A notion of transform of the OVq-module M⊂OV to an OV′q-module M′⊂OV′ will be defined in such a way that δ′:X′→V′ is the radicial morphism defined by M′. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on OVq-submodules M⊂OV.

Classification of singular differential invariants in ()-dimensional space and integrability.

Singularity is one of the important features in invariant structures in several physical phenomena reflected often in the associated invariant differential equations. The classification problem for singular differential invariants in (1+3)-dimensional space associated with Lie algebras of dimension 4 is investigated. The formulation of singular invariants for a Lie algebra of dimension possessed by the underlying system of three second-order ordinary differential equations is studied in detail and the corresponding canonical forms for these systems are deduced. Furthermore, the categorization of singular invariants on the basis of conditional singularity, weak uncoupling, weak linearization, partial uncoupling and partial linearization are described for the underlying canonical forms. In addition, those cases of classified canonical forms are also mentioned which do not lead to singular invariant systems of three second-order ODEs for a Lie algebra of dimension 4. The integrability aspect of these classified singular-invariant systems in (1+3)-dimensional space is discussed in a detailed manner for a Lie algebra of dimension 4. Finally, two physical systems from mechanics are presented to illustrate the utilization of the physical aspect of these singular invariants.

Landau–Ginzburg/Calabi–Yau Correspondence for a Complete Intersection via Matrix Factorizations

Abstract By generalizing the Landau–Ginzburg/Calabi–Yau correspondence for hypersurfaces, we can relate a Calabi–Yau complete intersection to a hybrid Landau–Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader–Ross have provided an equivalence between these invariants and Gromov–Witten invariants of complete intersections, in this way quantum cohomology yields an identification of the cohomology groups of the Calabi–Yau and of the hybrid Landau–Ginzburg model. It is not clear how to relate this to the known isomorphism descending from derived equivalences (due to Segal and Shipman, and Orlov and Isik). We answer this question for Calabi–Yau complete intersections of two cubics.

Topological and Differential Invariants of Singularities of Contact Structure on a Three-Dimensional Manifold

A contact structure on a three-dimensional manifold is a two-dimensional distribution on this manifold which satisfies the condition of complete non-integrability. If the distribution fails to satisfy this condition at points of some submanifold, we have a contact structure with singularities. The singularities of contact structures were studied by J. Martinet, B. Jakubczyk and M. Zhitomirskii. We consider a contact structure with singularities as a $$G$$ -structure with singularities, we find some topological and differential invariants of singularities of contact structure and establish their relation to the invariants found by B. Jakubczyk and M. Zhitomirskii.

TH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE

AbstractLet $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$, that is, $L(V)=\text{Der}(A(V),A(V))$. It is known that $L(V)$ is finite dimensional and its dimension $\unicode[STIX]{x1D706}(V)$ is called the Yau number. We introduce a new series of Lie algebras, that is, $k$th Yau algebras $L^{k}(V)$, $k\geq 0$, which are a generalization of the Yau algebra. The algebra $L^{k}(V)$ is defined to be the Lie algebra of derivations of the $k$th moduli algebra $A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$, that is, $L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$, where $m$ is the maximal ideal of ${\mathcal{O}}_{n}$. The $k$th Yau number is the dimension of $L^{k}(V)$, which we denote by $\unicode[STIX]{x1D706}^{k}(V)$. In particular, $L^{0}(V)$ is exactly the Yau algebra, that is, $L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$. These numbers $\unicode[STIX]{x1D706}^{k}(V)$ are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that $\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$ We prove this conjecture for a large class of singularities.

Introduction to Heegaard Floer homology

Heegaard Floer homology is an invariant of knots, links, and 3-manifolds introduced by Ozsváth and Szabó about 15 years ago. This survey defines Heegaard Floer homology and describes its basic properties. Also discussed is the relation between Heegaard Floer homology and invariants of singularities of curves and surfaces. Bibliography: 72 titles.

Singularities of metrics on Hodge bundles and their topological invariants

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.

On weight truncations in the motivic setting

Using punctual gluing of t-structures, we construct an analogue of S. Morel's weight truncation functors (for certain weight profiles) in the setting of motivic sheaves. As an application we construct a canonical motivic analogue of the intersection complex for an arbitrary threefold over any field k. We are also able to recover certain invariants of singularities motivically.

On the Evaluation of Singular Invariants for Canonical Generators of Certain Genus One Arithmetic Groups

ABSTRACTLet N be a positive square-free integer such that the discrete group Γ0(N)+ has genus one. In a previous article, we constructed canonical generators xN and yN of the holomorphic function field associated with Γ0(N)+ as well as an algebraic equation PN(xN, yN) = 0 with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to xN and yN, by which we mean the arithmetic nature of the numbers xN(τ) and yN(τ) for any CM point τ in the upper half plane . If τ is any CM point which is not equivalent to an elliptic point of Γ0(N)+, we prove that the complex numbers xN(τ) and yN(τ) are algebraic integers. Going further, we characterize the algebraic nature of xN(τ) as the generator of a certain ring class field of of prescribed order and discriminant depending on properties of τ and level N. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various N and τ, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of whose coefficients are less than 2.2 × 104, whereas the minimal polynomial obtained from the Hauptmodul of have coefficients as large as 6.6 × 1073.

Contraction algebra and invariants of singularities

In [7], Donovan and Wemyss introduced the contraction algebra of flop- ping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A_\infty-structure recovers various invariants associated to the underlying singularity and its small resolution, including the derived category of singularities and the genus zero Gopakuma-Vafa invariants.

Bernstein polynomial of $2$-Puiseux pairs irreducible plane curve singularities

In 1982, Tamaki Yano proposed a conjecture predicting the set of b-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In \cite{ACLM-Yano2} we proved the conjecture for the case in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In this article we aim to study the Bernstein polynomial for any function with the hypotheses above. In particular the set of all common roots of those Bernstein polynomials is given. We provide also bounds for some analytic invariants of singularities and illustrate the computations in suitable examples.

Singular invariants and coefficients of harmonic weak Maass forms of weight 5/2

Abstract We study the coefficients of a natural basis for the space of harmonic weak Maass forms of weight 5 / 2 ${5/2}$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function p ⁢ ( n ) ${p(n)}$ . We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoḡlu, and Tóth on cycle integrals of the j-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

Differential Forms on Quasihomogeneous Noncomplete Intersections

In this article, we discuss a few simple methods for computing the Poincare series of modules of differential forms given on quasihomogeneous noncomplete intersections of various types. Among them are curves associated with a semigroup, bouquets of such curves, affine cones over rational or elliptic curves, and normal determinantal and toric varieties, including some types of quotient singularities, as well as cones over the Veronese embedding of projective spaces or over the Segre embedding of products of projective spaces, rigid singularities, fans, etc. In many cases, correct formulas can be derived without resorting to analysis of complicated resolvents or using computer systems of algebraic calculations. The obtained results allow us to compute the basic invariants of singularities in an explicit form by means of elementary operations on rational functions.

Equivariant Hirzebruch class for quadratic cones via degenerations

The difference between the Hirzebruch class of the generic fiber and the Hirzebruch class of the special fiber is measured by the appropriate version of Milnor class, studied in [CMSS] for hypersurfaces and in [MSS] the general case. The same phenomenon happens for the equivariant Hirzebruch class developed in [We3], compare also with [Oh, Sec.4] for the equivariant Hirzebruch class in the context of quotient stacks. We fix our attention on the varieties with torus action. If we are interested in local invariants of singularities, we study the localization of the equivariant Hirzebruch class tdy (Y → X) at a fixed point. The bottom degree of the Hirzebruch class is the equivariant fundamental class, also called the multi-degree of the variety. It does not change in the deformation class. For example, let Qn ⊂ Cn be the cone over a quadric in Pn−1, in other words, Qn in some coordinates is ∗Supported by NCN grant 2013/08/A/ST1/00804

Monoidal transforms and invariants of singularities in positive characteristic

AbstractThe problem of resolution of singularities in positive characteristic can be reformulated as follows: fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps. The first consists of the transformation of $X$ to a hypersurface with $n$-fold points in the so-called monomial case. The second step consists of the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to present a notion of strong monomial case which parallels that of monomial case in characteristic zero: if a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.

Weights on cohomology, invariants of singularities, and dual complexes

We study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size, when it is the exceptional divisor of a singularity. We also give bounds for the cohomology of links. The invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities.

On Rees algebras and invariants for singularities over perfect fields

On Rees algebras and invariants for singularities over perfect fields