Quasi-homogeneous Singularities Research Articles

FJRW-Rings and Mirror Symmetry

We verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group, and include a discussion of eight axioms which facilitate the computation of FJRW-rings.

Geometry of obstructed equisingular families of algebraic hypersurfaces

We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with quasihomogeneous singularity with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructedness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the whole family becomes irreducible. As an application we show that if the equisingular family of a projective hypersurface H has a reduced component of expected dimension then the deformation of H induced by the equisingular family | H | is complete with respect to one-parameter deformations.

Riemann surfaces, integrable hierarchies, and singularity theory

In 1991, a celebrated conjecture of Witten [Wi1] asserted that the intersection theory of Deligne-Mumford moduli space is governed by KdVhierarchies. His conjecture was soon proved by Kontsevich [Ko]. Since then, the Witten-Kontsevich theorem has introduced the seemingly alien concept of integrable hierarchies to the geometry. Immediately after, a great deal of effort was spent in investigating other integrable hierarchies in GromovWitten theory. A much studied example is 2-Toda hierarchies for P1 by Okounkov-Pandharipande. It was generalized recently to orbifold P1 [J], [MT], [PR]. A famous problem of similar flavor is the Virasoro constraint for Gromov-Witten theory for an arbitrary target. The common characteristics of these problems are: (1) all of them are very difficult; (2) all of them are mysterious. In particular, the choice of integrable hierarchies seems to be matter of luck and there is no general pattern to predict the hierarchies for a given geometrical problem. Therefore, it is particularly important to explore the relation of integrable hierarchies to Gromov-Witten theory systematically. This is the main focus of this survey. In fact, this question was very much in Witten’s mind when he proposed his famous conjecture in the first place. Around the same time, he also proposed a sweeping generalization of his conjecture [Wi2, Wi3]. The core of his generalization is a remarkable first order nonlinear elliptic PDE associated to an arbitrary quasihomogeneous singularity. During the last few years, Witten’s generalization has been explored and a new Gromov-Witten type theory has been constructed by Fan-Jarvis-Ruan [FJR1, FJR2, FJR3]. In particular, Witten’s conjecture for ADE-integrable hierarchies has been verified. It is important to mention that the geometry behind these integrable

A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give a sufficient criterion for the property, a simple proof of F.J. Calderón-Moreno’s theorem that free divisors have the property, a proof that divisors in dimension $3$ with only isolated quasi-homogeneous singularities have the property, an example of a nonfree divisor with nonisolated singularity having the property, an example of a divisor not having the property, and an algorithm to compute the V-filtration along a divisor up to a given order.

Kohn-Rossi cohomology and its application to the complex plateau problem, II

Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n − 1 in C n+1 . Tanaka introduced a spectral sequence E (p,q) r (X) with E (p,q) 1 (X) being the Kohn-Rossi cohomology group and E (k,0) 2 (X) being the holomorphic De Rham cohomology denoted by H k h (X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated singularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has remained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dim X = 3 and H 2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.

Geometry of obstructed families of curves

We study certain equisingular families of curves with quasihomogeneous singularity of minimal obstructness (i.e. h 1 = 1 ). We show that our families always have expected codimension. Moreover they are either non-reduced with smooth reduction or decompose into two smooth components of expected codimension that intersect non-transversally or are reduced irreducible non-smooth varieties which have smooth singular locus with sectional singularity of type A 1 . On the other hand there is an example of an equisingular family of curves with multiple quasihomogeneous singularities of minimal obstructness which is smooth but has wrong codimension. We use algorithms of computer algebra as a technical tool.

Deformations with Section: Cotangent Cohomology, Flatness Conditions, and Modular Subgerms

We study modular subspaces corresponding to two deformation functors associated with an isolated singularity X0: the functor \(Def_{X_0 }\) of deformations of X0 and the functor \(Def_{X_0 }^s\) of deformations with section of X0. After recalling some standard facts on the cotangent cohomology of analytic algebras and the general theory of deformations with section, we give several criteria for modularity in terms of the relative cotangent cohomology modules of a deformation. In particular, it is shown that the modular strata for the functors \(Def_{X_0 }\) and \(Def_{X_0 }^s\) of quasi-homogeneous complete intersection singularities coincide. Then flatness conditions for the first cotangent cohomology modules of the deformation functors under consideration are compared.

PROJECTIVE HYPERSURFACES WITH MANY SINGULARITIES OF PRESCRIBED TYPES

Patchworking of singular hypersurfaces is used to construct projective hypersurfaces with prescribed singularities. For all , an asymptotically proper existence result is deduced for hypersurfaces in with singularities of corank at most 2 prescribed up to analytical or topological equivalence. In the case of -smooth hypersurfaces with only simple singularities, the result is even asymptotically optimal, that is, the leading coefficient in the sufficient existence condition cannot be improved, which is new even in the case of plane curves. Furthermore, an asymptotically proper existence result is proved for hypersurfaces in with quasihomogeneous singularities. The estimates substantially improve all known (general) existence results for hypersurfaces with these singularities.

Orbifolding Frobenius Algebras

We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super-graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi-homogeneous singularities and their symmetries.

Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity

Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity

Computing Gauss–Manin Systems for Complete Intersection Singularities Sμ

Abstract The Gauss–Manin systems with coefficients having logarithmic poles along the discriminant sets of the principal deformations of complete intersection quasihomogeneous singularities Sμ are calculated. Their solutions in the form of generalized hypergeometric functions are presented.

Computing Gauss-Manin systems for complete intersection singularitiesS μ

The Gauss-Manin systems with coecients having log- arithmic poles along the discriminant sets of the principal deforma- tions of complete intersection quasihomogeneous singularities Sn are calculated. Their solutions in the form of generalized hypergeometric functions are presented.

Micro-local characterization of quasi-homogeneous singularities

A moduli algebra A ( V ) of hypersurface singularity ( V , 0) is a finite dimensional C -algebra. In 1982, Mather and Yau proved that two germs of complex analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebra are isomorphic. It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasi-homogeneous in terms of its moduli algebra. In this paper we prove that ( V , 0) admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite dimensional nonnegatively graded algebra. In 1983, Yau introduced a finite dimensional Lie algebra L ( V ) to an isolated hypersurface singularity ( V , 0). L ( V ) is defined to be the algebra of derivations of the moduli algebra A ( V ) and is finite dimensional. We prove that ( V , 0) is quasi-homogeneous singularity if (1) L ( V ) is isomorphic to a nonnegatively graded Lie algebra without center, (2) There exists E in L ( V ) of degree zero such that [ E, D i ] = i D i for any D i in L ( V ) of degree i , and (3) For any element a ∈ m - m 2 where m is the maximal ideal of A ( V ), aE is not in degree zero part of L ( V ).

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

The moduli dependence of (2,2) superstring compactifications based on Calabi-Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with c = 9 whose potential is a sum of A-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at c = 9. We use minor symmetry to derive the dependence of the models on the complexified Kähler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (“twisted”) deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.

Cyclic quotients of 2-dimensional quasi-homogeneous hypersurface singularities

Cyclic quotients of 2-dimensional quasi-homogeneous hypersurface singularities

Algebraic geometry and effective lagrangians

N=2 supersymmetric Landau-Ginsburg fixed points describe nonlinear models whose target spaces are algebraic varieties in certain generalized projective spaces; the defining equation is precisely the zero set of the superpotential, considered as a condition in the projective space. The ADE classification of modular invariants arises as the classification of projective descriptions of P 1; in general, the hierarchy of fixed points is conjectured to be isomorphic to the classification of quasihomogeneous singularities. The condition of vanishing first Chern class is an integrality condition on the Virasoro central charge; the central charge is determined by the superpotential. The operator algebra is given by the algebra of Wick contractions of perturbations of the superpotential.

Deformations of quasi-homogeneous surface singularities

Let A = ~ A~ be a finitely generated normal graded domain over (E, so that i=0 V= Spec A has an isolated singularity. We say V is a quasi-homogeneous singularity, or admits a good IE*-action, or is defined by weighted homogeneous polynomials. We are concerned with two questions about the deformations of V. There are two natural notions of "equisingular deformation" of V. First, one has deformations of weight > 0; that is, one perturbs the defining equations of Vby terms of weight at least as big. To make a good notion out of this, Dock Sang Rim observed one is really deforming A and the weight filtration {Ij}, where Ij = ~ Ai ;

Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularit�ten

We describe a new construction to obtain a simple hypersurface singularity from the corresponding simple complex Lie-groupG. LetX be the closed orbit in the projective space attached to the Lie algebra $$\mathfrak{g}$$ ofG. Consider a regular nilpotent element $$y_0 \in \mathfrak{g}$$ and denote byH y 0 the hyperplane orthogonal toy 0 with respect to the Killing form. Then the hyperplane sectionX⊃H y 0, has exactly one singularity which is simple of desired type. By variation of the pointy 0 we obtain a versal deformation. The construction generalizes with minor modifications to any characteristicp of the basefield. Even in bad characteristic we recover at least the positive part of the semiuniversal deformation. We prove that forp=2 a simple, quasihomogeneous singularity of type A7 resp. D8 is adjacent to E7 resp. E8 provided its dimension is even. Furthermore A8 is adjacent to E8 forp=3.

COHOMOLOGY OF A QUASIHOMOGENEOUS COMPLETE INTERSECTION

In this paper are computed the Poincare series of the highest local cohomology of the modules of regular forms on a nondegenerate quasihomogeneous singularity and the dimensions of the cohomology spaces of the bidual sheaves of holomorphic forms on a quasismooth complete intersection. Theorems are proved about the structure of the module of vector fields on a nondegenerate quasihomogeneous singularity and about whether the maximal modular stratum of a versal deformation of such a singularity is reduced. Bibliography: 47 titles.

Constant milnor number implies constant multiplicity for quasihomogeneous singularities

We show that for certain topologically trivial deformations of an isolated hypersurface singularity the multiplicity does not change. This applies to all μ-constant “first order” deformations and to all μ-constant deformations of a quasihomogeneous singularity.