Isolated Hypersurface Singularities Research Articles

Logarithmic comparison theorem versus Gauss–Manin system for isolated singularities

Abstract For quasihomogeneous isolated hypersurface singularities, the logarithmic comparison theorem has been characterized explicitly by Holland and Mond. In the nonquasihomogeneous case, we give a necessary condition for the logarithmic comparison theorem in terms of the Gauss–Manin system of the singularity. It shows in particular that the logarithmic comparison theorem can hold for a nonquasihomogeneous singularity only if 1 is an eigenvalue of the monodromy.

Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.

A solvability criterion for the Lie algebra of derivations of a fat point

We consider the Lie algebra of derivations of a zero-dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result was motivated by and generalizes the solvability of the Yau algebra of an isolated hypersurface singularity.

Curvature of classifying spaces for Brieskorn lattices

We study t t ∗ -geometry on the classifying space for regular singular TERP-structures, e.g., Fourier–Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the holomorphic sectional curvature of this hermitian metric, which is the analogue of a similar result for classifying spaces of pure polarized Hodge structures.

A note on resolution of rational and hypersurface singularities

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.

The Lê numbers of the square of a function and their applications

Lê numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non-isolated hypersurface singularities and describing the topology associated with a function with non-isolated singularities. They are a generalization of the Milnor number for isolated hypersurface singularities. In this note the authors investigate the composite of an arbitrary square-free f and z2. They get a formula for the Lê numbers of the composite, and consider two applications of these numbers. The first application is concerned with the extent to which the Lê numbers are invariant in a family of functions which satisfy some equisingularity condition, the second is a quick proof of a new formula for the Euler obstruction of a hypersurface singularity. Several examples are computed using this formula including any X defined by a function which only has transverse D(q, p) singularities off the origin.

Characterization of isolated homogeneous hypersurface singularities in ℂ4

Let V be a hypersurface with an isolated singularity at the origin in ℂ n+1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero.

Quasihomogeneity of isolated hypersurface singularities and logarithmic cohomology

We characterize quasihomogeneity of isolated hypersurface singularities by the injectivity of the map induced by the first differential of the logarithmic differential complex in the top local cohomology supported in the singular point.

A remark on lower bound of Milnor number and characterization of homogeneous hypersurface singularities

Let f : (Cn+1, 0) → (C, 0) be a holomorphic germ defining an isolated hypersurface singularity V at the origin. Let μ and ν and pg be the Milnor number, multiplicity and geometric genus of (V, 0), respectively. We conjecture that μ ≥ (ν − 1)n+1 and the equality holds if and only if f is a semi-homogeneous function. We prove that this inequality holds for n = 1, and also for n = 2 or 3 with additional assumption that f is a quasihomogeneous function. For n = 1, if V has at most two irreducible branches at the origin, or if f is a quasi-homogeneous function, then μ = (ν − 1)2 if and only if f is a homogeneous polynomial. For n = 2, if f is a quasihomogeneous function, then μ = (ν − 1)3 iff 6pg = ν(ν − 1)(ν − 2) iff f is a homogeneous polynomial after biholomorphic change of variables. For n = 3, if f is a quasi-homogeneous function, then μ = (ν − 1)4 iff 24pg = ν(ν − 1)(ν − 2)(ν − 3) iff f is a homogeneous polynomial after biholomorphic change of variables.

On the defining equations of hypersurface purely elliptic singularities

We investigate a class of isolated hypersurface singularities, the so-called purely elliptic singularities, of complex algebraic varieties of dimension greater than or equal to two. We show that, for hypersurface purely elliptic singularities defined by nondegenerate polynomials, Calabi-Yau varieties arising among the irreducible components of the essential divisors are concretely associated with the defining equations of these singularities, and that the birational class of the Calabi-Yau varieties does not depend on the irreducible components.

The link of {f(x, y) + zn = 0} and Zariski's conjecture

We consider suspension hypersurface singularities of type g = f(x,y) + z n , where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated hypersurface singularity is a rational homology 3-sphere then it determines the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

Lower Bound of Newton Number

We show a lower estimate of the Milnor number of an isolated hypersurface singularity, via its Newton number. We also obtain analogous estimate of the Milnor number of an isolated singularity of a similar complete intersection variety.

A normal form algorithm for the Brieskorn lattice

This article describes a normal form algorithm for the Brieskorn lattice of an isolated hypersurface singularity. It is the basis of efficient algorithms to compute the Bernstein–Sato polynomial, the complex monodromy, and Hodge-theoretic invariants of the singularity such as the spectral pairs and good bases of the Brieskorn lattice. The algorithm is a variant of Buchberger’s normal form algorithm for power series rings using the idea of partial standard bases and adic convergence replacing termination.

Bernstein Polynomials of a Smooth Function Restricted to an Isolated Hypersurface Singularity

Let f, g be two germs of holomorphic functions on ℂ\_n\_ such that f is smooth at the origin and (f, g) defines an analytic complete intersection (Z, 0) of codimension two. We study Bernstein polynomials of f associated with sections of the local cohomology module with support in X = \_g\_−1(0), and in particular some sections of its minimal extension. When (X, 0) and (Z, 0) have an isolated singularity, this may be reduced to the study of a minimal polynomial of an endomorphism on a finite dimensional vector space. As an application, we give an effective algorithm to compute those Bernstein polynomials when f is a coordinate and g is non-degenerate with respect to its Newton boundary.

On the index of a vector field tangent to a hypersurface with non‐isolated zero in the embedding space

AbstractWe give a generalization of an algebraic formula of Gomez‐Mont for the index of a vector field with isolated zero in (ℂn, 0) and tangent to an isolated hypersurface singularity. We only assume that the vector field has an isolated zero on the singularity here.

Punctured local holomorphic de Rham cohomology

Let V be a complex analytic space and x be an isolated singular point of V. We define the q-th punctured local holomorphic de Rham cohomology Hhq(V,x) to be the direct limit of Hhq(U-{x}) where U runs over strongly pseudoconvex neighborhoods of x in V, and Hhq(U-{x}) is the holomorphic de Rahm cohomology of the complex manifold U-{x}. We prove that punctured local holomorphic de Rham cohomology is an important local invariant which can be used to tell when the singularity (V,x) is quasi-homogeneous. We also define and compute various Poincaré number p˜x(i) and p¯x(i) of isolated hypersurface singularity (V,x).

The Poincaré Series of Some Special Quasihomogeneous Surface Singularities

In (E6) a relation is proved between the Poincare series of the coordi- nate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. We study this relation for Fuchsian singularities and show that it is con- nected with the mirror symmetry of K3 surfaces and with automorphisms of the Leech lattice. We also indicate relations between other singularities and Conway's group.

On the geometry of Sasakian-Einstein 5-manifolds

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [24]. We expand on the recent work of Demailly and Kollar [14] and Johnson and Kollar [20] who give methods for constructing Kahler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors [9], circle V-bundles over log del Pezzo surfaces with Kahler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [36] together with [11] must be diffeomorphic to S5#l(S2×S3). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S2×S3 and infinite families of such structures on #l(S2×S3) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.

Monodromy of Hypersurface Singularities

We describe algorithmic methods for the Gauss-Manin connection of an isolated hypersurface singularity based on the microlocal structure of the Brieskorn lattice. They lead to algorithms for computing invariants like the monodromy, the spectrum, and the spectral pairs. These algorithms use a normal form algorithm for the Brieskorn lattice, standard basis methods for localized polynomial rings, and univariate factorization. We give a detailed description of the algorithm to compute the monodromy.

Limits of tangents and minimality of complex links

We show that the complex link of a large class of space germs ( X, x 0) is characterized by its “simplicity”, among the Milnor fibres of functions with isolated singularity on X. This amounts to the minimality of the Milnor number, whenever this number is defined. Such a phenomenon has been first pointed out in case ( X, x 0) is an isolated hypersurface singularity, by Teissier (Cycles évanescents, sections planes et conditions de Whitney, in: Singularités à Cargèse 1972, Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285–362).