On Chern numbers of algebraic varieties with arbitrary singularities

In 1965 the author introduced the notion of Chern classes for an algebraic variety with arbitrary singularities. Based on this definition the well-known Miyaoka-Yao inequalities have been proved and extended by quite simple direct computations.     

Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].     

On nonisolated hypersurface singularities

We study germs of holomorphic functions whose singular sets are hypersurfaces with isolated singularity in the cases where the transversal singularity is A ₁. For these singularities, we completely describe the homotopy structure of the Milnor fibers. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x_r be closed points in general position in projective spacePn, then the linear subspaceV ofH⁰ (?ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ?ⁿ) formed by those polynomials which are singular at eachx_i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x_r. As such, the “expected” value for the dimension ofV is max(0,h⁰(O(d))?r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.     

Free deformations of hypersurface singularities

The article is devoted to the study of the classification problem for Saito free divisors making use of the deformation theory of varieties. In particular, in the quasihomogeneous case, we describe an approach for computation of free deformations of quasicones over quasismooth varieties based on properties of deformations of varieties with $ {\mathbb{G}_m} $ -action. We also discuss some applications including the problem of compactification of modular spaces and computation of free deformations for certain simple, unimodal, and unimodular singularities.     

Equivalences between isolated hypersurface singularities

Research by this author supported in part by N.S.F. Grant No. DMS-84114477     

On the commutativity of localized Chern characters with refined Gysin homomorphisms

We give a simple proof of the fact that the localized Chern characters of Baum, Fulton and MacPherson commute with the refined Gysin homomorphisms of [3]. This has been proved in [3] in the context of bivariant intersection theory using the technique of deformation to the normal cone. Our proof is more elementary in the sense that it avoids such a deformation and relies on a commutativity formula of these Chern characters with effective Cartier divisors. From this formula we also derive easily that the localized Chern characters pass to rational equivalence.     

Irreducibility of analytic arc-sections of hypersurface singularities

Periodica Mathematica Hungarica - We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we...     

The existence of algebraic surfaces with preassigned Chern numbers

To Cao Xi-Hua on his 70th Birthday