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.     

Equimultiple deformations of isolated singularities

We study equimultiple deformations of isolated hypersurface singularities, introduce a blow-up equivalence of singular points, which is intermediate between topological and analytic ones, and give numerical sufficient conditions for the blow-up versality of the equimultiple deformation of a singularity or multisingularity induced by the space of algebraic hypersurfaces of a given degree. For singular points, which become Newton nondegenerate after one blowing up, we prove that the space of algebraic hypersurfaces of a given degree induces all the equimultiple deformations (up to the blow-up equivalence) which are stable with respect to removing monomials lying above the Newton diagrams. This is a generalization of a theorem by B. Chevallier. This work was partially supported by Grant No.6836-1-9 of the Israeli Ministry of Sciences. The second author thanks the Max-Planck Institut (Bonn) for hospitality and financial support.     

Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and G⊂Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology of any smooth symplectic resolution X?V/G (multiplicative McKay correspondence). We prove further that if is an irreducible Weyl group and , then no smooth symplectic resolution of V/G exists unless G is of types .     

Isolated singularities of nonlinear parabolic inequalities

We study C ^2,1 nonnegative solutions u(x,t) of the nonlinear parabolic inequalities

On equisingular deformations of cone-like surface singularities

In this note we study the rigidity-problem in the equisingular deformation theory for normal surface singularities whose exceptional sets of their minimal resolutions are smooth. We show that they admit non-trivial equisingular deformations if they are non-rational and if their analytic structures are not too different from those of cones. Latter condition is e.g. automatically satisfied if the absolute value of the selfintersection number of the exceptional set A is not less than the genus of A.     

One-parameter toric deformations of cyclic quotient singularities

In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.     

On deformations of maps and curve singularities

We study several deformation functors associated to the normalization of a reduced curve singularity . The main new results are explicit formulas, in terms of classical invariants of (X, 0), for the cotangent cohomology groups T ⁱ , i  =  0,1,2, of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas, respectively, estimates for the -codimension of a parametrized curve singularity, where denotes the Mather–Wall group of left-right equivalence.     

Infinitesimal deformations and obstructions for toric singularities

The obstruction space T² and the cup product T¹ × T¹ → T² are computed for toric singularities.     

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularities Vn(?)C2/Zn, n∈{2,3,4,…} defined by the immersions of C2 into Cn 1 Xn:(z,w)→(zn,zn-1w…,zwn-1,wn).     

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularitiesV _n ? ?²/? _n ,n ∈ {2,3,4,?} defined by the immersions of ?² into ? ⁿ⁺¹ X ⁿ : (z, w) → (z ⁿ ,z ^n?1 w, ?,zw ^n?1 ,w ⁿ )     

Removable singularities of solutions of second-order parabolic equations

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 9–17, November, 1991.     

First order deformations of schemes with normal crossing singularities

We study the sheaf T ¹(X) of first order deformations of a reduced scheme with normal crossing singularities. In particular, we obtain a formula for T ¹(X) in a suitable log resolution of X.