On the string-theoretic Euler number of a class of absolutely isolated singularities

An explicit computation of the so-called string-theoretic E-function of a normal complex variety X with at most log-terminal singularities can be achieved by constructing one snc-desingularization of X, accompanied with the intersection graph of the exceptional prime divisors, and with the precise knowledge of their structure. In the present paper, it is shown that this is feasible for the case in which X is the underlying space of a class of absolutely isolated singularities (including both usual ? _n -singularities and Fermat singularities of arbitrary dimension). As byproduct of the exact evaluation of lim, for this class of singularities, one gets counterexamples to a conjecture of Batyrev concerning the boundedness of the string-theoretic index. Finally, the string-theoretic Euler number is also computed for global complete intersections in ℙ_ℂ ^N with prescribed singularities of the above type. Received: 2 January 2001 / Revised version: 22 May 2001     

Determination of the poles of the topological zeta function for curves

Tof ∈ℂ[x ₁…,x _n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f ⁻¹{0} inf ⁻¹{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x ₁,x ₂]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution. The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research N.F.W.O.     

On the Jacobian Newton polygon of plane curve singularities

We investigate the Jacobian Newton polygon of plane curve singularities. This invariant was introduced by Teissier in the more general context of hypersurfaces. The Jacobian Newton polygon determines the topological type of a branch (Merle’s result) but not of an arbitrary reduced curve (Eggers example). Our main result states that the Jacobian Newton Polygon determines the topological type of a non-degenerate unitangent singularity. The Milnor number, the Łojasiewicz exponent, the Hironaka exponent of maximal contact and the number of tangents are examples of invariants that can be calculated by means of the Jacobian Newton polygon. We show that the number of branches and the Newton number defined by Oka do not have this property. Dedicated to Professor Arkadiusz Płoski on his 60th birthday     

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety X, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ‘stringy’ Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern–Schwartz–MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change-of-variable’ formula holds.     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

Fano 3-folds with divisible anticanonical class

We show the nonvanishing of H ⁰(X,−K _X ) for any a Fano 3-fold X for which −K _X is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, -factorial terminal singularities and −K _X  = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H ⁰(X,−K _X ) and the sharp bound (−K _X )³≥ 8/165. We find the families that can be realised in codimension up to 4.     

Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.     

Removability of singularities for a class of fully non-linear elliptic equations

In this paper we address the problem of understanding the singularities of the fully non-linear elliptic equation σ _k (v) = 1. These σ _k curvature are defined as the symmetric functions of the eigenvalues of the Schouten tensor of a Riemannian metric and appear naturally in conformal geometry, in fact, σ₁ is just the scalar curvature.Here we deal with the local behavior of isolated singularities. We give a sufficient condition for the solution to be bounded near the singularity. The same result follows for a more general singular set Λ as soon as we impose some capacity conditions. The main ingredient is an estimate of the L ^∞ norm in terms of a suitable L ^p norm. Mathematics Subject Classification (2000) 35J60, 53A30     

Halves of a real Enriques surface

The real partE _ℝ of a real Enriques surfaceE admits a natural decomposition in two halves,E _ℝ=E _ℝ ⁽¹⁾ ∪E _ℝ ⁽²⁾ , each half being a union of components ofE _ℝ. We classify the triads (E _ℝ;E _ℝ ⁽¹⁾ ,E _ℝ ⁽²⁾ ) up to homeomorphism. Most results extend to surfaces of more general nature than Enriques surfaces. We use and study in details the properties of Kalinin's filtration in the homology of the fixed point set of an involution, which is a convenient tool not widely known in topology of real algebraic varieties.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

The contact boundary of a complex polynomial

 We define the contact boundary of a complex polynomial f : ℂ ⁿ → ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ ⁿ . We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of polynomials, which are not necessarily topologically trivial. Received: 15 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 32S55, 53D15, 32S50     

Adaption allows efficient integration of functions with unknown singularities

We study numerical integration for functions f with singularities. Nonadaptive methods are inefficient in this case, and we show that the problem can be efficiently solved by adaptive quadratures at cost similar to that for functions with no singularities. Consider first a class of functions whose derivatives of order up to r are continuous and uniformly bounded for any but one singular point. We propose adaptive quadratures Q*_n, each using at most n function values, whose worst case errors are proportional to n^−r. On the other hand, the worst case error of nonadaptive methods does not converge faster than n⁻¹. These worst case results do not extend to the case of functions with two or more singularities; however, adaption shows its power even for such functions in the asymptotic setting. That is, let F^∞_r be the class of r-smooth functions with arbitrary (but finite) number of singularities. Then a generalization of Q*_n yields adaptive quadratures Q**_n such that |I(f)−Q**_n(f)|=O(n^−r) for any f ∈ F^∞_r. In addition, we show that for any sequence of nonadaptive methods there are `many' functions in F^∞_r for which the errors converge no faster than n⁻¹. Results of numerical experiments are also presented. The authors were partially supported, respectively, by the State Committee for Scientific Research of Poland under Project 1 P03A 03928 and by the National Science Foundation under Grant CCR-0095709.     

Averaging of Legendrian Submanifolds of Contact Manifolds

We give a procedure to ‘average’ canonically C¹-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby. Mathematics Subject Classification (2000): 53D10.     

Foliations with persistent singularities

Let ω be a differential q-form defining a foliation of codimension q in a projective variety. In this article we study the singular locus of ω in various settings. We relate a certain type of singularities, which we name persistent, with the unfoldings of ω, generalizing previous work done on foliations of codimension 1 in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of 1-forms defining the foliation.     

Topological equivalence of complex polynomials

The following numerical control over the topological equivalence is proved: two complex polynomials in n≠3 variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions f_s:Cⁿ→C with isolated singularities such that the degree, the number of vanishing cycles and the number of atypical values are constant in the family.     

Deformation of singular lagrangian subvarieties

We investigate deformations of lagrangian manifolds with singularities. We introduce a complex similar to the de Rham-complex whose cohomology calculates deformation spaces. This cohomology turns out to be constructible in many cases. Examples of singular lagrangian varieties are presented and deformations are calculated explicitly.Mathematics Subject Classification (2000): 14B05, 14B12, 58F05, 32S40, 32S60     

Invariant currents and dynamical Lelong numbers

Let ƒ be a polynomial automorphism of ℂ^k of degree λ, whose rational extension to ℙ^k maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙ^k of bidegree (1,1), we show that the sequence λ⁻ⁿ(ƒⁿ)*S converges in the sense of currents on ℙ^k to a linear combination of the Green current T₊ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for ƒ⁻¹.