We consider a generic complex polynomial in two variables and a basis in the first homology group of a nonsingular level curve. We take an arbitrary tuple of homogeneous polynomial 1-forms of appropriate degrees so that their integrals over the basic cycles form a square matrix (of multivalued analytic functions of the level value). We give an explicit formula for the determinant of this matrix.

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective $4$-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Soit $X$ une variété analytique complexe lisse et $D\subset X$ un diviseur libre. Les connexions logarithmiques intégrables par rapport à $D$ peuvent être étudiées comme des ${𝒪}_X$-modules localement libres munis d’une structure de module (à gauche) sur l’anneau ${𝒟}_X(logD)$ des opérateurs différentiels logarithmiques . Dans cet article nous étudions deux résultats liés : la relation entre les duaux d’une connexion logarithmique intégrable sur les anneaux de base ${𝒟}_X$ et ${𝒟}_X(logD)$, et un critère différentiel pour le théorème de comparaison...

We give a criterion for a real-analytic function defined on a compact nonsingular real algebraic set to be analytically equivalent to a rational function.

We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with the Poincaré series in equivariant homology for compact nonsingular sets, but is different in general. We put emphasis on the particular case of $\mathbb{Z}/2\mathbb{Z}$, and give an application to the study of the singularities of Nash function germs via an analog of the motivic zeta function of Denef and Loeser.

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the corresponding irregular value is regular at infinity. We apply these results to the study of vanishing and invariant cycles.

A complex hypersurface $D$ in ${ℂ}^n$ is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for $n$ at most $4$.By analogy with Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for $D$ if the complex of global logarithmic differential forms computes the complex cohomology of ${ℂ}^n\setminus D$. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the...

We derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities ${\Sigma }^{i,j}$. The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.

We associate to a given polynomial map from ${ℂ}^2$ to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Using the notion of the maximal polar quotient we characterize the critical values at infinity of polynomials in two complex variables. As an application we give a necessary and sufficient condition for a family of affine plane curves to be equisingular at infinity.

Let $X_1$ and $X_2$ be two compact strongly pseudoconvex CR manifolds of dimension $2n-1\ge 5$ which bound complex varieties $V_1$ and $V_2$ with only isolated normal singularities in ${ℂ}^{N1}$ and ${ℂ}^{N2}$ respectively. Let $S_1$ and $S_2$ be the singular sets of $V_1$ and $V_2$ respectively and $S_2$ is nonempty. If $2n-N_2-1\ge 1$ and the cardinality of $S_1$ is less than 2 times the cardinality of $S_2$, then we prove that any non-constant CR morphism from $X_1$ to $X_2$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR manifold of...

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.

By a classical formula due to Enriques, the Euler number χ(X) of the non-singular normalization X of an algebraic surface S with ordinary singularities in P³(ℂ) is given by χ(X) = n(n²-4n+6) - (3n-8)m + 3t - 2γ, where n is the degree of S, m the degree of the double curve (singular locus) $D_S$ of S, t is the cardinal number of the triple points of S, and γ the cardinal number of the cuspidal points of S. In this article we shall give a similar formula for an algebraic threefold with ordinary singularities...