Characteristic classes of complex hypersurfaces

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schürmann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schürmann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schürmann-Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces.     

Boundary manifolds of projective hypersurfaces

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge-theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.     

Vanishing cycles of pencils of hypersurfaces

We prove an extended Lefschetz principle for a large class of pencils of hypersurfaces having isolated singularities, possibly in the axis, and show that the module of vanishing cycles is generated by the images of certain variation maps.     

Monodromy eigenvalues and zeta functions with differential forms

For a complex polynomial or analytic function f, there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫|f|^2sω, where the ω are C^∞ differential forms with compact support, and eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form , where s₀ is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.     

A bound for the orders of the torsion groups of surfaces with c 1 2 = 2 - 1

We shall give a bound for the orders of the torsion groups of minimal algebraic surfaces of general type whose first Chern numbers are twice the Euler characteristics of the structure sheaves minus 1, where the torsion group of a surface is the torsion part of the Picard group. Namely, we shall show that the order is at most 3 if the Euler characteristic is 2, that the order is at most 2 if the Euler characteristic is greater than or equal to 3, and that the order is 1 if the Euler characteristic is greater than or equal to 7.     

Invariant hypersurfaces of endomorphisms of projective varieties

We consider surjective endomorphisms f of degree >1 on projective manifolds X   of Picard number one and their f⁻¹$f^{-1}$-stable hypersurfaces V, and show that V   is rationally chain connected. Also given is an optimal upper bound for the number of f⁻¹$f^{-1}$-stable prime divisors on (not necessarily smooth) projective varieties.     

Topological methods for complex-analytic Brauer groups

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-analytic surfaces.     

On normal surface singularities and a problem of enriques

We construct families of normal surface singularities with the following property: given any fiat projective connected family V →B of smooth, irreducible, minimal algebraic surfaces, the general singularity in one of our families cannot occur, analytically, on any algebraic surfaces which is Irrationally equivalent to a surface in V→B. In particular this holds for V→B consisting of a single rational surface, thus answering negatively to a long standing problem posed by F. Enriques. In order to prove the above mentioned results, wo develop a general, though elementary, method, based on the consideration of suitable correspondences, for comparing a given family of minimal surfaces with a family of surface singularities. Specifically the method in question gives us the possibility of comparing the parameters on which the two families depend, thus leading to the aforementioned results.     

Automorphism groups of positive entropy on minimal projective varieties

We determine the geometric structure of a minimal projective threefold having two ‘independent and commutative’ automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X,G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fitting the ‘boundary case’.     

Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

Limits of tangents and minimality of complex links

We show that the complex link of a large class of space germs (X,x₀) 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,x₀) 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).     

Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations'. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y₀, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G. Dedicated to Professor Jonathan Wahl on his sixtieth birthday. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Generic linear sections of complex hypersurfaces and monomial ideals

Let f:(Cn,0)→(C,0) be an analytic function germ. Under the hypothesis that f is Newton non-degenerate, we compute the μ^?-sequence of f in terms of the Newton polyhedron of f. This sequence was defined by Teissier in order to characterize the Whitney equisingularity of deformations of complex hypersurfaces.     

Topological finite-determinacy of functions with non-isolated singularities

We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal I, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion. Then A has infinite codimension in I in a suitable sense. We also prove the existence of generic topological types of families of germs of I parametrized by an irreducible analytic set.