Volume 7 (2007), Number 1. Abstracts

M. Bobieński and Żołądek. A Counterexample to a Multidimensional Version of the Weakened Hilbert's 16th Problem [PDF]

In the weakened 16th Hilbert's Problem one asks for a bound on the number of limit cycles which appear after a polynomial perturbation of a planar polynomial Hamiltonian vector field. It is known that this number is finite for an individual vector field. In the multidimensional generalization of this problem one considers polynomial perturbation of a polynomial vector field with an invariant plane supporting a Hamiltonian dynamics. We present an explicit example of such perturbation with an infinite number of limit cycles which accumulate at some separatrix loop.

Keywords. Polynomial vector field, limit cycle, invariant manifold, Abelian integral

2000 Mathematics Subject Classification. 34C07, 34C08

D. Cerveau, A. Lins-Neto, F. Loray, J. Pereira, and F. Touzet. Complex Codimension One Singular Foliations and Godbillon–Vey Sequences [PDF]

Let F be a codimension one singular holomorphic foliation on a compact complex manifold M. Assume that there exists a meromorphic vector field X on M generically transversal to F. Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N, or F is transversely projective (in the sense of Scárdua), improving our previous work.

Such a vector field insures the existence of a global meromorphic Godbillon–Vey sequence for the foliation F. We derive sufficient conditions on this sequence insuring such alternative. For instance, if there exists a finite Godbillon–Vey sequence or if the Godbillon–Vey 3-form ω₀∧ω₁∧ω₂ is zero, then F is the pull-back of a foliation on a surface, or F is transversely projective (in the sense of \cite{Scardua}). We illustrate these results with many examples.

Keywords. Holomorphic foliations, algebraic reduction, transversal structure

2000 Mathematics Subject Classification. 37F75

R. Efendiev. The Characterization Problem for One Class of Second Order Operator Pencil with Complex Periodic Coefficients [PDF]

The purpose of the present work is solving the characterization problem, which consists of identification of necessary and sufficient conditions on the scattering data ensuring that the reconstructed potential belongs to a particular class.

Keywords. Inverse problem, characterization problem, scattering data, transformation operator

2000 Mathematics Subject Classification. 34B25, 34L05, 34L25, 47A40, 81U40

B. Helffer and T. Hoffmann-Ostenhof. Converse Spectral Problems for Nodal Domains [PDF]

We consider two-dimensional Schrödinger operators in bounded domains. Abstractions of nodal sets are introduced and spectral conditions for them ensuring that they are actually zero sets of eigenfunctions are given. This is illustrated by an application to optimal partitions.

Keywords. Schrödinger operator, Nodal domain, Spectral theory

2000 Mathematics Subject Classification. 35B05

S. Lando and D. Zvonkine. Counting Ramified Coverings and Intersection Theory on Spaces of Rational Functions I (Cohomology of Hurwitz Spaces) [PDF]

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko–Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP¹ by CP¹ with prescribed ramification points and ramification types is related to the degree of the Lyashko–Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko–Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations between the homology classes represented by various strata.

Keywords. Riemann surfaces, moduli space, ramified coverings, Lyashko–Looijenga map, Hurwitz space, Hurwitz numbers

2000 Mathematics Subject Classification. 05A, 14C, 14D22, 30F

A. Polishchuk. Constant Families of t-Structures on Derived Categories of Coherent Sheaves [PDF]

We generalize the construction (due to D. Abramovich and the author) of a “constant” t-structure on the bounded derived category of coherent sheaves D(X×lS) starting with a t-structure on D(X). Namely, we remove smoothness and quasiprojectivity assumptions on X and S and work with t-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced t-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of L. Alonso Tarrío, A. Jeremías López, M.-J. Souto Salorio. As an application of the “constant” t-structures techniques we prove that every bounded nondegenerate t-structure on D(X) with Noetherian heart is invariant under the action of a connected group of autoequivalences of D(X). Also, we show that if X is smooth then the only local t-structures on D(X), i.e., those for which there exist compatible t-structures on D(U) for all open U⊂X, are the perverse t-structures considered by R. Bezrukavnikov.

Keywords. t-structures, triangulated categories, derived categories, coherent sheaves

2000 Mathematics Subject Classification. Primary: 14F05; Secondary: 18E30

D. Zvonkine. Counting Ramified Coverings and Intersection Theory on Hurwitz Spaces II (Local Structure of Hurwitz Spaces Combinatorial Results) [PDF]

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.

Keywords. Riemann surfaces, moduli space, ramified coverings, Lyashko–Looijenga map, Hurwitz space, Hurwitz numbers

2000 Mathematics Subject Classification. 05A, 14C, 14D22, 30F

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