Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré-Dulac.     

Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (², 0)(, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f _t : (², 0)(, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.     

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space of curves of genus 3.

     

Feuilletages en surfaces, cycles évanouissants et variétés de Poisson

Surface foliations, vanishing cycles and Poisson manifolds. Afoliated cylinder of a foliated manifold (M,F) is a path of integral loopsc _t forF. Such a cylinder defines anon-trivial vanishing cycle c ₀ ifc _t is null-homotopic in its supportF _t for eacht>0, butc ₀ is not null-homotopic in its supportF ₀. Vanishing cycles were introduced by S. P. Novikov to study qualitative aspects of codimension one foliations. In this paper we apply this notion to the study of higher codimensional foliations.The first aim is to show the influence that the triviality of vanishing cycles exerts on the topology of thehomotopy groupoid ofF. It is natural to try to reduce the study of triviality to more regular vanishing cycles, as well as to obtain a nice criterion of triviality. In this way, we introduce the notion ofregular vanishing cycle as the orthogonal version (for a riemannian metric onM) of the classical notion of immersed vanishing cycle and the notion ofcoherent vanishing cycle, i.e. an integral discD ₁ with boundaryc ₁ extends to a global foliated homotopyD _t such thatc _t is the boundary ofD _t for eacht>0. We also prove that the triviality of these vanishing cycles implies the triviality of all vanishing cycles. For compact foliated manifolds, we obtain the following criterion: a regular coherent vanishing cycle is non-trivial if and only if the area of the discsD _t converges to infinity.Finally, we give two applications of these results to surface foliations: we generalize the Reeb stability theorem to higher codimensions and we resolve the problem of the symplectic realization of Poisson structures supported by surface foliations.

     

Systems of Roots and Topology of Complex Flag Manifolds

In this paper we study the topology of a complex homogeneous space M = G/H of complex dimension n, with non vanishing Euler characteristic and G of type A, D, E by means of a topological invariant ₂, which is related to the Poincaré polynomial of M. We introduce the function Q = ₂/n and we examine how it varies as one passes from a principal orbit of the adjoint representation of a compact Lie group G to a more singular one. Moreover, it is proved that if M is a principal orbit G/T then Q depends only on the Weyl group of G.     

Rigidity of certain holomorphic foliations

There is a well-known rigidity theorem of Y. Ilyashenko for (singular) holomorphic foliations in and also the extension given by Gómez-Mont and Ortíz-Bobadilla (1989). Here we present a different generalization of the result of Ilyashenko: some cohomological and (generic) dynamical conditions on a foliation on a fibred complex surface imply the d-rigidity of , i.e. any topologically trivial deformation of is also analytically trivial. We particularize this result to the case of ruled surfaces. In this context, the foliations not verifying the cohomological hypothesis above were completely classified in an earlier work by X. Gómez-Mont (1989). Hence we obtain a (generic) characterization of non-d-rigid foliations in ruled surfaces. We point out that the widest class of them are Riccati foliations.

     

On some geometrical properties of Seifert bundles

In this paper we use the integration along the leaves introduced by Haefliger in 1980 to obtain a differentiable version of the Gysin sequence and Euler class for compact Hausdorff orientable foliations with generic leaf the sphereS ^p. From this we give a geometrical significance to the vanishing of the Euler class on Seifert bundles. We also obtain an integral formula on Seifert bundles similar to the Gauss-Bonnet one.     

Deformations of twisted harmonic maps and variation of the energy

We study the deformations of twisted harmonic maps \(f\) with respect to the representation \(\rho \). After constructing a continuous “universal” twisted harmonic map, we give a construction of every first order deformation of \(f\) in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional \(E\) coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kähler form of the “Betti” moduli space; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of \(E\) at critical points.     

Semi-complete foliations associated to Hamiltonian vector fields in dimension 2

We classify the foliations associated to Hamiltonian vector fields on C², with an isolated singularity, admitting a semi-complete representative. In particular we also classify semi-complete foliations associated to the differential equation .     

Flache T1-Stabilität in der Strati-fizierung verseller Entfaltungen

Let 0 be the local ring of a simple singularity defined over the complex numbers and the dimension of its versal deformation space. Than it is well known that any nearby singularity in this space is also simple and has smaller unfolding dimension in the hierarchy of simple singularities. In particular this implies that the =max-stratum consists just of one point namely the given singularity. We want to generalize this concept as we are interested in families of varieties with formal unchanged singularities. For this we introduce in quite generality the notion of flat T¹-stabi1ity which may be checked for any k- algebra 0 where k is for simplicity an algebraically closed field of à priori arbitrary characteristics. We call 0 formal flat T¹ stable or for short flat T¹-stable if the following is true: if R is any deformation of 0 over an Artin local finite k-algebra A and if T¹(R/A,R) is A-flat than R is isomorphic to the trivial deformation . T¹(R/A,R) is the first cotangent module of R over A with values in R. Obviously the simple singularities A_k, D_k, E₆, E₇, E₈ fulfill this criterion over C but we look also at fibres of arbitrary stable map germs, generic singularities of algebraic varieties where we have to modify this notion in order to deal with wild ramification and to quasihomo-genous hypersurface singularities where it functorializes because in this case T¹ commutes with arbitrary base change. The notion of flat T¹-stable singularities is closely related to questions of existence of equisingular families and is used in[12] and [5], [6] to stratify certain Hilbert schemes.     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.     

On the topology of foliations with a first integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle generates the first homology group of a generic fiber of . In particular, we will prove that for any two Lefschetz vanishing cycles ₀ and ₁ in a regular compact fiber of , there exists a mondromyh such thath( ₀)=± ₁.Partially supported by CNPq-Brazil.     

Equivariant Embeddings of the n-Dimensional Space into a Hilbert Space

The present paper is devoted to the study of equivariant embeddings of the n-dimensional space into a Hilbert space. We consider a representation of a group of similarities. The existence of a cocycle for this representation implies the existence of an isometric embedding of a metric group into the Hilbert space. Then we describe all cocycles of a representation of the additive group of real numbers and construct an embedding of the n-dimensional space with metric d(x,y)=|x-y| into the Hilbert space. Bibliography: 5 titles.     

Moduli spaces of principal <Emphasis Type="Italic">F</Emphasis>-bundles

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation .of of Our spaces generalize moduli spaces of F-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case G = GL^r and is the tensor product of the standard representation and its dual. The importance of the moduli spaces of F-bundles is due to the belief that Langlands correspondence is realized in their cohomology.     

Cyclic Group Actions on 4-Manifolds

Let X be a closed, oriented Riemannian 4-manifold. Suppose that a cyclic group Z( _p (p is prime) acts on X by an orientation preserving isometry with an embedded Riemann surface as fixed point set. We study the representation of Z _p on the Spin^c-bundles and the Z _p-invariant moduli space of the solutions of the Seiberg–Witten equations for a Spin^c-structure X. When the Z _p action on the determinant bundle det L acts non-trivially on the restriction L| over the fixed point set , we consider -twisted solutions of the Seiberg-Witten equations over a Spin^c-structure ' on the quotient manifold X/Z _p X', (0,1). We relate the Z _p -invariant moduli space for the Spin^c-structure on X and the -twisted moduli space for the Spin^c-structure on X'. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the -twisted moduli space. When Z _p acts trivially on L|, we prove that there is a one-to-one correspondence between the Z _p -invariant moduli space M( ^Zp and the moduli space M (") where ' is a Spin^c-structure on X' associated to the quotient bundle L/Z _p X'. vskip0pt When p = 2, we apply the above constructions to a Kahler surface X with b ₂ ⁺ (X) > 3 and H ₂(X;Z) has no 2-torsion on which an anti-holomorphic involution acts with fixed point set , a Lagrangian surface with genus greater than 0 and []2H ₂(H ;Z). If _K ^X 2 > 0 or K _X ² = 0 and the genus g()> 1, we have a vanishing theorem for Seiberg–Witten invariant of the quotient manifold X'. When K _X ² = 0 and the genus g()= 1, if there is a Z ²-equivariant Spin^c-structure on X whose virtual dimension of the Seiberg–Witten moduli space is zero then there is a Spin^c-structure " on X' such that the Seiberg-Witten invariant is ±1.     

Analytic surface germs with minimal Pythagoras number

We determine all complete intersection surface germs whose Pythagoras number is 2, and find that they are all embedded in ³ and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs. In addition, we discuss completely these properties for mixed surface germs in ³. Finally, we find in higher embedding dimension three different families with these same properties. Partially supported by DGICYT, BFM2002-04797 and HPRN-CT-2001-00271 Mathematical Subject Classification (2000): 11E25, 14P15.An erratum to this article can be found at     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.     

Normal forms for singularities of pedal curves produced by non-singular dual curve germs in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual curve germs are non-singular. As an application of our list, we characterize C ^∞ left equivalence classes of pedal curve germs (I, s ₀) → S ⁿ produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.