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We compute the integral homology of the space of paths in $\mathbb{C}P^n$ with endpoints in $\mathbb{R}P^n$, $n \ge 1$ and its algebra structure with respect to the Pontryagin-Chas-Sullivan product with $\mathbb{Z}/2$-coefficients., 33 pages, 6 figures

We examine the action of the fundamental group $\Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $\widetilde{\mathcal B}$ on $m$ strings on a suitable space of $m\times m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${\mathcal S}_c:\widetilde{\mathcal B}\to GL_m({\mathbb{Z}}[\Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information., Comment: 62 pages, 7 figures; v5: minor modifications with respect to v4

International audience

We study, in this thesis, some numerical block Krylov subspace methods. These methods preserve geometric properties of the reduced matrix (Hamiltonian or skew-Hamiltonian or symplectic). Among these methods, we interest on block symplectic Arnoldi, namely block J-Arnoldi algorithm. Our main goal is to study this method, theoretically and numerically, on using ℝ²nx²s as free module on (ℝ²sx²s, +, x) with s ≪ n the size of block. A second aim is to study the approximation of exp (A)V, where A is a real Hamiltonian and skew-symmetric matrix of size 2n x 2n and V a rectangular matrix of size 2n x 2s on block Krylov subspace Km (A, V) = blockspan {V, AV,...Am-1V}, that preserve the structure of the initial matrix. this approximation is required in many applications. For example, this approximation is important for solving systems of ordinary differential equations (ODEs) or time-dependant partial differential equations (PDEs). We also present a block symplectic structure preserving Lanczos method, namely block J-Lanczos algorithm. Our approach is based on a block J-tridiagonalization procedure of a structured matrix. We propose algorithms based on two normalization methods : the SR factorization and the Rj R factorization. In the last part, we proposea generalized algorithm of Greville method for iteratively computing the Moore-Penrose inverse of a rectangular real matrix. our purpose is to give a block version of Greville's method. All methods are completed by many numerical examples., Nous nous intéressons dans cette thèse, à l'étude de certaines méthodes numériques de type krylov dans le cas symplectique, en utilisant la technique de blocs. Ces méthodes, contrairement aux méthodes classiques, permettent à la matrice réduite de conserver la structure Hamiltonienne ou anti-Hamiltonienne ou encore symplectique d'une matrice donnée. Parmi ces méthodes, nous nous sommes intéressés à la méthodes d'Arnoldi symplectique par bloc que nous appelons aussi bloc J-Arnoldi. Notre but essentiel est d’étudier cette méthode de façon théorique et numérique, sur la nouvelle structure du K-module libre ℝ²nx²s avec K = ℝ²sx²s où s ≪ n désigne la taille des blocs utilisés. Un deuxième objectif est de chercher une approximation de l'epérateur exp(A)V, nous étudions en particulier le cas où A est une matrice réelle Hamiltonnienne et anti-symétrique de taille 2n x 2n et V est une matrice rectangulaire ortho-symplectique de taille 2n x 2s sur le sous-espace de Krylov par blocs Km(A,V) = blockspan {V,AV,...,Am-1V}, en conservant la structure de la matrice V. Cette approximation permet de résoudre plusieurs problèmes issus des équations différentielles dépendants d'un paramètre (EDP) et des systèmes d'équations différentielles ordinaires (EDO). Nous présentons également une méthode de Lanczos symplectique par bloc, que nous nommons bloc J-Lanczos. Cette méthode permet de réduire une matrice structurée sous la forme J-tridiagonale par bloc. Nous proposons des algorithmes basés sur deux types de normalisation : la factorisation S R et la factorisation Rj R. Dans une dernière partie, nous proposons un algorithme qui généralise la méthode de Greville afin de déterminer la pseudo inverse de Moore-Penros bloc de lignes par bloc de lignes d'une matrice rectangulaire de manière itérative. Nous proposons un algorithme qui utilise la technique de bloc. Pour toutes ces méthodes, nous proposons des exemples numériques qui montrent l'efficacité de nos approches.

Résumé: Soit $$(N,c_N)$$ un fibré vectoriel complexe muni d'une structure réelle au-dessus d'une courbe réelle $$(\Sigma _g,c_\Sigma )$$ de genre $$g\in \mathbb{N }$$ . Nous étudions le signe de l'action des automorphismes de $$(N,c_N)$$ relevant l'identité de $$\Sigma _g$$ sur les orientations du fibré déterminant au-dessus de l'espace des opérateurs de Cauchy-Riemann réels de $$(N,c_N)$$ . Ce signe s'obtient comme le produit de deux termes. Le premier calcule la signature des permutations induite par les automorphismes sur les structures $$Pin^\pm $$ de la partie réelle de $$(N,c_N)$$ . Le second provient de l'action des automorphismes du fibré sur les classes de bordisme de structures $$Spin$$ réelles de $$(\Sigma _g,c_\Sigma )$$

The main objective of this thesis is to present a geometry given by Cartan in 1933 \cite{Cartan1933}. The Finsler geometry has many analogies with this theory. We studied the outline of this geometry. The starting point of Cartan which is similar to that which leads to the Finsler geometry, is to imagine the space to be made of ''contact elements'', an element being given by a point $M$ in $\mathcal{M}$ and an oriented hyperplan passing through this point in the tangent space $T_M\mathcal{M}^n$. Thus we have defined \textit{Cartan geometry based on the concept of area}. In a first step, I was interested in the notion of orthogonality in this geometry. Cartan's method to state the equivalence problem is a crucial tool. After, we applied this method to Monge-Ampère equations (elliptic case). Hence following the work of R. Bryant, D. Grossman and P. Griffiths in the years 2002-2005 in order to clarify the strategy of Cartan. However, many points have to be explored in order to have a clear dictionary between a modern language as the one used by Bryant, Griffiths and Grossman and that of Cartan., Mon travail de thèse consiste à comprendre une géométrie introduite par Cartan en 1933 \cite{Cartan1933}. \textit{La géométrie de Finsler} présente de nombreuses analogies avec cette théorie. Nous avons étudié les grandes lignes de cette géométrie. Le point de départ de Cartan qui est analogue à celui qui conduit à la géométrie finslerienne, est d'imaginer l'espace comme étant un lieu ''d'éléments de contact'', un élément étant la donnée d'un point $M\in\mathcal{M}^n$ et d'un hyperplan $H$ passant par ce point et orienté dans l'espace tangent $T_M\mathcal{M}^n$. Nous avons ainsi défini \textit{la géométrie de Cartan fondée sur la notion d'aire} dans un premier temps, je me suis intéressé à la notion d'orthogonalité dans cette géométrie. La méthode de Cartan pour étudier le problème d'équivalence est un outil puissant qui est implicitement décrit dans cette géométrie. Nous avons ensuite appliqué cette méthode aux équations de Monge-Ampère (cas elliptique), en s'inspirant des travaux de R. Bryant, D. Grossmann et P. Griffiths. Plusieurs faits ne sont pas encore suffisamment clairs pour disposer d'un dictionnaire évident entre ces travaux et celui donné par Cartan.

Added Theorems 1.1 and 1.2, which are corollaries of the other results in the paper. These results apply to general d-capacities, not only 2-capacities. 27 pages, 13 figures; We show that the cylinder B^2(1) \times \mathbb{R}^{2(n-1)} embeds symplectically into the product B^{2(n-1)}(R) \times \mathbb{R}^2 for some finite R>0, which answers a question posed by H. Hofer in 1989. This implies the nonexistence of symplectic d-capacities, 1\sqrt{3}. We supply the final arguments to answer Hofer's questions. The main tool is a construction to remove singular limits of families of embeddings.

For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compact support on T^*N. These satisfy properties analogous to those of partial quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity., 41 pages. This is a new substantially revised version of our older paper 1104.4928, which contained unverifiable assumptions. The current version is not based on those assumptions

These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and describe an integrable system for the representation space of a surface. Finally, we explain some basic ideas on geometric quantization. We apply them to the case of representation spaces by computing Bohr-Sommerfeld orbits with metaplectic correction., 47 pages, 8 figures

In a first part, we develop the theory of symplectic homogenezation and its application to the Aubry Mather theory and also to symplectic rigidity. Spectral invariants are the main tool in this work. In a second part, we recall all the new applications from sheaves theoretic methods to the study of non-displacability problem . We formulate what we think to be the equivalent object to the lagrangian Floer homology and its spectral invariants. Then, using this tools, we prove non-displacability of non exact lagrangian submanifolds into the cotangent bundle. After, we discuss the applications to $C^0$ symplectic topology and to non smooth optimization., Dans une première partie, nous développerons la théorie de l'homogénéisation symplectique ainsi que ses applications à la théorie de Mather et à la rigidité symplectique. Les invariants spectraux lagrangiens seront l'outil de base de ce travail. Dans une seconde partie, nous rappelerons les toutes nouvelles applications de la théorie des faisceaux aux problèmes de non déplaçabilité. Nous formulerons ce que nous pensons être l'équivalent de l'homologie de Floer dans ce cas là et les invariants spectraux. Puis, à l'aide de ces outils nous prouverons la non-déplaçabilité de sous-variétés lagrangiennes non exactes du cotangent. Ensuite, nous parlerons des applications à la topologie symplectique $C^0$ et à l'optimisation non lisse.

We consider a complex vector bundle N equipped with a real structure cN over a real curve. The set of all real Cauchy-Riemann operators on (N, cN) is an infinite dimensional affine space. The union of all the determinant lines of such operators is a real line bundle Det(N) over this space. The aim of this work is to study the action of the the automorphism group of (N, cN) on the orientations of Det(N) and to deduct some results concerning the orientability of the moduli spaces of real curves in a real symplectic manifold. We first compute the action of the automorphisms which induce the identity on the complex line bundle det(N) in terms of Pin± structures on the real part of (N, cN). We then note that an automorphism of (N, cN) which lifts the identity of the curve acts on the bordism classes of real Spin structures on the curve. We use this action to give a topological description of the action on the orientations of Det(N). Finally, we introduce the notion of a lift of a diffeomorphism of the curve associated to a divisor compatible with (N, cN). Those are automorphisms of (N, cN) for which we can compute explicitly the action on the orientations of Det(N). The last part is devoted to some applications to the study of the orientability of moduli spaces of real curves in a real symplectic manifold. We compute for example, the first Stiefel-Whitney class of such moduli spaces in the case of the complex projective space of dimension three., L'ensemble des opérateurs de Cauchy-Riemann réels sur un fibré vectoriel complexe N muni d'une structure réelle cN au-dessus d'une courbe réelle est un espace affine de dimension infinie. L'union des déterminants de ces opérateurs est un fibré en droites réelles au-dessus de cet espace. L'objet de cette thèse est l'étude de l'action des automorphismes du fibré (N, cN) sur les orientations de ce fibré déterminant ainsi que de ses conséquences sur l'orientabilité des espaces de modules de courbes réelles dans une variété symplectique réelle. Nous commençons par interpréter l'action des automorphismes qui induisent l'identité sur le fibré en droites complexes det(N) en termes d'action sur les structures Pin± de la partie réelle de N. Nous remarquons ensuite qu'un automorphisme au-dessus de l'identité agit sur les classes de bordisme de structures Spin réelles de la courbe et nous utilisons cette action afin d'obtenir une description en termes topologiques de l'action sur les orientations du fibré déterminant. Enfin, pour comprendre l'action des automorphismes de (N, cN) qui ne relèvent pas l'identité, nous introduisons la notion de relevé d'un difféomorphisme de la courbe associé à un diviseur compatible avec (N, cN) et nous calculons le signe de l'action d'un tel relevé sur les orientations du fibré déterminant. Dans une dernière partie, nous appliquons les résultats obtenus à l'étude de l'orientabilité des espaces de modules de courbes réelles dans des variétés symplectiques réelles. Nous calculons en particulier la première classe de Stiefel-Whitney de l'espace de modules des courbes réelles dans l'espace projectif complexe de dimension trois.

Consider the Chern-Simons topological quantum field theory with gauge group SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the SU(2)-character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures., 47 pages, 2 figures

We shall give the geometric definition of several holomorphic curve invariants for symplectic and contact manifolds, such as contact homology, symplectic homology and some of their variants. We shall then explain the relationship between these theories, leading to a common algebraic framework. These relations can also be used to translate structural results from one of these invariants to the other ones. This can be illustrated with the effect of some geometric operations on these invariants, such as Legendrian surgery.

International audience

In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of Poisson, presymplectic, but also complex structures. Proofs are given in detail. The second part deals with sigma models. Topological ones, in particular the AKSZ and the Dirac sigma models, as generalizations of the Poisson sigma models to higher dimensions and to Dirac structures, respectively, but also physical ones, that reduce to standard Yang Mills theories for the "flat" choice of a Lie algebra: Lie algebroid Yang Mills theories and possible action functionals for nonabelian gerbes and general higher gauge theories. Characteristic classes associated to Dirac structures and to higher principal bundles are also mentioned., 52 pages, Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry, ed. V. Cortes

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts., 16 pages. v2: Final version to be published in Theor. Math. Phys. (incorporates only very minor changes)

Invited lecture; Invited lecture at "Homological Mirror Symmetry and Related Topics" workshop, January 18-23, 2010, University of Miami, USA; International audience

10 pages.; Dans ce texte, nous cherchons à apporter quelques éléments de réflexion sur la bibliométrie et ses conséquences.

I first recall the various problems of real enumerative geometry out of which I could extract some integer valued invariants, providing some real counterpart to Gromov-Witten invariants. I then discuss sharpness of the lower bounds given by these invariants and some of their arithmetical properties. Finally, I present further results which guaranty the presence or absence of pseudo-holomorphic discs with boundary on a Lagrangian submanifold of a given symplectic manifold., Comment: 26 pages, 2 figures

The Rabinowitz-Floer homology groups $RFH_*(M,W)$ are associated to an exact embedding of a contact manifold $(M,\xi)$ into a symplectic manifold $(W,\omega)$. They depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz-Floer homology $RFH_*(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RFH_*(ST^*L,T^*L)$, where $ST^*L$ is the unit cosphere bundle of a closed manifold $L$. As an application, we prove that the image of an exact contact embedding of $ST^*L$ (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided $\dim L\ge 4$ and the embedding induces an injection on $\pi_1$. In particular, $ST^*L$ does not admit an exact contact embedding into a subcritical Stein manifold if $L$ is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings., Comment: 59 pages, 8 figures

We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space., Comment: v3: 48 pages. Added Lemma 5.8 relating the long exact sequence of the cone and the Gysin sequence. Various other modifications throughout the text following referee's comments. This is the final version published in J. Topol. Anal

Gromov has shown how to construct holomorphic maps of the plane to a complex manifold with prescribed values on a lattice. We prove a similar interpolation theorem for pseudo-holomorphic maps from a cylinder to an almost-complex manifold. An infinite number of surgeries are required. On the way, we need a refinement of the gluing process for two pseudo-holomorphic curves.

Using Sympelctic Field Theory as a computational tool, we compute Gromov-Witten theory of target curves using gluing formulas and quantum integrable systems. In the smooth case this leads to a relation of the results of Okounkov and Pandharipande with the quantum dispersionless KdV hierarchy, while in the orbifold case we prove triple mirror symmetry between GW theory of target P^1 orbifolds of positive Euler characteristic, singularity theory of a class of polynomials in three variables and extended affine Weyl groups of type ADE.

We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and re-discovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra U'_q(sp_{2n}). We use the Casimir elements of both twisted quantized enveloping algebras to re-produce some well-known and construct some new polynomial invariants of the corresponding Poisson algebras., 29 pages, more references added

International audience; We investigate the relation between two notions of integrability, to have enough first integrals on the one hand, and to have meromorphic solutions on the other, that are present in Kowalevskaya's famous mémoire on the rigid body problem. We concentrate on the examples of the rigid body and of the system of Hénon-Heiles.

L'algèbre et le groupe de Virasoro jouent un rôle important dans un grand nombre de branches des mathématiques et de la physique théorique, comme les systèmes intégrables, les classes caractéristiques, la quantification, les systèmes dynamiques, les théories de cordes et la théorie des champs conformes.Ce livre constitue un exposé mathématique détaillé de leurs multiples aspects, accompagné des notions algébriques et géométriques indispensables à l'autonomie de l'ouvrage : théorie des groupes de Lie, éléments de géométrie symplectique et algèbre homologique.Les trois derniers chapitres sont consacrés à diverses généralisations, comme les algèbres-W et les algèbres superconformes. Une notice historique illustrant les développements successifs de la théorie ainsi qu'un appendice de Vlad Sergiescu sur le groupe de Thompson font office d'épilogue temporaire à l'un des domaines de recherche les plus féconds de la physique théorique actuelle, domaine qui est encore loin d'avoir livré tous ses secrets.

We consider the product of two projective lines equipped with the complex conjugation transforming $(x,y)$ into $(\bar{y},\bar{x})$ and blown up in at most two real, or two complex conjugate, points. For these four surfaces we prove the logarithmic equivalence of Welschinger and Gromov-Witten invariants., 13 pages, 6 figures

This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kahler manifold M with a Hamiltonian torus action. Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced Toeplitz operator, a result related to the Riemann-Roch-Kawazaki theorem., Comment: corrected typos, accepted for publication in J. Funct. Anal

We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the theorem implies a conjecture of J Harer, namely, that any fibered link in the three-sphere can be obtained from the unknot by a sequence of plumbings and deplumbings of Hopf bands. The proof presented here involves contact geometry in an essential way., This is the version published by Geometry & Topology on 4 March 2006

Let $(X, \omega, c_X)$ be a real symplectic 4-manifold with real part $R X$. Let $L \subset R X$ be a smooth curve such that $[L] = 0 \in H_1 (R X ; Z / 2Z)$. We construct invariants under deformation of the quadruple $(X, \omega, c_X, L)$ by counting the number of real rational $J$-holomorphic curves which realize a given homology class $d$, pass through an appropriate number of points and are tangent to $L$. As an application, we prove a relation between the count of real rational $J$-holomorphic curves done in math.AG/0303145 and the count of reducible real rational curves done in math.SG/0502355. Finally, we show how these techniques also allow to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics., Comment: 21 pages, 8 figures

In this paper, we describe an example of a hyperkaehler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can furthermore be identified either with the cotangent space of a connected component of the restricted grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkaehler manifolds. In addition, Kaehler potentials are computed using Kostant-Souriau's theory of prequantization.

A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such ``almost-toric 4-manifolds'' which admits a Hamiltonian $S^1$-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system., Comment: finally a revision of the 2003 preprint. 29 pages, 8 figures

We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some mutiplicity. In the case of the complex projective plane equipped with its standard symplectic form and real structure, these invariants coincide with the ones previously constructed in math.AG/0303145. This leads to a relation between the count of real rational J-holomorphic curves done in math.AG/0303145 and the count of real rational reducible J-holomorphic curves presented here., Comment: 27 pages, 5 figures

We say that a contact manifold is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). Generalizing results of Milnor and Giroux, we associate to each holomorphic function f defined on X, with isolated singularity at x, an open book which supports the contact structure. Moreover, we prove that any 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. * * * * * * * * In the first version of the paper, we showed that the open book associated to f carries the contact structure only up to an isotopy. Here we drop this restriction. Following a suggestion of Janos Kollar, we also give a simplified proof of the algebro-geometrical theorem 4.1, central for the uniqueness result., 17 pages

The purpose of this paper is to give a survey of the various versions of Floer homology for manifolds with contact type boundary that have so far appeared in the literature. Under the name of ``Symplectic homology'' or ``Floer homology for manifolds with boundary'' they bear in fact common features and we shall try to underline the principles that unite them. Once this will be accomplished we shall proceed to describe the peculiarity of each of the constructions and the specific applications that unfold out of it: classification of ellipsoids and polydiscs in $\CC^n$, stability of the action spectrum for contact type boundaries of symplectic manifolds, existence of closed characteristics on contact type hypersurfaces and obstructions to exact Lagrange embeddings. The computation of the Floer cohomology for balls in $\CC^n$ is carried by explicitly perturbing the nondegenerate Morse-Bott spheres of closed characteristics., Comment: 32 pages, 5 figures

The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of K\"ahler differentials. Here we generalize it for simplicial, graded and anticommutative objects in ``good pairs of categories''. We apply this generalization to complex spaces and noetherian schemes and deduce two decomposition theorems for their (relative) Hochschild cohomology (special cases of those were recently shown by Buchweitz-Flenner and Yekutieli). The first one shows that Hochschild cohomology contains tangent cohomology: $\HH^n(X/Y,\sM)=\coprod_{i-j=n}\Ext^i(\dach^j\LL(X/Y),\sM)$. The left side is the n-th Hochschild cohomology of $X$ over $Y$ with values in $\sM$. The right hand-side contains the $n$-th relative tangent cohomology $\Ext^n(\LL(X/Y),\sM)$ as direct factor. The second consequence is a decomposition theorem for Hochschild cohomology of complex analytic manifolds and smooth schemes in characteristic zero: $\HH^n(X)=\coprod_{i-j=n}H^i(X,\dach^j\sT_X).$ On the right hand-side we have the sheaf cohomology of the exterior powers of the tangent complex., Comment: 41 pages

This paper is concerned with the Hamiltonian actions of a torus on a symplectic manifold. We are interested here in two global invariants: the Duistermaat-Heckman measure DH(M), and the Riemann-Roch chatacters RR(M,L^k),k>0, which are defined when the symplectic manifold is prequantized by a Kostant-Souriau line bundle L. We can associate to each connected component C of regular values of the moment map the following local invariants: the polynomial DH_c which coincides with DH(M) on C, and the periodic polynomial m_c which computes the multiplicity of RR(M,L^k), k>0, in the cone generated by C. The purpose of this paper is to compute the differences DH_c - DH_c' and m_c - m_c' when C and C' are two adjacent connected components of regular values of the moment map., Latex, 40 pages

Introducing the notion of stabilized fundamental group for the complement of a branch curve in $CP^2$, we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples., Comment: 34 pages, 9 figures

We recently constructed examples of compact Kaeler manifolds which do not have the homotopy type of a projective complex manifold. They were however obtained by blowing-up certain complex tori, which are themselves deformation equivalent to complex projective manifolds. Thus it remained possible that in higher dimension, a birational version of Kodaira's theorem still holds. We construct however in this paper compact Kaehler manifolds, no birational model of which has the homotopy type of a projective manifold.

On a complex symplectic manifold, we construct the stack of quantization-deformation modules, that is, (twisted) modules of microdifferential operators with an extra central parameter, a substitute to the lack of homogeneity. We also quantize involutive submanifolds of contact manifolds., In this new version, we have deleted from the previous one Lemma 7.6, Proposition 7.7 and Proposition 7.8 which were erroneous

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $��^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge theoretical Gromov-Witten type invariants of arbitrary genus associated with certain symplectic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects. The main tool for describing moduli spaces associated with symplectic factorization problems is the "universal Kobayashi-Hitchin correspondence", which gives canonical isomorphisms ${\cal M}^*\to{\cal M}^{\rm st}$ between gauge theoretic moduli spaces of irreducible solutions of certain PDE's and complex geometric moduli spaces of stable framed holomorphic objects. We state a conjecture for the general situation: When the gauge theoretic problem is of Fredholm type, and the data for ${\cal M}^{\rm st}$ are algebraic, then ${\cal M}^{\rm st}$ admits a canonical perfect obstruction theory in the sense of Behrend-Fantechi, and the Kobayashi-Hitchin isomorphism ${\cal M}^*\to{\cal M}^{\rm st}$ identifies the gauge theoretic and the algebraic virtual fundamental classes. The conjecture was checked for the symplectic factorization problems which yield the toric varieties., LaTeX, 33 pages. Comments are welcome! Some references were corrected

We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational $J$-holomorphic curves in a given homology class passing through a given real configuration of points., 31 pages, 8 figures

In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants. This paper is based on a part of my PhD Thesis (see math/9912150)., Comment: 36 pages

We show that in every dimension greater than or equal to 4, there exist compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds. Thus they a fortiori are not deformation equivalent to a projective manifold, which solves negatively Kodaira's problem. We give both non simply connected (of dimension at least 4) and simply connected (of dimension at least 6) such examples., Comment: To appear in Inventiones Math

In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group., Comment: 42 pages, new introduction