22 research outputs found

    Dual elliptic structures on CP2

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    We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that the space of E-lines is again a CP2 with a tame elliptic structure E^*, and that each E-curve has an associated dual E^*-curve. This implies that the E-curves, and in particular the J-curves, satisfy the Pl\"ucker formulas, which restricts their possible sets of singularities.Comment: 18 pages The only difference with the first version is the mention of the thesis of Benjamin MacKay ("Duality and integrable systems of pseudoholomorphic curves", Duke University, 1999), which I did not know at the time, and which contains a large part of the results of my pape

    Bounds on primitives of differential forms and cofilling inequalities

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    We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area.Comment: The new features of the main result are its sharpness and the fact that the manifold is not assumed have bounded geometry, nor even to be complete. This paper corresponds to a part of a talk given in January 2004 in Haifa, at a workshop in memory of Robert Brooks. The other part, which is the "translation" in the framework of geometric group theory, will soon be deposited on arxi

    Growth of a primitive of a differential form

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    Points fixes de difféomorphismes symplectiques, intersections de sous-variétés lagrangiennes, et singularités de un-formes fermées

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    This thesis consists of four distinct parts.In the first part, we give a proof of the Arnold conjecture for surfaces and a generalisation to some other symplectic manifolds, using the variational approach introduced by Conley and Zehnder for tori.In the second part, we show that the zero-section M ⊂ T*M, the cotangent bundle, can never be disconnected from itself by a Hamiltonian isotopy, and give a lower bound for the number of intersection points; the proof, which is elementary, uses generating functions for Lagrangian submanifolds. As an application, we give a much simpler proof of the Arnold conjecture.In the third part, we extend the problem studied in the second part to symplectic isotopies, using results on the Novikov homology associated to a cohomology class of degree one; in particular, we show that if disconnection is possible with dim M >= 6 et π1M = Z, then M carries a nonsingular closed one-form, and thus fibers over S1.In the fourth part, we study some properties of the Novikov homology, and prove the same result as in the third part for irreducible manifolds of dimension three.Cette thèse est formée de quatre parties distinctes.Dans la première partie, on donne une preuve de la conjecture d'Arnold pour les surfaces et une généralisation à certaines autres variétés symplectiques, en utilisant la méthode variationnelle introduite par Conley et Zehnder.Dans la seconde partie, on montre que la section nulle M ⊂ T*M du fibré cotangent ne peut jamais être disjointe d'elle-même par une isotopie hamiltonienne, et l'on minore le nombre de points d'intersection ; la preuve, qui est élémentaire, utilise les fonctions génératrices d'immersions lagrangiennes. Comme corollaire, on obtient une preuve beaucoup plus simple de la conjecture d'Arnold.Dans la troisième partie, on étend aux isotopies symplectiques le problème étudié dans la seconde partie, en utilisant l'homologie de Novikov associée à une classe de cohomologie de degré un ; en particulier, on montre que si la disjonction est possible avec dim M >= 6 et π1M = Z, alors M admet une un-forme fermée non singulière, et donc fibre sur S1.Dans la quatrième partie, on étudie quelques propriétés de l'homologie de Novikov, et l'on prouve le même résultat que dans la troisième partie si M est de dimension trois et irréductible


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    We prove that for "most" closed 3-dimensional manifolds N , the existence of a closed non singular one-form in a given cohomology class u ∈ H 1 (M, R) is equivalent to the non-vanishing modulo p of all twisted Alexander polynomials associated to finite Galois coverings of N. When u ∈ H 1(M,Z), a stronger version of this had been proved by S. Friedl and S. Vidussi in 2013, asking only the non-vanishing of the Alexander polynomials