Publications

2002

• Braids and symplectic four-manifolds with abelian fundamental group
  
  • Seidel Paul
  Turkish Journal of Mathematics, Scientific and Technical Research Council of Turkey, 2002, 26, pp.93-100. We explain how a version of Floer homology can be used as an invariant of symplectic manifolds with $b_1>0$. As a concrete example, we look at four-manifolds produced from braids by a surgery construction. The outcome shows that the invariant is nontrivial; however, it is an open question whether it is stronger than the known ones.
• Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)
  
  • Douai Antoine
  • Sabbah Claude
  , 2004, 36, pp.1-18. We give an explicit description of the canonical Frobenius structure attached (by the results of the first part of this article) to the polynomial f(u_0,...,u_n)=w_0u_0+...+w_nu_n restricted to the torus u_0^{w_0}...u_n^{w_n}=1, for any family of positive integers w_0,...,w_n such that gcd(w_0,...,w_n)=1.
• Vanishing cycles and Hermitian duality
  
  • Sabbah Claude
  Proceedings of the Steklov Institute of Mathematics, MAIK Nauka/Interperiodica, 2002, 238, pp.204-223. We show the compatibility between the moderate or nearby cycle functor for regular holonomic $\mathcal{D}$-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara.
• Fukaya categories and deformations
  
  • Seidel Paul
  , 2002, pp.351-360. This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.
• Kahler surfaces of finite volume and Seiberg-Witten equations
  
  • Rollin Yann
  Bulletin de la société mathématique de France, Société Mathématique de France, 2002, 130, pp.409-456. Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics.
• On the distribution of points of bounded height on equivariant compactifications of vector groups
  
  • Chambert-Loir Antoine
  • Tschinkel Yuri
  Inventiones Mathematicae, Springer Verlag, 2002, 148 numéro 2, pp.421-452. We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous établissons un développement asymptotique du nombre de points rationnels de hauteur bornée sur les compactifications équivariantes lisses de l'espace affine.)
• Champs de Hurwitz
  
  • Chambert-Loir Antoine
  , 2002. On construit les champs de Hurwitz et on en donne quelques propriétés, essentiellement contenues dans SGA 1. Quelques applications de nature arithmétique en sont déduites. We propose a construction of Hurwitz stacks and give some properties of them, most of which are consequences of SGA 1. Some applications of arithmetic flavor are then deduced.
• Théorèmes d'algébrisation en géométrie diophantienne (Algebricity theorems in diophantine geometry)
  
  • Chambert-Loir Antoine
  , 2002. In a recent paper, J.-B. Bost establishes a criterion for certain ``formal subvarieties'' of algebraic varieties to be algebraic. His theorem unifies and generalizes results of Chudnovsky's and Y. André, motivated by an arithmetic conjecture of Grothendieck that predicts that the solutions of certain differential equations are algebraic functions. The proof makes use of the diophantine approximation techniques introduced by these authors but with a systematic geometric point of view, notably via Arakelov geometry and the formalism of ``slopes''.
• The question of interior blow-up for an elliptic Neumann problem: the critical case
  
  • Rey Olivier
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2002, 81, pp.655-696. In contrast with the subcritical case, we prove that for any bounded domain $\Omega$ in $\mathbb{R}^3$, the Neumann elliptic problem with critical nonlinearity $-\Delta u + \mu u = u^{5}$, $u > 0$ in $\Omega$, $\partial u / \partial n= 0$ on $\partial\Omega$ has no solution blowing up at only interior points as μ goes to infinity.