arXiv subjects

Jean-Yves Welschinger

Publications and source records attributed to Jean-Yves Welschinger.

33 records · Page 2Linked to original sources

arXiv2010

Exponential rarefaction of real curves with many components

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L^d inherits for every positive integer d a L^2 scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.

math.AG↗

arXiv2010

Invariants entiers en géométrie énumérative réelle

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.

math.AG↗

arXiv2010

Do uniruled six-manifolds contain Sol Lagrangian submanifolds?

We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Kollár. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.

math.SG↗

arXiv2009

Associahedral categories, particles and Morse functor

Every smooth manifold contains particles which propagate. These form objects and morphisms of a category equipped with a functor to the category of Abelian groups, turning this into a 0+1 topological field theory. We investigate the algebraic structure of this category, intimately related to the structure of Stasheff's polytops, introducing the notion of associahedral categories. An associahedral category is preadditive and close to being strict monoidal. Finally, we interpret Morse-Witten theory as a contravariant functor, the Morse functor, to the homotopy category of bounded chain complexes of particles.

math.SG↗

arXiv2008

Open strings, Lagrangian conductors and Floer functor

We introduce a contravariant functor, called Floer functor, from the category of Lagrangian conductors of a symplectic manifold to the homotopy category of bounded chain complexes of open strings in this manifold. The latter two categories are defined for all symplectic manifolds, whereas Floer functor is defined for semipositive manifolds which are either closed or convex at infinity. We then prove that when the first Chern class of the symplectic manifold vanishes, Lagrangian spheres define Lagrangian conductors so that in particular their integral Floer cohomology is well defined. This requires the introduction of singular almost-complex structures given by symplectic field theory.

math.SG↗

arXiv2007

Enumerative invariants of stongly semipositive real symplectic six-manifolds

Following the approach of Gromov and Witten, we define invariants under deformation of stongly semipositive real symplectic six-manifolds. These invariants provide lower bounds in real enumerative geometry, namely for the number of real rational $J$-holomorphic curves which realize a given homology class and pass through a given real configuration of points.

math.AG↗

arXiv2007

Optimalité, congruences et calculs d'invariants des variétés symplectiques réelles de dimension quatre

This paper follows a previous one in which were introduced deformation invariants $χ^d_r$, $d \in H_2 (X ; \Z)$, $r \in \N$, of closed real symplectic four-manifolds $(X, ω, c_X)$, invariants which produced lower bounds in real enumerative geometry. We prove here using methods of symplectic field theory that the lower bounds are sharp when $r \leq 1$ and the real locus of the manifold contains a sphere, torus or real projective plane (under stronger assumptions in this last case). We also prove that a big power of two divides $χ^d_r$ as soon as r is not too big and when the real locus contains a sphere or real projective plane (under the same stronger assumptions in this last case). We finally present some explicit computations in the case of the projective plane or quadric ellipsoid surface as well as the general formulas used to get them, formulas which involve some relative invariants that we first define.

math.SG↗

arXiv2007

Effective classes and Lagrangian tori in symplectic four-manifolds

An effective class in a closed symplectic four-manifold $(X, ω)$ is a two-dimensional homology class which is realized by a $J$-holomorphic cycle for every tamed almost complex structure $J$. We prove that effective classes are orthogonal to Lagrangian tori in $H_2 (X ; \Bbb{Z})$.

math.SG↗

arXiv2005

Invariants of real symplectic 4-manifolds out of reducible and cuspidal curves

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.

math.SG↗

arXiv2005

Towards relative invariants of real symplectic 4-manifolds

Let $(X, ω, c_X)$ 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 $(X, ω, c_X, L)$ 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.

math.SG↗

arXiv2003

Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

Let $X$ be a real algebraic convex 3-manifold whose real part is equipped with a $Pin^-$ structure. We show that every irreducible real rational curve with non-empty real part has a canonical spinor state belonging to $\{\pm 1\}$. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real configuration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independantly of the choice of the real configuration of points.

math.AG↗

arXiv2003

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

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.

math.AG↗

arXiv2002

Deformation classes of real ruled manifolds

A complete description of the deformation classes of real ruled manifolds is given. In particular, we prove that once the complex deformation class is fixed, the real deformation class is prescribed by the topology of the real structure.

math.AG↗

arXiv2002

Real structures on minimal ruled surfaces

In this paper, we give a complete description of the deformation classes of real structures on minimal ruled surfaces. In particular, we show that these classes are determined by the topology of the real structure, which means that real minimal ruled surfaces are quasi-simple. As an intermediate result, we obtain the classification, up to conjugation, of real structures on decomposable ruled surfaces.

math.AG↗

arXiv1999

Orientations complexes des J-courbes reelles

The aim of this article is to obtain restrictions on complex orientations of dividing real J-curves of almost-complex manifolds. This problem comes from real algebraic geometry and the main result is a congruence generalising Arnol'd, Rokhlin, Mishachev, Zvonilov and Mikhalkin 's previous results. This congruence is obtained with a canonical map defined on homology groups of dimension half the dimension of the manifold with coefficients in $Z/lZ$, with values in $Z/2lZ$.

math.AG↗