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Register a new userLevi-flat filling of real two-spheres in symplectic manifolds (I)
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Herve Gaussier
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Let (M,J,w) be a manifold with an almost complex structure J tamed by a symplectic form w. We suppose that M has complex dimension two, is Levi convex and has bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of M may be foliated by the boundaries of pseudoholomorphic discs.
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For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauvilles. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandts theory of subnormal subgroups.
A complex manifold $X$ is emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $phi$. The minimal kernels $Sigma_X^k, k in [0,infty]$ (the loci where are all $mathcal{C}^k$ plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic),introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far $X$ is from being Stein. We compare these notions, prove that all Levi currents are supported by all the $Sigma_X^k$s, and give sufficient conditions for points in $Sigma_X^k$ to be in the support of some Levi current. When $X$ is a surface and $phi$ can be chosen analytic, building on previous work by the second author, Slodkowski, and Tomassini,we prove the existence of a Levi current precisely supported on $Sigma_X^infty$, and give a classification of Levi currents on $X$. In particular,unless $X$ is a modification of a Stein space, every point in $X$ is in the support of some Levi current.
We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing diagrams, and amenable to algorithmic classification. Applying the theory to four-dimensional projective space, we obtain a total of 40 irreducible components of the moduli space, most of which are new. Our main technique is a detailed analysis of the relevant deformation complex (the Poisson cohomology) as an object of the constructible derived category.
We investigate the $CR$ geometry of the orbits $M$ of a real form $G_0$ of a complex simple group $G$ in a complex flag manifold $X=G/Q$. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $G_0$-equivariant and Mostow fibrations, and topological properties of the orbits.
We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie group G in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to them some cross-marked diagrams that generalize those for minimal orbits that we introduced in a previous paper. By constructing canonical fibrations over real flag manifolds, with simply connected complex fibers, we are also able to compute their fundamental group.
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