Abstract:
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $\phi:M\to\R^n$, a convex polytope $P=\phi(M)\subset\R^n$. In this paper we show, using symplectic (action-angle) coordinates on $P\times \T^n$, how all $\om$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$. We also discuss some topics suited for application of this symplectic coordinates approach to K\"ahler toric geometry, namely: explicit construction of extremal K\"ahler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

Abstract:
We prove that the rational blowdown, a surgery on smooth 4-manifolds introduced by Fintushel and Stern, can be performed in the symplectic category. As a consequence, interesting families of smooth 4-manifolds, including the exotic $K3$ surfaces of Gompf and Mrowka, admit symplectic structures.

Abstract:
Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear plumbing configuration of spheres $C_n$ is replaced with a rational homology ball $B_n$, $n \geq 2$. Subsequently, Symington [Sy] defined this procedure in the symplectic category, where a symplectic $C_n$ (given by symplectic spheres) is replaced by a symplectic copy of $B_n$ to yield a new symplectic manifold. As a result, a symplectic rational blow-down can be performed on a manifold whenever such a configuration of symplectic spheres can be found. In this paper, we define the inverse procedure, the rational blow-up in the symplectic category, where we present the symplectic structure of $B_n$ as an entirely standard symplectic neighborhood of a certain Lagrangian 2-cell complex. Consequently, a symplectic rational blow-up can be performed on a manifold whenever such a Lagrangian 2-cell complex is found.

Abstract:
In this paper we provide a computational approach to the shape of curves which are rational in polar coordinates, i.e. which are defined by means of a parametrization (r(t),\theta(t)) where both r(t),\theta(t) are rational functions. Our study includes theoretical aspects on the shape of these curves, and algorithmic results which eventually lead to an algorithm for plotting the "interesting parts" of the curve, i.e. the parts showing the main geometrical features of it. On the theoretical side, we prove that these curves, with the exceptions of lines and circles, cannot be algebraic (in cartesian coordinates), we characterize the existence of infinitely many self-intersections, and we connect this with certain phenomena which are not possible in the algebraic world, namely the existence of limit circles, limit points, or spiral branches. On the practical side, we provide an algorithm which has been implemented in the computer algebra system Maple to visualize this kind of curves. Our implementation makes use (and improves some aspects of) the command polarplot currently available in Maple for plotting curves in polar form.

Abstract:
We study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that projective holomorphic symplectic fourfolds of K3^[2]-type contain uniruled divisors and rational Lagrangian surfaces.

Abstract:
Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= \pi r^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_{\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_{\omega}(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_{\omega}(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.

Abstract:
Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface E(1) (complex projective plane blown-up at nine branch points of a generic pencil of cubic curves). We also show how these tori can be non-isotopically and symplectically embedded in many other symplectic 4-manifolds.

Abstract:
A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,\omega)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional complete Kaehler manifolds with non-negative sectional curvature globally symplectomorphic to $R^{2n}$ through a symplectomorphism satisfying Ciriza's property.

Abstract:
We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\lambda}= (S^2 \times S^2, \mu \omega_0 \oplus \omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$. We show that, when $\mu$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < \mu < 2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{crit} = \mu - 1$, which is the difference of areas between the two $S^2$ factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of $c$ remain either below or above that critical value.