From Symplectic Packing to Algebraic Geometry and Back

  title={From Symplectic Packing to Algebraic Geometry and Back},
  author={Paul Biran},
In this paper we survey various aspects of the symplectic packing problem and its relations to algebraic geometry, going through results of Gromov, McDuff, Polterovich and the author. 

An explicit construction of a maximal relative symplectic packing of the Clifford torus

In this paper we present an explicit construction of a relative symplectic packing. This confirms the sharpness of the upper bound for the relative packing of a ball into the pair (CP^2, T^2) of the

Symplectic packings in dimension $4$ and singular curves

The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves,

Kähler packings of projective complex manifolds

We show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of Kähler balls and vice versa.

Topology of symplectomorphism groups and ball-swappings

We summarize some recent progress and problems on the symplectomorphism groups, with an emphasis on the connection to the space of ball-packings.

One Explicit Construction of a Relative Packing

In this paper we present an explicit construction of a relative symplectic packing. This confirms the precision of the upper bound for the relative packing of a ball into the pair (CP 2 , T 2 Clif f)

Morse Spectra, Homology Measures and Parametric Packing Problems

We suggest several mathematical counterparts to the idea of "effective degrees of freedom" and formulate specific questions, much of which are inspired by Larry Guth's results and ideas on the

The Space of Symplectic Structures on Closed 4-Manifolds

This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004

K\"ahler packings and Seshadri constants on projective complex surfaces

Quantum Structures for Lagrangian Submanifolds

We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.

Symplectic embedding problems, old and new

We describe old and new motivations to study symplectic embedding problems, and we discuss a few of the many old and the many new results on symplectic embeddings.



Algebraic Geometry

Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)

Connectedness of spaces of symplectic embeddings

We prove that the space of symplectic packings of ${\Bbb C}P^2$ by $k$ equal balls is connected for $3\leq k\leq 6$. The proof is based on Gromov-Witten invariants and on the inflation technique due

Symplectic Packing in Dimension 4

Abstract. We discuss closed symplectic 4-manifolds which admit full symplectic packings by N equal balls for large N's. We give a homological criterion for recognizing such manifolds. As a corollary

A stability property of symplectic packing

Abstract. We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H2(M, Q) there exists a number N0 such that for every N≥N0, (M,Ω) admits full symplectic packing by N equal balls. We also

Partial Differential Relations

1. A Survey of Basic Problems and Results.- 2. Methods to Prove the h-Principle.- 3. Isometric C?-Immersions.- References.- Author Index.

From symplectic deformation to isotopy

Let $X$ be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic

The classification of ruled symplectic $4$-manifolds

Let M be an oriented S2-bundle over a compact Riemann surface Σ. We show that up to diffeomorphism there is at most one symplectic form on M in each cohomology class. Since the possible cohomology

Constructing new ample divisors out of old ones

We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the

Symplectic submanifolds and almost-complex geometry

In this paper we develop a general procedure for constructing symplectic submanifolds. Recall that if (V, ω) is a symplectic manifold, a submanifold W C V is called symplectic if the restriction of ω

Lectures on Gromov invariants for symplectic 4-manifolds

Taubes’s recent spectacular work setting up a correspondence between J-holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg-Witten equations counts J-holomorphic curves in a