A guiding principle in Kahler geometry is that the infinite dimensional sym- metric space H of Kahler metrics in a fixed Kahler class on a polarized projective Kahler manifold M should be well approximated by finite dimensional submanifolds Bk ⊂ H of Bergman metrics of height k (Yau, Tian, Donaldson). The Bergman metric spaces are sym- metric spaces of type GC/G where G = U(dk +1) for certain dk. This article establishes the basic estimates for Bergman approximations for geometric families of toric Kahler manifolds. The approximation results are applied to the endpoint problem for geodesics of H, which are solutions of a homogeneous complex Monge-Ampere equation in A × X, where A ⊂ C is an annulus. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether H- geodesics with fixed endpoints can be approximated by geodesics of Bk. Phong-Sturm proved weak C 0 -convergence of Bergman to Monge-Ampere geodesics on a general Kahler manifold. Our approximation results show that one has C 2 (A × X) convergence in the case of toric Kahler metrics, extending our earlier result on CP 1 . In subsequent papers, the techniques of this article are applied to approximations for harmonic maps into H, to test configuration geodesic rays and to the smooth initial value problem.

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Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $$S^{2}$$
 or $$\mathbb{RP }^{2}$$
 , in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.

Spherical Lagrangians via ball packings and symplectic cutting

We find a non-displaceable Lagrangian torus fiber in a semi-toric system, which is superheavy with respect to certain symplectic quasi-state. In particular, this proves Lagrangian $\RR P^2$ is not a stem in $\CC P^2$, answering a question of Entov and Polterovich.

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On an exotic Lagrangian torus in $\mathbb{C}P^2$

We find a non-displaceable Lagrangian torus fiber in a semi-toric system which is superheavy with respect to a certain symplectic quasi-state. The proof employs both 4-dimensional techniques and those from symplectic field theory. In particular, our result implies Lagrangian $\mathbb{R}P^{2}$
 is not a stem in $\mathbb{C}P^{2}$
 , answering a question of Entov and Polterovich.

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On an exotic Lagrangian torus in

In this paper we introduce the notion of relative Kodaira dimension for a symplectic 4-manifold M with a possibly disconnected embedded symplectic surface F. One notable feature is that the sphere components of F have to be discarded, which resembles the definition of Thurston norm of 3-manifolds. 
To define it we need to establish several homological properties of embedded symplectic surfaces. We also formulate the notion of relative minimal model, and prove the existence and the somewhat surprising uniqueness.

Additivity and Relative Kodaira Dimensions

In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.

The Relative Symplectic Cone and T^2-Fibrations

In this note, we verify that the complex Kodaira dimension $\kappa^h$ equals the symplectic Kodaira dimension $\kappa^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations.

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The Kodaira Dimension of Lefschetz Fibrations

In this note we discuss the effect of the symplectic sum along spheres in symplectic four-manifolds on the Kodaira dimension of the underlying symplectic manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we are able to obtain precise results on the structure of the manifold obtained from the blow down of an embedded symplectic -4-sphere.

Kodaira Dimension of Fiber Sums along Spheres

In this note we discuss the effect of the symplectic sum along spheres in symplectic four-manifolds on the Kodaira dimension of the underlying symplectic manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we are able to obtain precise results on the structure of the manifold obtained from the blow down of an embedded symplectic $$-4$$
 -sphere.

Kodaira dimension of fiber sums along spheres

We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic $4-$manifolds with $\kappa=-\infty$. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.

Josef G. Dorfmeister

Papers

The Relative Symplectic Cone and T^2-Fibrations

The Kodaira Dimension of Lefschetz Fibrations

Kodaira Dimension of Fiber Sums along Spheres

Kodaira dimension of fiber sums along spheres

Stability and Existence of Surfaces in Symplectic 4-Manifolds with $b^+=1$