arXiv subjects
Publications and source records attributed to Josef G. Dorfmeister.
A key question for $4$-manifolds $M$ admitting symplectic structures is to determine which cohomology classes $α\in H^2(M,\mathbb R)$ admit a symplectic representative. The collection of all such classes, the symplectic cone $\mathcal C_M$, is a basic smooth invariant of $M$. This paper describes the symplectic cone for elliptic surfaces with positive Euler number.
This paper gives a survey of the progress on the minimal genus problem since Lawson's survey.
Let $(M,ω)$ be a symplectic 4-manifold of negative Kodaira dimension. Let $C$ be an $ω$-symplectic curve, $J$-holomorphic for some $J$ tamed by $ω$. Then $[C]^2$ is bounded below by a constant depending only on $ω$. Related bounded negativity problems for other structures are also briefly discussed. In particular, the symplectic result implies the bounded negativity conjecture for complex projective surfaces with Kodaira dimension $κ=-\infty$.
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 $κ=-\infty$. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.
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.
In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non-minimal if X contains a certain configuration of exceptional spheres in relation to this -4-sphere.
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.