Symplectic couples on 4-manifolds
Let \(M\) be a smooth connected oriented 4-manifold. A pair of symplectic forms \((\omega_1, \omega_2)\) on \(M\) is called a symplectic couple if \(\omega_1 \wedge \omega_2 \equiv 0\) and \(\omega^2_i\) is a volume form defining the positive orientation for each \(i=1,2\). The study of symplectic couples on smooth 4-manifolds was essentially motivated by the fact that the condition \(\omega_1\wedge \omega_2\equiv 0\) is equivalent to a certain pair of first-order partial differential equations, whose solution surfaces are Lagrangian with respect to both \(\omega_1\) and \(\omega_2\). Furthermore, symplectic couples may be regarded as symplectic generalizations of holomorphic symplectic structures. The main result of the paper is the following: Let \(M\) be as above. Then \(M\) admits a conformal symplectic couple if and only if \(M\) is diffeomorphic to either a complex torus, a K3 surface, or a primary Kodaira surface. The author also proves that if \(M\) admits a symplectic form whose compatible almost complex structure has nonvanishing first Chern class, then it does not admit a symplectic couple. Finally, he gives examples of symplectic couples on open 4-manifolds of type \(N^3\times \mathbb{R}\) having certain invariance properties, and proves that no conformal symplectic couples with the same invariance properties can exist on these manifolds.
scientific article; zbMATH DE number 987005