Manchester Geometry Seminar 2004/2005

The Seiberg-Witten equation gives some new invariants of smooth structures on 4-dimensional manifolds. As it was established by Witten himself, every algebraic surface has a non-trivial Seiberg-Witten invariant. Taubes generalized this fact and established that every symplectic manifold with b₂⁺ > 1 has a non-trivial invariant. So it is a natural problem to find some criterion for the non-triviality of the invariants in terms of almost complex (hermitian) geometry in dimension 4. In this talk we present some remarks on the necessary part of the criterion which is related to some canonical map from the space of hermitian triples to the real second cohomology of an underlying almost complex 4-dimensional manifold.