The geometry of the Seiberg-Witten invariants (Q1126773)

The argument of this expository paper is the geometry of the Seiberg-Witten invariants of a compact smooth oriented 4-manifold, first introduced by \textit{E. Witten} in [Math. Res. Lett. 1, No. 6, 769-796 (1994; Zbl 0867.57029)]. The aim of the author is to clarify and to specify a geometric interpretation of the above-mentioned invariants as an algebraic count of certain distinguished submanifolds with boundary in the given smooth 4-manifold. This point of view would suggest a new approach to the existence question for symplectic forms on smooth compact 4-manifolds. Some illustrative examples complete the survey paper.