The canonical class of a symplectic 4-manifold (Q2719823)

The paper has as a goal to understand the Poincaré dual of the canonical class in a simply connected smooth symplectic 4-manifold. \textit{C. H. Taubes} [J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025)] proved that the Poincaré dual of the canonical class of a simply connected smooth symplectic 4-manifold can be represented by an embedded symplectic surface (which may have different components). The authors of the paper under review prove that the representative of such a Poincaré dual of the canonical class can be a disjoint union of embedded symplectic surfaces with different genus and finite-multiplicity prescribed, i.e., for any given finite sequence of a certain genus and multiplicity, there is a simply connected minimal symplectic 4-manifold whose canonical class is represented by the disjoint union of embedded symplectic surfaces with this genus and multiplicity assigned. The technique to construct such a manifold is somehow standard by employing the Gompf symplectic sum in [\textit{R. E. Gompf}, Ann. Math. (2) 142, No. 3, 527-595 (1995; Zbl 0849.53027)] and the authors' rational blowdown in [J. Differ. Geom. 46, No. 2, 181-235 (1997; Zbl 0896.57022)]. The construction is clearly described in details and steps, and some interesting questions related to the paper are also posed.