##
**Symplectic genus, minimal genus and diffeomorphisms.**
*(English)*
Zbl 1008.57024

Let \(M\) be a smooth, closed oriented \(4\)-manifold. The minimal genus of a \(2\)-homology class \(e\) is the minimal genus of a smoothly embedded connected surface representing \(e\).

The symplectic cone of \(M\) is the set of cohomology classes represented by symplectic forms for \(M\) compatible with the orientation. If the symplectic cone is non-empty, we define the symplectic genus of \(e\) as the maximum of the values \((e^{2} +K \cdot e)/2+1\), for \(K\) the canonical class of a symplectic structure \(\omega\) with \([\omega]\cdot e>0\). This number is invariant under \(\text{Diff}(M)\).

The symplectic genus is a lower bound for the minimal genus, and, if \(e\) is represented by a connected symplectic surface, then both are equal.

In this paper, the relation between the symplectic and the minimal genus is studied for \(4\)-manifolds with \(b^{+}=1\). Special attention is paid to the case where \(M\) is a blow-up of a rational or irrational ruled surface. The arguments are based on the work of C. H. Taubes on Seiberg-Witten theory [Math. Res. Lett. 2, No. 2, 221-238 (1995; Zbl 0854.57020)].

Finally, the authors also study the orbits of \(\text{Diff}(M)\) on the set of classes represented by embedded spheres of square \(s\geq -1\) in any \(4\)-manifold admitting a symplectic structure.

The symplectic cone of \(M\) is the set of cohomology classes represented by symplectic forms for \(M\) compatible with the orientation. If the symplectic cone is non-empty, we define the symplectic genus of \(e\) as the maximum of the values \((e^{2} +K \cdot e)/2+1\), for \(K\) the canonical class of a symplectic structure \(\omega\) with \([\omega]\cdot e>0\). This number is invariant under \(\text{Diff}(M)\).

The symplectic genus is a lower bound for the minimal genus, and, if \(e\) is represented by a connected symplectic surface, then both are equal.

In this paper, the relation between the symplectic and the minimal genus is studied for \(4\)-manifolds with \(b^{+}=1\). Special attention is paid to the case where \(M\) is a blow-up of a rational or irrational ruled surface. The arguments are based on the work of C. H. Taubes on Seiberg-Witten theory [Math. Res. Lett. 2, No. 2, 221-238 (1995; Zbl 0854.57020)].

Finally, the authors also study the orbits of \(\text{Diff}(M)\) on the set of classes represented by embedded spheres of square \(s\geq -1\) in any \(4\)-manifold admitting a symplectic structure.

Reviewer: Vicente Muñoz (Madrid)

### MSC:

57R57 | Applications of global analysis to structures on manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |