Instanton invariants of \(\mathbb{C} P^ 2\) via topology
The paper contains a calculation of Donaldson invariants of \(\mathbb{C} P^2\) and a proof of the fact that \(\mathbb{C} P^2\) is not of simple type in the sense formulated by P. Kronheimer and T. Mrowka in the context of Donaldson invariants. The authors study the version of Donaldson invariants for 4-manifolds with \(b^+_2 = 1\) introduced in [\textit{S. K. Donaldson}, J. Differ. Geom. 26, 141-168 (1987; Zbl 0631.57010); the first author, Proc. Lond. Math. Soc., III. Ser. 63, 426-448 (1991; Zbl 0699.53036); the first author and \textit{J. W. Morgan}, J. Differ. Geom. 39, 433-456 (1994; Zbl 0828.57013)]. These invariants depend on a choice of chamber in the positive cone in the real 2-homology. The change of the invariant as one passes from a chamber to another one through a wall is described by a certain difference term. The authors prove that all the instanton invariants of \(\mathbb{C} P^2\) and \(S^2 \times S^2\) are uniquely determined by the difference terms of SO(3)-invariants of \(\mathbb{C} P^2 \# k \mathbb{C} \overline {P}^2\) (where \(k \geq 2\)).
scientific article; zbMATH DE number 840189
scientific article