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Wall-crossing formulas, the Bott residue formula and the Donaldson invariants of rational surfaces. (English) Zbl 0951.57016

S. K. Donaldson first defined his invariant (which depends on the chamber structure of the positive cone) for manifolds with \(b_+^2\) (the rank of a maximal positive subspace for the intersection form) equal to 1 [J. Differ. Geom. 26, 141-168 (1987; Zbl 0631.57010)]. Then he extended this notion to an infinite family of invariants for simply connected 4-manifolds with \(b_+^2\) odd and \( > 1\) [Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)]. For the case \(b_+^2 =1\), Kotschick and Morgan showed that the Donaldson invariants only depend on the chamber of the period point of the metric used in the definition, and analyzed the change of the degree \(N\) Donaldson invariant when the period point passes through a wall [D. Kotschick and J. W. Morgan, J. Differ. Geom. 39, No. 2, 433-456 (1994; Zbl 0828.57013)]. The paper under review is a continuation of the authors’ previous work [G. Ellingsrud and L. Göttsche, J. Reine Angew. Math. 467, 1-49 (1995; Zbl 0834.14005)]; most of the results were independently obtained by R. Friedman and Z. Qin [Commun. Anal. Geom. 3, No. 1, 11-83 (1995; Zbl 0861.14032)]. It is specialized to rational surfaces (all walls are good in the authors’ sense). Note that a rational surface can be deformed to a surface \(S\) with an action of a two-dimensional algebraic torus \({\Gamma}\) with only a finite number of fixed points. Such an action can be lifted to the Hilbert schemes \(\text{Hilb}^{d_{\xi, N}}(S \bigsqcup S)\) with \(d_{\xi, N} = (N+3+\xi^2)/4\) and \(\xi \in H^2(S, Z)\), and to the standard bundles \(v_{\xi, N}\) as well. The lifted action has a finite number of fixed points on the Hilbert schemes. The same is true for a general 1-parameter subgroup \(T\) of \(\Gamma\). The weights of the \(T\)-action on the tangent space of \(\text{Hilb}^{d_{\xi, N}}(S \bigsqcup S)\) and on the fibers of \(v_{\xi, N}\) at the fixed points are completely determined by the weights of the \(\Gamma\)-action on the algebraic surface \(S\). This is why one can employ the Bott residue formula to this particular situation. Indeed, there is an explicit algorithm to compute the change of the wall-crossing term. Applications of results:
(1) The Donaldson invariants of \(\mathbb{C}\mathbb{P}^2\) are computed in terms of modular forms with the aid of the blow-up formula;
(2) Kotschick and Morgan conjectured that the wall-crossing terms restricted to \(\text{Sym}^N(H_2(S, Q))\) are polynomials of linear form and quadratic form whose coefficients depend on \(N, \xi^2\) and the homotopy type of the surface. The authors prove the conjecture for rational ruled surfaces with \(N \leq 40\) and \(d_{\xi, N} \leq 8\);
(3) the Donaldson invariants can be determined for all generic polarizations of rational ruled surfaces, for polarizations in a big subcone of the ample cone of rational surfaces.

MSC:

57R57 Applications of global analysis to structures on manifolds
14J26 Rational and ruled surfaces
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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