On the Donaldson polynomials of elliptic surfaces. (English) Zbl 0807.57016

The Donaldson invariants of smooth simply connected minimal elliptic surfaces over \(\mathbb{C} \mathbb{P}^ 1\) are polynomials in the intersection form \(q\) and canonical class \(k\). We compute, using Donaldson-Floer theory, the second nonzero coefficient in their expansion in \(q\) and \(k\) when the geometric genus is odd. We point out how this result (recently obtained also by Morgan and Mrowka, using different methods, for any geometric genus) yields the classification of smooth simply connected elliptic surfaces over \(\mathbb{C} \mathbb{P}^ 1\) up to diffeomorphism.
Reviewer: P.Lisca (Roma)


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
14J15 Moduli, classification: analytic theory; relations with modular forms
81T13 Yang-Mills and other gauge theories in quantum field theory
57R55 Differentiable structures in differential topology
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