Homotopy \(K3\) surfaces containing \(\Sigma{} (2,3,7)\) (Q1176355)

Let \(\Sigma(2,3,7)\) be the Brieskorn homology 3-sphere (that is the link of the complex algebraic singularity \(\{(x,y,z)\in\mathbb{C}^ 3:x^ 2+y^ 3+z^ 7=0\}\cap S^ 5)\). Main result: Any homotopy \(K3\) surface which admits an embedding of \(\Sigma(2,3,7)\) has a nontrivial Donaldson invariant polynomial of degree ten. The result has other interesting consequences. The paper depends heavily on the Donaldson theory.