Gluing construction of compact complex surfaces with trivial canonical bundle
Let \(X\) (resp., \(X'\)) be a compact complex surface with a smooth irreducible anticanonical divisor \(D\) (resp., \(D'\)). Suppose there exists an isomorphism \(f:D \to D'\) such that the corresponding normal bundles \(N_{D/X}\) and \(N_{D'/X'}\) are dual to each other via \(f\). Then the author shows the existence of a pair of tubular neighborhoods \((W_1,W_2)\) for \(D\) in \(X\) (resp., \((W'_1,W'_2)\) for \(D'\) in \(X'\)) with \(\overline W_1 \subset W_2\), \(\overline W'_1 \subset W'_2\) and a diffeomorphism \(h:W_2 \setminus \overline W_1 \to W'_2 \setminus \overline W'_1\) such that the gluing of \( X \setminus \overline W_1 \) with \( X' \setminus \overline W'_1 \) using \(h\) yields a complex surface with trivial canonical bundle. Applications are given to new constructions of complex tori, Kodaira surfaces and K3 surfaces, as well as the construction of smoothings of some normal crossing surfaces.
scientific article; zbMATH DE number 5603964