The topological types of some irregular Kähler surfaces (Q5928172)

The paper studies the topological type of irregular Kähler surfaces using a fibration structure admitting a simply connected piece with odd intersection form and a trivial fibre space over a base with one hole. The author uses this to give explicit examples of pairwise non-diffeomorphic irregular Kähler surfaces that are homeomorphic. He shows: Any set of mutually homeomorphic but non-diffeomorphic Kähler surfaces of positive irregularity \(q \geq 1\) is finite, but there is no upper bound for the cardinality of such sets even for surfaces of fixed \(q.\) He also shows that there is a pair \(X,X'\) of homeomorphic irregular Kähler surfaces with Kodaira dimensions \(\kappa_X,\kappa_{X'}\) and irregularity \(q\) iff \(\kappa_X = \kappa_{X'}\) or \(\kappa_X,\kappa_X' > 0.\) A key role is played by a fibre surgery construction which leads to connected sum decomposition due to \textit{Akbulut} and \textit{R. Mandelbaum} [Contemp. Math. 44, 291-310 (1985; Zbl 0577.14006)]. The technique used for the topological classification relies on results of \textit{S. Boyer} [Trans. Am. Math. Soc. 298, 331-357 (1986; Zbl 0615.57008)] on simply connected 4-manifolds with boundary. The diffeomorphism classification of the examples uses results of \textit{R. Friedman} and \textit{J. W. Morgan} [J. Algebr. Geom. 6, No. 3, 445-479 (1997; Zbl 0896.14015)] that the number of exceptional fibers is a diffeomorphism invariant. Results from \textit{R. Friedman} and \textit{J. W. Morgan} [Smooth four-manifolds and complex surfaces, Ergeb. Math. Grenzgeb. 3. Folge, Band 27 (1994; Zbl 0817.14017)] also play a key role in both theorems.