Dolgachev surface

The blowup ${\displaystyle X_{0}}$ of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface ${\displaystyle X_{q}}$ is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some ${\displaystyle q\geq 3}$.

The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature ${\displaystyle (1,9)}$ (so it is the unimodular lattice ${\displaystyle I_{1,9}}$). The geometric genus ${\displaystyle p_{g}}$ is 0 and the Kodaira dimension is 1.

Simon Donaldson (1987) found the first examples of homeomorphic but not diffeomorphic 4-manifolds ${\displaystyle X_{0}}$ and ${\displaystyle X_{3}}$. More generally the surfaces ${\displaystyle X_{q}}$ and ${\displaystyle X_{r}}$ are always homeomorphic, but are not diffeomorphic unless ${\displaystyle q=r}$.

Selman Akbulut (2012) showed that the Dolgachev surface ${\displaystyle X_{3}}$ has a handlebody decomposition without 1- and 3-handles.

• Akbulut, Selman (2012). "The Dolgachev surface. Disproving the Harer–Kas–Kirby conjecture". Commentarii Mathematici Helvetici. 87 (1): 187–241. arXiv:0805.1524. Bibcode:2008arXiv0805.1524A. doi:10.4171/CMH/252. MR 2874900.
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004). Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). 4. Springer-Verlag, Berlin. doi:10.1007/978-3-642-96754-2. ISBN 978-3-540-00832-3. MR 2030225.
• Dolgachev, Igor (2010), "Algebraic surfaces with ${\displaystyle p_{g}=g=0}$", Algebraic Surfaces, C.I.M.E. Summer Schools, 76, Heidelberg: Springer, pp. 97–215, doi:10.1007/978-3-642-11087-0_3, MR 2757651
• Donaldson, Simon K. (1987). "Irrationality and the h-cobordism conjecture". Journal of Differential Geometry. 26 (1): 141–168. doi:10.4310/jdg/1214441179. MR 0892034.