In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by Igor Dolgachev (1981). They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic.
Properties
The blowup X 0 {\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 X q {\displaystyle X_{q}} is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some q ≥ 3 {\displaystyle q\geq 3} .
The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature ( 1 , 9 ) {\displaystyle (1,9)} (so it is the unimodular lattice I 1 , 9 {\displaystyle I_{1,9}} ). The geometric genus p g {\displaystyle p_{g}} is 0 and the Kodaira dimension is 1.
Simon Donaldson (1987) found the first examples of simply-connected homeomorphic but not diffeomorphic 4-manifolds X 0 {\displaystyle X_{0}} and X 3 {\displaystyle X_{3}} . More generally the surfaces X q {\displaystyle X_{q}} and X r {\displaystyle X_{r}} are always homeomorphic, but are not diffeomorphic unless q = r {\displaystyle q=r} .
Selman Akbulut (2012) showed that the Dolgachev surface X 3 {\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). Vol. 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 p g = g = 0 {\displaystyle p_{g}=g=0} ", Algebraic Surfaces, C.I.M.E. Summer Schools, vol. 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.