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
[edit]The blowup of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some .
The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature (so it is the unimodular lattice ). The geometric genus is 0 and the Kodaira dimension is 1.
Simon Donaldson (1987) found the first examples of homeomorphic but not diffeomorphic 4-manifolds and . More generally the surfaces and are always homeomorphic, but are not diffeomorphic unless .
Selman Akbulut (2012) showed that the Dolgachev surface has a handlebody decomposition without 1- and 3-handles.
References
[edit]- 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 ", 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.