Dolgachev surface explained

In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . 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

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} is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some q\ge 3 . ( 1 , 9 ) (so it is the unimodular lattice I _{1,9} ). The geometric genus p _{g} is 0 and the Kodaira dimension is 1.

found the first examples of homeomorphic but not diffeomorphic 4-manifolds

X _{0} and X _{3} . More generally the surfaces X _{q} and X _{r} are always homeomorphic, but are not diffeomorphic unless q = r .

showed that the Dolgachev surface

X _{3} has a handlebody decomposition without 1- and 3-handles.

References

• 0805.1524. The Dolgachev surface. Disproving the Harer–Kas–Kirby conjecture . Selman . Akbulut. Selman Akbulut. 2008arXiv0805.1524A . . 87 . 2012 . 1 . 187–241. 2874900 . 10.4171/CMH/252 .
• Book: Barth . Wolf P. . Wolf Barth. Hulek . Klaus . Klaus Hulek. Peters . Chris A.M. . Van de Ven . Antonius . Compact Complex Surfaces . Springer-Verlag, Berlin . Ergebnisse der Mathematik und ihrer Grenzgebiete (3) . 978-3-540-00832-3 . 2030225 . 2004 . 4. 10.1007/978-3-642-96754-2.
  • Simon Donaldson . Donaldson . Simon K. . Irrationality and the h-cobordism conjecture . 892034 . 1987 . . 26 . 1 . 141–168. 10.4310/jdg/1214441179. free .