Dolgachev surface

In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by 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 X₀ 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 ≥ 3.

The Dolgachev surfaces are simply connected and the bilinear form on the second cohomology group is odd of signature (1, 9) (so it is the unimodular lattice I_1,9). The geometric genus p_g is 0 and the Kodaira dimension is 1.

Donaldson (1987) found the first examples of homeomorphic but not diffeomorphic 4-manifolds X₀ and X₃. More generally the surfaces X_q and X_r are always homeomorphic, but are not diffeomorphic unless q = r.

Akbulut (2008) showed that the Dolgachev surface X₃ has a handlebody decomposition without 1- and 3-handles.

References

• Akbulut, Selman (2008), The Dolgachev surface, arXiv:0805.1524 
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225 
• Dolgachev, I. (1981), "Algebraic surfaces with p_g = g = 0", Algebraic Surfaces, CIME 1977, Cortona, Liguori Napoli, pp. 97–215 
• Donaldson, S. K. (1987), "Irrationality and the h-cobordism conjecture", Journal of Differential Geometry 26 (1): 141–168, MR 892034