Dolgachev surface


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

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 . The geometric genus is 0 and the Kodaira dimension is 1.
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.
showed that the Dolgachev surface has a handlebody decomposition without 1- and 3-handles.