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 is given by applying logarithmic transformations of orders 2 and q.

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 is1.

.

has a handlebody decomposition without 1- and 3-handles.

Detection of exotic smooth structures

According to Freedman's classification, a simply connected oriented closed topological 4-manifold with intersection form. Any such smooth manifold with a non-vanishing Seiberg–Witten invariant with the induced smooth structure allows a Riemannian metric of positive scalar curvature (psc metric) and therefore has only vanishing Seiberg–Witten invariants.

References