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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 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.

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.

found the first examples of simply-connected 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

  • {{cite journal|arxiv=0805.1524|title=The Dolgachev surface. Disproving the Harer–Kas–Kirby conjecture | first=Selman | last=Akbulut| authorlink=Selman Akbulut| bibcode=2008arXiv0805.1524A | journal=Commentarii Mathematici Helvetici
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