Dolgachev surface

In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by Igor 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 ${\displaystyle X_{0}}$ of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface ${\displaystyle X_{q}}$ is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some ${\displaystyle q\geq 3}$.

The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature ${\displaystyle (1,9)}$ (so it is the unimodular lattice ${\displaystyle I_{1,9}}$). The geometric genus ${\displaystyle p_{g}}$ is 0 and the Kodaira dimension is 1.

Simon Donaldson (1987) found the first examples of simply-connected homeomorphic but not diffeomorphic 4-manifolds ${\displaystyle X_{0}}$ and ${\displaystyle X_{3}}$. More generally the surfaces ${\displaystyle X_{q}}$ and ${\displaystyle X_{r}}$ are always homeomorphic, but are not diffeomorphic unless ${\displaystyle q=r}$.

Selman Akbulut (2012) showed that the Dolgachev surface ${\displaystyle X_{3}}$ 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 ${\displaystyle [+1]\oplus 9[-1]}$ is homeomorphic to ${\displaystyle \mathbb {C} P^{2}\#9{\overline {\mathbb {C} P^{2}}}}$. Any such smooth manifold with a non-vanishing Seiberg–Witten invariant has an exotic smooth structure since ${\displaystyle \mathbb {C} P^{2}\#9{\overline {\mathbb {C} P^{2}}}}$ with the induced smooth structure allows a Riemannian metric of positive scalar curvature (psc metric) and therefore has only vanishing Seiberg–Witten invariants.

References

• Akbulut, Selman (2012). "The Dolgachev surface. Disproving the Harer–Kas–Kirby conjecture". Commentarii Mathematici Helvetici. 87 (1): 187–241. arXiv:0805.1524. Bibcode:2008arXiv0805.1524A. doi:10.4171/CMH/252. MR 2874900.
• Akbulut, Selman (2016). "Chapter 7.2 Dolgachev surface". 4-Manifolds. Oxford Graduate Texts in Mathematics. Vol. 25. Oxford University Press. ISBN 9780198784869.
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004). Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 4. Springer-Verlag, Berlin. doi:10.1007/978-3-642-96754-2. ISBN 978-3-540-00832-3. MR 2030225.
• Dolgachev, Igor (2010), "Algebraic surfaces with ${\displaystyle p_{g}=g=0}$", Algebraic Surfaces, C.I.M.E. Summer Schools, vol. 76, Heidelberg: Springer, pp. 97–215, doi:10.1007/978-3-642-11087-0_3, MR 2757651
• Donaldson, Simon K. (1987). "Irrationality and the h-cobordism conjecture". Journal of Differential Geometry. 26 (1): 141–168. doi:10.4310/jdg/1214441179. MR 0892034.