Dolgachev surface

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 $q\backslash ge\; 3$.

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.

{{harvs|txt|first=Simon|last=Donaldson|authorlink=Simon Donaldson|year=1987}} 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 $q=r$.

{{harvs|txt|first=Selman|last=Akbulut|authorlink=Selman Akbulut|year=2012}} showed that the Dolgachev surface $X\_3$ has a handlebody decomposition without 1- and 3-handles.

• {{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

| volume= 87 | year=2012 | issue=1 | pages= 187–241| mr=2874900 | doi = 10.4171/CMH/252 }}

• {{cite book | last1=Barth | first1=Wolf P. | author1-link=Wolf Barth| last2=Hulek | first2=Klaus | author2-link=Klaus Hulek| last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) | isbn=978-3-540-00832-3 | mr=2030225 | year=2004 | volume=4|doi=10.1007/978-3-642-96754-2}}
• {{citation|authorlink=Igor Dolgachev|first=Igor|last= Dolgachev|chapter= Algebraic surfaces with $p\_g=\; g\; =\; 0$|title= Algebraic Surfaces|series= C.I.M.E. Summer Schools|volume=76| publisher=Springer|location= Heidelberg|year= 2010|pages= 97–215|doi=10.1007/978-3-642-11087-0_3|mr=2757651}}
• {{cite journal | authorlink = Simon Donaldson | last=Donaldson | first=Simon K. | title=Irrationality and the h-cobordism conjecture | url=http://projecteuclid.org/euclid.jdg/1214441179 | mr=892034 | year=1987 | journal=Journal of Differential Geometry | volume=26 | issue=1 | pages=141–168|doi=10.4310/jdg/1214441179| doi-access=free }}

Category:Algebraic surfaces

Category:Complex surfaces