In mathematics, Dolgachev surfaces are certain
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
elliptic surface In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed ...
s, introduced by . They can be used to give examples of an infinite family of
homeomorphic simply connected
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. ...
s, no two of which are
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
.
Properties
The
blowup
''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemmi ...
of the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
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
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
is odd of
signature
A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
(so it is the
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundam ...
). The
geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for comp ...
is 0 and the
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''Î ...
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.
References
*
*
*
*{{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
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in boo ...
, volume=26 , issue=1 , pages=141–168, doi=10.4310/jdg/1214441179, doi-access=free
Algebraic surfaces
Complex surfaces