In mathematics, Dolgachev surfaces are certain simply connected

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 fi ...

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 mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a ...

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 man ...

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 Hemming ...

X_0

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 d ...

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 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 viewed ...

is odd of

signature A signature (; from la, signare, "to sign") is a handwritten (and often 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 intent. The writer of a ...

(1,9)

(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 fundamen ...

I_

). 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 complex ...

p_g

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 ''κ''. ...

is 1. found the first examples of 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 , 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 b ...

, volume=26 , issue=1 , pages=141–168, doi=10.4310/jdg/1214441179, doi-access=free Algebraic surfaces Complex surfaces