Fake projective space
Encyclopedia
In mathematics, a fake projective space is a complex algebraic variety
that has the same Betti number
s as some projective space
, but is not isomorphic to it.
There are exactly 50 fake projective plane
s. found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in dimensions other than 2 and 4.
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
that has the same Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s as some projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
, but is not isomorphic to it.
There are exactly 50 fake projective plane
Fake projective plane
In mathematics, a fake projective plane is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type....
s. found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in dimensions other than 2 and 4.