In mathematics, a fake projective space is a complex
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
that has the same
Betti numbers as some
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, but is not
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to it.
There are exactly 50
fake projective plane
In mathematics, a fake projective plane (or Mumford surface) 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 ...
s. found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in dimensions other than 2 and 4.
References
*{{citation, title=Fake Projective spaces , url=http://www.mpim-bonn.mpg.de/preprints/send?bid=3351 , first=Gopal, last= Prasad , last2=Yeung, arxiv=math/0602144, publisher=MPIM-Bonn, series= preprint, year=2006, bibcode=2006math......2144P
Algebraic geometry