Another example is a smooth model of the Barth–Nieto quintic. One example of a three-dimensional Calabi–Yau manifold is a non-singular quintic threefold in CP 4, which is the algebraic variety consisting of all of the zeros of a homogeneous quintic polynomial in the homogeneous coordinates of the CP 4.

In turn, it has also been conjectured by Miles Reid that the number of topological types of Calabi–Yau 3-folds is infinite, and that they can all be transformed continuously (through certain mild singularizations such as conifolds) one into another-much as Riemann surfaces can. In three complex dimensions, classification of the possible Calabi–Yau manifolds is an open problem, although Yau suspects that there is a finite number of families (albeit a much bigger number than his estimate from 20 years ago). However, the Enriques surface subset do not conform entirely to the SU(2) subgroup in the String theory landscape. Abelian surfaces are sometimes excluded from the classification of being Calabi–Yau, as their holonomy (again the trivial group) is a proper subgroup of SU(2), instead of being isomorphic to SU(2). Enriques surfaces and hyperelliptic surfaces have first Chern class that vanishes as an element of the real cohomology group, but not as an element of the integral cohomology group, so Yau's theorem about the existence of a Ricci-flat metric still applies to them but they are sometimes not considered to be Calabi–Yau manifolds. These conditions imply that the first integral Chern class c 1 ( M ) equipped with a complex manifold structure. M has a Kähler metric with global holonomy contained in SU( n).The structure group of the tangent bundle of M can be reduced from U( n) to SU( n).M has a holomorphic n-form that vanishes nowhere.This section summarizes some of the more common definitions and the relations between them.Ī Calabi–Yau n-fold or Calabi–Yau manifold of (complex) dimension n is sometimes defined as a compact n-dimensional Kähler manifold M satisfying one of the following equivalent conditions: There are many other definitions of a Calabi–Yau manifold used by different authors, some inequivalent. The motivational definition given by Shing-Tung Yau is of a compact Kähler manifold with a vanishing first Chern class, that is also Ricci flat.

Calabi yau variety