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Calabi-Yau manifolds are a certain class of smooth complex manifolds that have proved to play an important role in several areas of mathematics and physics. Calabi-Yau manifolds have proved to occupy a central place in string theory and are the subject of the important mirror symmetry conjecture. Research here has in turn led to surprising results in the study of algebraic geometry and related fields.
To be Calabi-Yau, a manifold must first have the structure of Kähler manifold -- this means that it is a manifold with some notion of distance provided by a Riemannian manifold (called the Kähler metric), satisfying certain additional conditions. Then a Kähler manifold M is said to be Calabi-Yau if it carries a non-vanishing holomorphic differential form of maximal dimension (in more sophisticated language, one says that "the canonical bundle is trivial", and this is turn is equivalent to the vanishing of the first Chern class). In the case that the manifold in question is compact, this has many several equivalent statements, which may be interpreted directly as conditions imposed on the Kähler metric: M is Calabi-Yau if and only if the Kähler metric has zero Ricci curvature.
All elliptic curves are Calabi-Yau, since when regarded as quotients of the complex plane by a lattice, they carry a non-vanishing holomorphic 1-form dz.
A degree n + 1 hypersurface M in is Calabi-Yau. This is a simple object, nothing more than the vanishing set of a polynomial of degree n + 1 in n dimensional space. The proof is easy, but requires the development of some machinery: one computes the first Chern class c1(TM) by using the normal bundle exact sequence and the adjunction formula, which gives:
by additivity of the total Chern class this yields , where ω is the hyperplane class, a generator of the cohomology ring . But this implies that
from which one easily concludes that c1(TM) = c1(KM) = 0.
There are many other ways to construct Calabi-Yau manifolds. In any example similar to that above, one may start with a hypersurface that is not smooth (it might intersect itself, or have something like a corner: these are called singularities). By smoothing out these manifolds (more technically, taking a resolution of singularities), one then obtains a smooth Calabi-Yau. A vast array of other examples, useful for computations, arisse as so-called toric varieties.
The first Chern class of a Kähler manifold is homologous to the Ricci curvature. For this reason, Calabi conjectured that when c1(TM) = 0, there exists a metric on the manifold whose Ricci form vanishes identically. This was the Calabi Conjecture, eventually proven by Shing-Tung Yau.