Complex varieties with a vanishing canonical class are called Calabi-Yau varieties. In case of smooth varieties (manifolds), the vanishing of the canonical class is equivalent to the vanishing of the 1st Chern class, which is the cohomology class of the Ricci 2-form. That is, a Calabi-Yau manifold admits (but for a special type of exception) a Kähler metric for which the Ricci 2-form (the Ricci tensor contracted with two coordinate differentials) is a total differential. E. Calabi [kah-Lah-bee] has conjectured, and S.-T. Yau [Yah-oo] proved the structural theorem of these spaces, hence their name.
In complex 1 dimension (real 2 dimensions) the only topological class of such compact spaces are the tori - the surfaces of doughnuts/bagels. Note that these size and shape (thickness relative to overall size) of these spaces may be changed continuously. It turns out that discontinuous contortions, such as pinching and splitting whereby the doughnut is turned into a shape of a bent sausage, would change the 1st Chern class and with it the Ricci-flatness (the admittance of a Kähler metric whose Ricci 2-form is a total derivative). A careful study shows that the continuous changes of the shape a torus may be parametrized by a single complex variable, making the (complex structure) moduli space for the torus (complex) 1-dimensional.
In complex 2 dimensions, there is again a single topological class of compact Calabi-Yau spaces, called the K3 surface. This time, however, the different shapes are parametrized by 20 complex parameters. The (complex structure0 moduli space for the K3 surface thus is (complex) 20-dimensional.
In complex 3 dimensions, many thousands of topologically distinct Calabi-Yau compact varieties have been constructed, the shape of each being parametrized by a varying number of parameters. The Calabi-Yau 3-folds' (complex structure) moduli spaces then come in varying dimensionalities, from 0 (point) to many hundreds. Certain cohomology rings turn out to describe the local geometry of these moduli spaces
The ground states of certain, so-called gauged linear sigma models turn out to be functions over a restricted part of the field space. In part, this restricted field space is a Calabi-Yau variety. The other components in this restricted field space may be reached through a form of analytical continuation; Landau-Ginzburg orbifolds are, in a very certain sense, the opposite component.
Certain nontrivial transformations of the underlying gauged linear sigma model turn a Calabi-Yau variety (perhaps with additional structure) into a completely different Calabi-Yau variety (with a very different additional structure). These are examples of the mirror and the duality transforms, the latter involving a much more extensive array of additional structures. Once such a mirror- or dual-pair is established, it is then possible to translate a hopefully easily completed calculation on one end into a formidably complicated calculation on the other
© Tristan Hübsch, 2008