An -dimensional Calabi-Yau variety is an -dimensional Kähler manifold with (holomorphically, rather than just topologically) trivial canonical bundle. This is equivalent to saying that it is real Riemannian manifold of even dimension which has special holonomy in the subgroup .
Is it also true for non-compact?
Note that implies in general that the canonical bundle is topologically trivial. But if is a compact Kähler manifold, implies further that the canonical bundle is holomorphically trivial.
The language used in this article is implicitly analytic, rather than algebraic. Is this OK? Or should I make this explicit?
A Calabi-Yau variety can be described algebraically as a smooth proper variety of dimension over a field (not necessarily algebraically closed and not necessarily of characteristic ) in which and also for all .
Beware that there are slighlty different (and inequivalent) definitions in use. Notably in some contexts only the trivialization of the canonical bundle is required, but not the vanishing of the . To be explicit on this one sometimes speaks for emphasis of “strict” CY varieties when including this condition.
See also (Geer-Katsura 03).
|G-structure||special holonomy||dimension||preserved differential form|
|Kähler manifold||U(k)||Kähler forms|
|quaternionic Kähler manifold||Sp(k)Sp(1)|
|Spin(7) manifold||Spin(7)||8||Cayley form|
|G2 manifold||G2||associative 3-form|
The original articles are
Surveys and reviews include
Discussion of the relation between the various shades of definitions includes
Discussion of CYs in positive characteristic includes