CalabiYau

Calabi-Yau manifold is a compact Kähler manifold with vanishing first Chern class, equivalently a Ricci-flat Kähler manifold. It possesses a nowhere vanishing holomorphic n-form, where n is the complex dimension, and its canonical bundle is trivial. This condition implies a reduction of the Riemannian holonomy group to SU(n). In particular, when the manifold is simply connected, its holonomy is exactly SU(n), and it admits a parallel holomorphic volume form.

The existence of Ricci-flat metrics in a given Kähler class was established by Shing-Tai Yau in his solution to the Calabi conjecture (1976). Thus every Calabi-Yau manifold supports a Ricci-flat Kähler metric, and the metric's holonomy is contained in SU(n).

The topology of a Calabi-Yau manifold is reflected in its Hodge numbers, with h^{p,0}=0 for 0<p<n and h^{0,0}=h^{n,0}=1 when the canonical bundle is trivial.

Famous examples include Calabi-Yau threefolds, such as the quintic hypersurface in complex projective 4-space, and K3 surfaces, which are Calabi-Yau in complex dimension 2. Calabi-Yau manifolds play a central role in string theory, where they are used to compactify extra dimensions while preserving a portion of supersymmetry. They also feature prominently in mathematics through mirror symmetry, relating pairs of Calabi-Yau manifolds with exchanged Hodge numbers.

Historically, the term arises from Eugenio Calabi’s conjecture, later proven by Yau. The study of these manifolds connects complex geometry, differential geometry, algebraic geometry, and theoretical physics.