a Calabi-Yau manifold
/ kah-LAH-bee YOW /
Among Kähler manifolds there is a most-symmetric, most-balanced class: those whose Ricci curvature is exactly zero — the gravitational vacuum, in physics language, of complex geometry. A Calabi-Yau manifold is a compact Kähler manifold admitting such a Ricci-flat Kähler metric. They are the complex-geometric analogue of flat space, but with rich topology, and they are simultaneously the central objects of string theory's extra dimensions and a deep triumph of geometric analysis.
Precisely, a Calabi-Yau manifold is a compact Kähler manifold whose first Chern class c_1 vanishes (in H^2(M, R)); equivalent characterizations include having a trivial canonical bundle K_M, or admitting a nowhere-vanishing global holomorphic n-form (a holomorphic volume form), or having holonomy contained in SU(n). The depth comes from the Calabi conjecture, proved by Yau: on a compact Kähler manifold, every Ricci form representing 2pi c_1 is the Ricci form of a unique Kähler metric in each Kähler class. Specializing to c_1 = 0, this yields a unique Ricci-flat Kähler metric in every Kähler class — the existence is guaranteed but the metric is essentially never known in closed form (no explicit Ricci-flat metric on a compact Calabi-Yau is known beyond tori). The proof reduces the geometric statement to solving a complex Monge-Ampere equation det(g_{i,j-bar} + (mixed second derivatives of phi)) = e^F times det(g_{i,j-bar}) for a potential phi, via the partial partial-bar lemma.
Why they matter: Calabi-Yau threefolds (complex dimension 3) compactify the extra six dimensions in superstring theory, and their geometry encodes the resulting physics; they are also the stage for mirror symmetry, a startling duality pairing topologically different Calabi-Yaus. The honesty caveats: Ricci-flatness (a condition on Ricci curvature) does not mean flatness — the full Riemann curvature is generally nonzero, so a Calabi-Yau is curved, just Ricci-flat. And the celebrated metrics exist by Yau's theorem but are non-explicit; the quintic threefold in CP^4, the standard example, has a Ricci-flat metric guaranteed to exist that no one can write down.
The quintic threefold — the zero locus in CP^4 of a generic degree-5 homogeneous polynomial in five variables — is the textbook Calabi-Yau. Its canonical bundle is trivial (by adjunction, since 5 = 4 + 1), so c_1 = 0, and Yau's theorem guarantees a Ricci-flat Kähler metric on it. That metric is known to exist but has never been written explicitly, and the quintic is the most-studied example in mirror symmetry.
The quintic in CP^4 is the standard Calabi-Yau threefold: c_1 = 0, with a guaranteed but unknown Ricci-flat metric.
Ricci-flat is not flat: a Calabi-Yau has nonzero full Riemann curvature in general. And while Yau's theorem guarantees the metric exists, it is essentially never explicit — beyond flat tori, no compact Calabi-Yau metric has been written in closed form.