Complex Monge–Ampère and the Calabi–Yau theorem

Let X be a connected compact Kähler manifold without boundary, with smooth Kähler form omega and complex dimension n. Seek a smooth real potential phi such that omega plus i partial bar-partial phi remains positive. This deformation stays in the same de Rham class. Stokes' theorem then fixes the total top-wedge integral, so a proposed positive target volume form must have the same total integral. In exponential notation the compatibility condition concerns the integral of exponential F, not the ordinary average of F.

The smooth Calabi–Yau theorem states that every smooth positive target volume form satisfying that compatibility is the top wedge of a unique Kähler form in the given class. The potential is unique up to an additive constant; requiring its integral against the original volume to be zero makes it unique. Connectedness matters for that constant statement and for using one total-volume condition: on a disconnected space one instead checks components and has one constant per component. The theorem is invoked here, not established by the explicit one-dimensional solution below.

In a holomorphic chart our background form has coefficient matrix G and the deformed form has G plus the complex Hessian of phi. The volume ratio is the determinant ratio. Positivity requires the whole deformed Hermitian matrix to be positive definite, not merely its determinant to be positive. In even complex dimension a negative definite matrix also has positive determinant, but it lies outside the Kähler branch. The linearization of the logarithmic equation contracts the Hessian of a variation against the inverse deformed metric, giving an elliptic operator on that branch.

Ellipticity at a candidate solution does not by itself prove global existence or the estimates needed for a continuity method. Yau's proof requires substantial a priori control and regularity, which this lesson does not replace by a scalar root search. Uniqueness uses comparison or integration arguments and compactness, leaving only constants in the potential ambiguity. Local algebraic solutions need not descend to a compact manifold or preserve its Kähler class, so a uniform scaling example must be labeled local unless its cohomological effect is checked.

On a compact Kähler manifold the partial-bar-partial lemma expresses the difference between two real Ricci-class representatives as i partial bar-partial of a global function. Choosing its additive constant enforces volume compatibility, after which the theorem yields the desired Ricci representative. If the real first Chern class vanishes, choosing target Ricci form zero produces a unique Ricci-flat Kähler metric in each Kähler class. This assertion does not require every original metric in that class to be Ricci flat.

The name Calabi–Yau is used with additional hypotheses in different contexts, such as a trivial canonical bundle or a particular holonomy group. Real first-Chern-class vanishing and the metric theorem alone should not be presented as proving all those stronger properties. The explicit torus example instead focuses on what is directly guaranteed: a positive periodic potential deformation, the correct total volume, and uniqueness after a constant normalization. It also shows that a density's normalization must be computed before interpreting its logarithm as the right-hand side.