Ricci forms and the first Chern class

For a Kähler metric the Chern connection of the holomorphic tangent bundle agrees with the induced complex Levi-Civita connection. Trace of its curvature is the curvature of its determinant line. With curvature defined as connection squared, that trace is minus partial bar-partial of the log determinant of the Hermitian coefficient matrix. Multiplying by i gives the Ricci form. Equivalently, on real vectors the Ricci form is the Ricci tensor evaluated after J acts on its first argument, consistent with the chosen fundamental-form convention.

The canonical bundle is the determinant of the holomorphic cotangent bundle, dual to the tangent determinant. Its curvature therefore has the opposite trace, and its first Chern class is minus the tangent first Chern class. The normalized Ricci class below is the image of the integral class in real de Rham cohomology. A differential form does not detect torsion in the integral class, so statements about real vanishing should not be silently strengthened to triviality of the canonical holomorphic line or existence of a global nowhere-vanishing holomorphic volume form.

Under holomorphic coordinate changes the metric determinant acquires a squared modulus of a nonvanishing holomorphic Jacobian factor. The partial-bar-partial of its logarithm is zero locally, so the Ricci expressions agree across overlaps. This proves the formula defines a global form without requiring a global log determinant function. Closedness follows from the differential identities. A constant positive rescaling of the metric adds only a constant to the logarithmic determinant, leaving the Ricci form unchanged even though the fundamental form and volume scale.

For two Kähler metrics, the ratio of their top volume forms is a global positive function. Taking its logarithm f gives the displayed change formula for the Ricci form. Thus different metrics represent the same real first Chern class. Exact changes need not vanish pointwise. On a compact Kähler manifold the partial-bar-partial lemma provides the converse representation needed to prescribe a new Ricci representative in that class; solving for a positive metric with the corresponding volume is the separate analytic problem of the next lesson.

On the projective line a Fubini-Study potential yields a positive Ricci form whose normalized integral equals two, the tangent degree. On a complex torus, a globally varying conformal metric in complex dimension one can have positive curvature in some places and negative curvature in others while its Ricci integral remains zero. Every Hermitian metric on a complex curve is Kähler, so this example does not leave the assumptions of the determinant formula. It instead exhibits the distinction between representative and class within those assumptions.

A matrix instrument can check determinant factors in constant-coefficient examples, but curvature requires spatial derivatives. Interpreting a selected positive constant Gram matrix as Hermitian coefficients produces a flat metric, regardless of how anisotropic it is. Its log determinant can be a nonzero constant with zero Ricci form. This provides a useful failure case for the temptation to identify positive determinant with positive Ricci curvature. The matrix's displayed singular stretches measure one tangent transformation; they do not encode the derivatives of a metric field.