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Kähler class, canonical bundle, and Ricci form

Definition

Assume the full Axiom of Choice for the characteristic-class identification below. Let $n \geq 1$ , and let $X$ be a compact complex manifold of complex dimension $n$ . Write $J$ for its complex structure, and put $T X_{C} : = T X \otimes_{R} C$ with the complex-linear extension $J_{C}$ of $J$ . Define $T_{X}^{1, 0} : = \ker (J_{C} - i I)$ ; in a holomorphic chart $(z^{1}, \dots, z^{n})$ its holomorphic frame is $\partial / \partial z^{1}, \dots, \partial / \partial z^{n}$ , whose transition matrices are holomorphic Jacobians. Define the holomorphic cotangent bundle $Ω_{X}^{1} : = (T_{X}^{1, 0})^{*}$ , with dual local frame $d z^{1}, \dots, d z^{n}$ . The canonical bundle is the holomorphic line bundle $K_{X} : = Λ^{n} Ω_{X}^{1} .$

A Hermitian metric $g$ is a positive Riemannian metric with $g (J u, J v) = g (u, v)$ . Its real fundamental form is $ω (u, v) = g (J u, v)$ . It is a Kähler form when it is a smooth real positive $(1, 1)$ -form and $d ω = 0$ . A Kähler class is a real de Rham cohomology class represented by such a form; for a specified Kähler form it is $[ω] \in H_{d R}^{2} (X; R)$ .

In holomorphic coordinates $(z^{1}, \dots, z^{n})$ use the repository convention $ω = i \sum_{j, k = 1}^{n} h_{j \overset{ˉ}{k}} d z^{j} \land d {\overset{ˉ}{z}}^{k}, h_{j \overset{ˉ}{k}} = g_{C} ⁣ (\frac{\partial}{\partial z^{j}}, \frac{\partial}{\partial {\overset{ˉ}{z}}^{k}}),$ where $g_{C}$ is the complex-bilinear extension of $g$ and $H = (h_{j \overset{ˉ}{k}})$ is positive Hermitian. Define the Ricci form $ρ (ω) : = - i \partial \overset{ˉ}{\partial} \log \det H .$ This expression is independent of the holomorphic coordinates and is a smooth real closed $(1, 1)$ -form. Under the Chern–Weil normalization of the imported characteristic-class suppliers, $\frac{[ρ (ω)]}{2 π} = (c_{1} (T_{X}^{1, 0}))_{R} in H_{d R}^{2} (X; R),$ where the right side is the image of the integral first Chern class under change of coefficients $Z \to R$ .

Definition references

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Compact Kähler and rational Hodge polarization conventions

De rham cohomology

Characteristic forms represent topological characteristic classes over the reals

First Chern form agrees with the topological line class

The Axiom of Choice

Demailly, Chapter VI §6.1 and Chapter VIII §6

Facts & Assumptions

Given: Full AC; a compact smooth complex $n$ -manifold $X$ , $n \geq 1$ , and a supplied Kähler metric with the real fundamental-form convention stated above.

[A1]

Full AC is the choice-function principle of The Axiom of Choice. The imported Chern–Weil and line-class suppliers explicitly assume it.

[A2]

Use the local coordinate convention declared in this Definition: $ω = i h_{j \overset{ˉ}{k}} d z^{j} \land d {\overset{ˉ}{z}}^{k}$ with $H = (h_{j \overset{ˉ}{k}})$ positive Hermitian.

[F1]

A complex manifold has holomorphic charts, complex structure $J$ , holomorphic tangent and cotangent bundles, and Hermitian metrics as in Complex manifolds, holomorphic vector bundles and Hermitian metrics.

[F2]

In holomorphic charts, complex-valued forms decompose by bidegree and $d = \partial + \overset{ˉ}{\partial}$ ; compact Kähler manifolds use the supplied positive closed real fundamental form (Compact Kähler and rational Hodge polarization conventions).

[F3]

For a supplied Hermitian connection on a smooth complex bundle, the normalized Chern forms represent the real coefficient images of the topological Chern classes (Characteristic forms represent topological characteristic classes over the reals).

[F4]

For a Hermitian connection on a complex line, the first Chern form is $- Ω / (2 π i)$ and represents the real image of the integral line class (First Chern form agrees with the topological line class).

[F5]

Real de Rham cohomology is the cohomology of the smooth real de Rham complex; a real closed form that has a real primitive represents zero (De rham cohomology).

[F6]

The total Chern form is $c (\nabla) = \det (I - Ω / (2 π i))$ ; its degree-two term is the normalized first Chern form (Chern, Pontryagin, and Euler characteristic forms).

Proof

technique · direct

1.1givenA2F1algebra

On an overlap, write the coordinate change as $z = z (w)$ and let $A = (\partial z_{j} / \partial w_{a})_{j, a}$ . The Hermitian metric matrices satisfy $H^{(w)} = A^{T} H^{(z)} \overline{A}$ , so $\log \det H^{(w)} = \log \det H^{(z)} + \log ∣ \det A ∣^{2}$ . The holomorphic function $\det A$ is nowhere zero; locally it has a holomorphic logarithm, whence $\partial \overset{ˉ}{\partial} \log ∣ \det A ∣^{2} = 0$ . Thus the displayed local forms agree on overlaps and define a global $(1, 1)$ -form.

2.1step 1.1F2algebra

Since $\det H > 0$ , $f = \log \det H$ is real. Complex conjugation gives $\overline{- i \partial \overset{ˉ}{\partial} f} = - i \partial \overset{ˉ}{\partial} f$ , so $ρ$ is real. Also $\partial ρ = - i \partial^{2} \overset{ˉ}{\partial} f = 0$ and $\overset{ˉ}{\partial} ρ = - i \overset{ˉ}{\partial} \partial \overset{ˉ}{\partial} f = i \partial {\overset{ˉ}{\partial}}^{2} f = 0$ ; hence $d ρ = 0$ .

2.2F1A2F2step 1.1algebra

In a holomorphic frame $e_{j} = \partial / \partial z^{j}$ of $T_{X}^{1, 0}$ , let $H_{j \overset{ˉ}{k}} = g_{C} (e_{j}, {\overset{ˉ}{e}}_{k})$ . Define a connection $\nabla = d + A$ by the matrix $A^{T} = (\partial H) H^{- 1}$ , whose $(0, 1)$ -part is zero in this holomorphic frame and hence equals the holomorphic structure $\overset{ˉ}{\partial}$ . The relation $\partial H = A^{T} H$ and its conjugate give metric compatibility; under a holomorphic frame change $e^{'} = e B$ , the formula transforms as $A^{'} = B^{- 1} A B + B^{- 1} \partial B$ , so these local formulas define a global Hermitian connection. Its determinant connection has trace form $tr A = \partial \log \det H$ . For its curvature $F = d A + A \land A$ , $tr (A \land A) = 0$ , and therefore $tr F = d tr A = \overset{ˉ}{\partial} \partial \log \det H = - \partial \overset{ˉ}{\partial} \log \det H$ .

3.1F3F4F6step 2.2algebra

The normalized first Chern form of this Hermitian connection on $T_{X}^{1, 0}$ is $- tr F / (2 π i)$ by [F6]. By step 2.2 it equals $- i \partial \overset{ˉ}{\partial} \log \det H / (2 π) = ρ / (2 π)$ ; the line normalization in [F4] gives the same sign on the determinant connection.

4.1A1F3F5step 3.1∎

Apply the Chern–Weil statement [F3] to $E = T_{X}^{1, 0}$ and this connection. It identifies the real cohomology class of the normalized form in step 3.1 with the real image of $c_{1} (T_{X}^{1, 0})$ , giving the asserted equality.

Depends on

Dependency tree · 2 levels

Sources

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, author manuscript (2012)