Chern-Weil forms and hermitian metrics [02F0]

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Chern-Weil forms and hermitian metrics

Let $\mathcal{PH}_{V}$ be the sheaf of real-valued pluriharmonic functions on $V$ . By definition, a closed (1,1)-form on $V$ is a section of the sheaf ${\mathcal{C}}^{\infty}_{V}/\mathcal{PH}_{V}$ . We have the exact sequence:

{\mathcal{C}}^{\infty}(V)\to\Gamma(V,{\mathcal{C}}^{\infty}_{V}/\mathcal{PH}_{V})\mathrel{\mathop{\kern 0.0pt\longrightarrow}\limits^{[\ \ .\ \ ]}}H^{1}(V,\mathcal{PH}_{V})\to 0.

A class in $H^{1}(X,\mathcal{PH}_{X})$ will be called Kähler, if it is in the $[\ \ .\ \ ]$ image of a smooth Kähler metric.

Remark 5.10.

Assume $X$ is smooth. A class $[\omega]$ in $H^{1}(X,\mathcal{PH}_{X})$ will be called numerically base point free iff there exists a proper surjective holomorphic mapping $X\to Y$ , $Y$ normal, such that $[\omega]$ is the pull back of a Kähler class on $Y$ . This is a stronger condition than being semi-Kähler.

In the non-big case (i.e.: $\int_{X}\omega^{n}=0$ ), it is straightforward to construct semi-Kähler classes that are not numerically base point free (e.g. on complex tori). On the other hand, it is still unknown whether there exists a smooth projective variety $X$ and a semi-Kähler form $\omega$ which is big without being numerically base point free.

Let $L$ be a holomorphic line bundle on $V$ . The notion of smooth hermitian metric on $(V,L)$ is defined as in the smooth case. Let $h$ be such a metric on $(V,L)$ .

Let $s\in H^{0}(U,L)$ be a nowhere zero local holomorphic section of $L$ (a local generator of $L$ ) defined over the open subset $U\subset V$ . Set $e^{-\varphi_{s}}:=||s||_{h}^{2}$ , where $\varphi_{s}$ is a ${\mathcal{C}}^{\infty}$ -smooth function on $U$ . The current $dd^{c}\varphi_{s}$ is a smooth closed (1,1)-form on $V$ which does not depend on $s$ ; it is a semi-Kähler current if $\varphi_{s}$ is psh.

More generally, let $(U_{i})_{i}$ be an open covering of $V$ and $s_{i}\in H^{0}(U_{i},\mathcal{O}_{V}(L))$ a local generator of $L$ . Let $\varphi_{i}=\varphi_{s_{i}}$ . The datum $(U_{i},\varphi_{i})$ defines a smooth closed (1,1)-form on $V$ .

Definition 5.11.

The Chern-Weil form of $(V,L,h)$ (or of $h$ ) is the $\sim$ -equivalence class of the data $(U_{i},\varphi_{i})$ constructed above. We will denote it by $c_{1}(L,h)$ .

It is immediate that $[c_{1}(L,h)]$ is independent of $h$ . Hence there is a linear map $c_{1}:Pic(V)\to H^{1}(V,\mathcal{PH}_{V})$ . The connection with the more widely known smooth case is made by the observation that, if $X$ is a compact Kähler manifold, $H^{1,1}(X,\mathbb{R})=H^{1}(X,\mathcal{PH}_{X})$ .

Proposition 5.12.

Let $V$ a compact normal complex analytic variety.

The space $H^{1}(V,\mathcal{PH}_{V})$ is finite dimensional.

Let $L$ a holomorphic line bundle on $V$ . Every representative of $c_{1}(L)$ in $H^{1}(V,\mathcal{PH}_{V})$ is the Chern-Weil form of a smooth hermitian on $L$ .

If there exists a smooth hermitian metric $h$ such that $c_{1}(L,h)$ is Kähler, then $V$ is projective-algebraic and $L$ is ample.

Proof.

The most difficult task is to show that, in the last assertion, $V$ is Moishezon. This follows from Siu’s solution of the Grauert-Riemenschneider conjecture [Siu]. ∎

A singular metric on $L$ is an expression $h=e^{-\varphi}h_{sm}$ , $\varphi$ being a locally smooth + psh function and $h_{sm}$ a smooth hermitian metric. Its Chern-Weil form is the quasi-positive current $c_{1}(L,h_{sm})+dd^{c}\varphi$ .