Some formulas related to complex transgression. (English) Zbl 0203.54202

Bott, Raoul; Chern, S. S.

Essays Topol. Relat. Top., Mém. dédiés à Georges de Rham 48-57 (1970).

Let \( X \) be complex manifold and \( \pi: E \rightarrow X \) a holomorphic vector-bundle of rank \( n \). A hermitian metric on the fibres of \( E \) defines an affine connection of type \( (1,0) \) in \( E \) in a natural way; more generally one can define, for all \( p, q, r, s \geq 0 \), operators \[ d^{\prime}: \tilde{\Lambda}_{s}^{p q} \rightarrow \tilde{\Lambda}_{r+1, s}^{p q}, d^{\prime \prime}: \tilde{\Lambda}_{r}^{p q} \rightarrow \tilde{\Lambda}_{r}^{p q} q+1 \] (where \( \tilde{\Lambda}_{r}^{p q}: \) = space of \( C^{\infty} \)-forms of bidegree \( (r, s) \) with coefficients in \( \left.\Lambda E \otimes \Lambda \bar{E}\right) \), which reduce to the exterior differentiation operators for \( p=q=0 \), such that \( d=d^{\prime}+d^{\prime \prime} \) becomes a derivation of the algebra \( \sum \tilde{\Lambda}_{\mathbf{r}}^{p q} \). The \( n \)-th Chern form \( C_{n}(E) \) of the connection in \( E \) is an exterior differential form on \( X \) of bidegree \( (n, n) \). In this paper the authors explicitly construct a real form \( \rho \) of bidegree \( (n-1 \), \( n-1 \) ) on \( B^{*}(E):= \) complement of the zero-section in \( E \), such that \( \pi^{*}\left(C_{n}(E)\right)=i \) d’d” \( P \). The existence of such a \( \rho \) had already been proved by the authors in Acta Math. 114, 11-112 (1965, this ZbI. 148, 319). The proof is via the curvature \( K \) of the connection in \( E \), which is an element of \( \tilde{\Lambda}_{11}^{11} \). The authors consider the bundle \( \pi^{*}(E) \mid B^{*}(E) \rightarrow B^{*}(E) \) with the connection induced by that in \( E \); \( \bar{M}_{\mathrm{r}}^{p q} \) denoting the space corresponding to \( \tilde{\Lambda}_{r}^{p q} \) for this bundle, they construct \( \sigma \in \widetilde{M}_{\mathfrak{n}}^{m}-1, n-1 \) such that \( \pi^{*}\left(K^{n}\right)=d\left(d^{\prime}-d^{\prime \prime}\right) p \). The main result follows immediately. The proof is elementary and computational.

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