Celebratio Mathematica

R. Bott and S. S. Chern: “Some formulas related to complex transgression,” pp. 48–57 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham]. Edited by A. Haefliger and R. Narasimhan. Springer (New York), 1970. MR 0264715 Zbl 0203.54202 incollection

Let \( X \) be a complex manifold of complex dimension \( n \) and \( \pi:E\to X \) a holomorphic vector bundle whose fiber dimension is also \( n \). On \( E \) we introduce a positive definite hermitian norm \( N \) and denote by \[ B^*(E) = \{e\in E\mid 0 < N(e)\} \] the subset of non-zero vectors of \( E \). Let \( c_n(E) \) be the Chern form of the hermitian structure (to be described again below). In [1965] we proved the theorem: There exists a real-valued form \( \rho \) of type \( (n-1, \) \( n-1) \) on \( B^*(E) \) such that \begin{equation*}\tag{1} \pi^* c_n(E) = \frac{dd^c}{4\pi}\rho. \end{equation*} In this paper we wish to give a proof of (1) based on an explicit construction. We will follow and extend a formalism developed in the real case by H. Flanders [1953]. It is possible that this formalism will be useful for later purposes.

S. S. Chern	Related	Volume
Georges de Rham	Related
Raghavan Narasimhan	Related
André Haefliger	Related

P. F. Baum and R. Bott: “On the zeroes of meromorphic vector-fields,” pp. 29–47 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham] (Geneva, 26–28 March 1969). Edited by A. Haefliger and R. Narasimhan. Springer (Berlin), 1970. MR 261635 Zbl 0193.52201 incollection

Let \( M \) be a compact complex analytic manifold and let \( x \) be a holomorphic vector-field on \( M \). In an earlier paper by one of us (see [Bott 1967]) it was shown that the behavior of \( x \) near its zeroes determined all the Chern numbers of \( M \) and the nature of this determination was explicitly given where \( x \) had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections \( s \) of \( T\otimes L \) where \( T \) is the holomorphic tangent bundle to \( M \) and \( L \) is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of \( s \), but we treat only the case where \( s \) vanishes at isolated points \( \{p\} \).

Georges de Rham	Related
Paul Baum	Related	Volume
Raghavan Narasimhan	Related
André Haefliger	Related

R. Bott: “Georges de Rham: 1901–1990,” Notices Am. Math. Soc. 38 : 2 (1991), pp. 114–115. MR 1090924 Zbl 1194.01069 article

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Raoul H. Bott

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