Holomorphic symmetries. (English) Zbl 0639.32015

Lawson, H. Blaine jun.; Yau, Stephen S.-T.

Ann. Sci. Éc. Norm. Supér. (4) 20, No. 4, 557-577 (1987).

Main results proved in this paper are the following: (1) A compact maximally complex \(C^ 1\)-class manifold M, \(\dim_{{\mathbb{R}}}M=2p-1\geq 3,\) with a transversal holomorphic \(S^ 1\)-action is realized as a real hypersurface in a p-dimensional affine algebraic variety with a linear \({\mathbb{C}}^*\)-action, if M is embedded in a Stein manifold. (2) Suppose that compact maximally complex smooth pseudoconvex manifolds \(M\), \(M'\), \(\dim_{{\mathbb{R}}}M=\dim_{{\mathbb{R}}}M'=\) \(2n-1\geq 3,\) with transversal holomorphic \(S^ 1\)-actions are embedded in \({\mathbb{C}}^{n+1}\) and have isomorphic Kohn-Rossi’s \({\bar \partial}_ b\)-cohomologies. Then there is a diffeomorphism \(f: {\mathbb{C}}^{n+1}\to {\mathbb{C}}^{n+1}\) such that \(f(M)=M'\). (3) Suppose that a compact complex space admits a holomorphic \(S^ 1\)-action. Then the Euler number of the fixed point set of the action is equal to that of the total space. To prove (1), first they show that the holomorphic \(S^ 1\)-action extends to a holomorphic semigroup action on \(Y\) which \(M\) bounds. Here the semigroup is the punctured unit disk in \(C\). The existence of \(Y\) is guaranteed by a result of F. R. Harvey and H. B. Lawsonjun. [Ann. Math., II. Ser. 102, 223-290 (1975; Zbl 0317.32017)]. Next they observe that this extended semigroup action induces a contracting holomorphic embedding of (the normalization of) \(Y\) into itself. From this observation (1) follows.
The proof of (2) is based on a previous result of one of the authors.
As an application of the result (3), they calculate the Euler characteristics of the Chow varieties of complex projective space. Moreover, for an arbitrary compact Kaehler manifold \(X\), they define a generating function whose coefficients are the Euler numbers of certain subsets of the Barlet space of X and show the rationality of the function by calculation in the cases \(X = P^ n\) and \(P^ m \times P^ n\).
In (1), the algebraicity of \(Y\), and hence of \(M\), in the sense of the paper under review, follows also from a reviewer’s result as a special case. See, Theorem 1 in J. Math. Soc. Japan 28, 550-576 (1976; Zbl 0323.32017) and the proof of Lemma 10 in Complex Anal., algebr. Geom., 191-206 (1977; Zbl 0354.32013).

Reviewer: M.Kato

MSC:

32J99

Compact analytic spaces

Keywords:

algebraicity criteria; CR-structure; \({\bar\partial}_ b\)-cohomology; fixed point formula; holomorphic \(S^ 1\)-action

Citations:

Zbl 0317.32017; Zbl 0323.32017; Zbl 0354.32013

References:

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