×

Ueda theory: theorems and problems. (English) Zbl 0704.32006

Mem. Am. Math. Soc. 415, 123 p. (1989).
The paper is a collection of two articles: “Geometric consequences of Ueda’s results” and “Problems in Ueda theory”.
Given M an arbitrary compact Kähler manifold, X a complex manifold of one dimension higher, i: \(M\hookrightarrow X\) an embedding such that the conormal bundle is topologically torsion then Ueda associates to this a certain cohomology class u(M,X) called by the author the Ueda class. Ueda theory shows that the nonvanishing of this class permits the construction of certain plurisubharmonic functions and thus the study of the convexity of X-M. In most cases it is shown that X-M is holomorphically convex.
Ueda theory can be applied to a problem rised by Hartshorne: Let X be a smooth projective surface and \(M\subset X\) a smooth projective curve of self-intersection zero. Then what is the convexity type of X-M?
There is an example of \(M\subset X\) with X-M Stein, non-affine deduced from the works of Serre and Hartshorne.
One of the main new results in the first article is the construction using Ueda theory of a totally new example of \(M\hookrightarrow X\) with X-M holomorphically convex.
The second article studies problems such as the variation of Ueda class in families and the classification of pairs \(M\hookrightarrow X\) with u(M,X)\(\neq 0\). For instance a classification of such pairs is obtained when M is a smooth elliptic curve on X.
Reviewer: N.Mihalache

MSC:

32E05 Holomorphically convex complex spaces, reduction theory
32E10 Stein spaces
32U05 Plurisubharmonic functions and generalizations
32L20 Vanishing theorems
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14H52 Elliptic curves
PDFBibTeX XMLCite
Full Text: DOI